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2014-08-26T14:45:06Z

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2014-08-22T10:37:17Z

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2014-08-11T12:21:59Z

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2014-02-26T15:00:27Z

Jitse Niesen: both formulas are almost the same; mention this in a footnote

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2014-02-25T15:46:35Z

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2014-02-08T21:41:58Z

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2014-02-07T12:15:04Z

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== Purpose ==

The purpose of click-through rates is to capture customers' initial response to websites. Most commercial websites are designed to elicit some sort of action, whether it be to buy a book, read a news article, watch a music video, or search for a flight. People generally don't visit a website with the intention of viewing advertisements, just as people rarely watch TV with the purpose of consuming commercials.<ref name=Marketing_Metrics>Farris, Paul W.; Neil T. Bendle; Phillip E. Pfeifer; David J. Reibstein (2010). ''Marketing Metrics: The Definitive Guide to Measuring Marketing Performance.'' Upper Saddle River, New Jersey: Pearson Education, Inc. ISBN 0-13-705829-2. The [[Marketing Accountability Standards Board (MASB)]] endorses the definitions, purposes, and constructs of classes of measures that appear in ''Marketing Metrics'' as part of its ongoing [http://www.themasb.org/common-language-project/ Common Language: Marketing Activities and Metrics Project].</ref>

Marketers want to know the reaction of the web visitor. Under current technology, it is nearly impossible to fully quantify the emotional reaction to the site and the effect of that site on the firm's brand. One piece of information that is easy to acquire, however, is the click-through rate. The click-through rate measures the proportion of visitors who initiated action with respect to an advertisement that redirected them to another page where they might purchase an item or learn more about a product or service. Here we have used "click their mouse" on the advertisement (or link) because this is the generally used term, although other interactions are possible.<ref name=Marketing_Metrics />

== Construction ==

The click-through rate is the number of times a click is made on the advertisement divided by the total impressions (the times an advertisement was served):

Click-through rate (%) = Click-throughs (#) / Impressions (#)<ref name=Marketing_Metrics />

===Online Advertising CTR===

The click-through rate of an advertisement is defined as the number of clicks on an ad divided by the number of times the ad is shown ([[impression (online media)|impressions]]), expressed as a percentage.<ref name=Marketing_Metrics /><ref>[http://adwords.google.com/support/aw/bin/answer.py?hl=en&answer=107955&from=6305&rd=1 Google AdWords Help: Clickthrough rate (CTR)]</ref><ref>[http://developer.searchmarketing.yahoo.com/docs/V7/gsg/glossary.php Yahoo Search Marketing Glossary]</ref><ref>[http://www.iab.net/media/file/click-measurement-guidelines2009.pdf IAB Click Measurement Guidelines]</ref><ref>[http://www.google.com/support/analytics/bin/answer.py?answer=57164 Google Analytics Help: What's the difference between clicks, visits, visitors, pageviews, and unique pageviews?]</ref><ref>[http://www.iab.net/media/file/GlossaryofInteractivAdvertisingTerms.pdf IAB Glossary of Interactive Advertising Terms]</ref> For example, if a [[banner ad]] is delivered 100 times (100 impressions) and receives one click, then the click-through rate for the advertisement would be 1%.

 
<math> CTR = {Clicks \over Impressions}</math>

Click-through rates for [[banner ads]] have fallen over time. When banner ads first started to appear, it was not uncommon to have rates above five percent. They have fallen since then, currently averaging closer to 0.2 or 0.3 percent.<ref name="iMedia">{{cite news |last=Stern |first=Andrew |url=http://www.imediaconnection.com/content/25781.asp|title=8 ways to improve your click-through rate |work=iMedia Connection |date=February 1, 2010 |accessdate=February 7, 2010}}</ref> In most cases, a 2% click-through rate would be considered very successful, though the exact number is hotly debated and would vary depending on the situation. The average click-through rate of 3% in the 1990s declined to 2.4%-0.4% by 2002.<ref name=li /> Since advertisers typically pay more for a high click-through rate, getting many click-throughs with few purchases is undesirable to advertisers.<ref name="iMedia"/> Similarly, by selecting an appropriate advertising site with high affinity (e.g., a [[Film criticism|movie magazine]] for a [[movie]] advertisement), the same banner can achieve a substantially higher CTR. Though personalized ads, unusual formats, and more obtrusive ads typically result in higher click-through rates than standard banner ads, overly intrusive ads are often avoided by viewers.<ref name=li>{{cite web|url=http://champtec.googlepages.com/ad_format_print.pdf|title=Internet Advertising Formats and Effectiveness|last=Li|first=Hairong|coauthors=Leckenby, John D.|year=2004|publisher=Center for Interactive Advertising|accessdate=26 February 2010}}</ref><ref>{{cite web|url=http://blog.buysellads.com/2010/12/how-to-price-and-place-your-ads/|title=How to Price and Place Your Ads|accessdate=21 October 2011}}</ref><ref>{{US patent application|20090157495}}</ref>

===Email CTR===

An email click-through rate is defined as the number of recipients who clicked one or more links in an email and landed on the sender's website, blog, or other desired destination. More simply, email click-through rates represent the number of clicks that your email generated.<ref>{{cite web | url = http://www.iab.net/guidelines/508676/508905/79176 | title = Email Campaign Performance Metrics Definitions | accessdate = December 18, 2012}}</ref><ref>{{cite web | url = http://emailmarketing.comm100.com/email-marketing-ebook/click-through-rates.aspx | title = Click Through Rates: Click Through Rates Numbers and Their Meaning | author = Kevin Gao | accessdate = December 18, 2012}}</ref>

Email click-through rate, is expressed as a percentage, and calculated by dividing the number of click throughs by the number of messages delivered.<ref>{{cite web | url = http://www.idealware.org/articles/email_metrics.php | title = The Basics of Email Metrics: Are Your Campaigns Working? | date = October 2008}}</ref><ref>{{cite web | url = http://www.dummies.com/how-to/content/calculating-the-clickthrough-rate-for-your-email-m.html | title = Calculating the Click-through Rate for Your E-Mail Marketing Campaign from E-Mail Marketing for Dummies, 2nd Ed. | author = John Arnold | date = April 2011}}</ref>

Most email marketers use this metrics along with [[open rate]], [[bounce rate]] and other metrics, to understand the effectiveness and success of their email campaign.<ref>{{cite web | url = http://www.michaelleander.com/blog/2010/01/email-marketing-metrics-click-through-rate-ctr-relevant-to-email-marketing-measurement/ | title = Email marketing metrics: Click through rate (CTR) relevant to email marketing measurement | date = January 17, 2010}}</ref> In general there is no ideal click-through rate. This metric can vary based on the type of email sent, how frequently emails are sent, how the list of recipients is segmented, how relevant the content of the email is to the audience, and many other factors.<ref>{{cite web | url = http://bluesite.lyris.com/blog/85-Average-Email-Click-Through-Rate | title = Average Email Click-Through Rate | accessdate = December 20, 2012}}</ref> Even time of day can affect click-through rate. Sunday appears to generate considerably higher click-through rates on average when compared to the rest of the week.<ref>{{cite web | url = http://www.websitemagazine.com/content/blogs/posts/archive/2011/07/21/email-marketing-metrics-2011-mailermailer.aspx | title = EMail Marketing Metrics 2011 | author = Pete Prestipino | date = July 21, 2011}}</ref>

Every year studies and various types of research are conducted to track the overall effectiveness of click-through rates in email marketing.<ref>{{cite web | url = http://marketingland.com/e-mail-open-rates-declining-click-through-rates-rising-study-17005 | title = E-mail Open Rates Declining, Click-Through Rates Rising [Study] | author = Matt McGee | date = July 23, 2012}}</ref><ref>{{cite web | url = http://econsultancy.com/us/blog/10404-email-marketing-stats-consumers-open-just-20-of-messages | title = Email marketing stats: consumers open just 20% of messages | author = David Moth | date= July 24, 2012}}</ref>

== See also==
* [[Abandonment rate]]
* [[Ad serving]]
* [[Banner blindness]]
* [[Click fraud]]
* [[Clicktag]]
* [[Cost per action]]
* [[Cost per click]]
* [[Cost per thousand]]
* [[Cost per impression|CPI]]
* [[Impression (online media)]]
* [[View-through rate]]
* [[Open rate]]
* [[Email marketing]]

==References==
<references/>

==Further reading==
*Sherman, Lee and John Deighton, (2001), "Banner advertising: Measuring effectiveness and optimizing placement," Journal of Interactive Marketing, Spring, Vol. 15, Iss. 2.
*Ward A. Hanson and Kirthi Kalyanam, (2007), Internet Marketing and eCommerce, Chapter8, Traffic Building, Thomson College Pub, Mason, Ohio.

== External links ==
* [http://www.themasb.org MASB Official Website]

{{DEFAULTSORT:Click-Through Rate}}
[[Category:Internet advertising]]
[[Category:Email]]
[[Category:Internet marketing terminology]]
[[Category:Consumer behaviour]]

Simpson's rule

2014-01-29T09:26:36Z

Jitse Niesen: revert per reference; note that b-a = 3h; also clean up some references

In mathematics, '''Thurston's geometrization conjecture''' states that certain three-dimensional [[topological space]]s each have a unique geometric structure that can be associated with them. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surfaces]], which states that every [[simply connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|hyperbolic]]).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed [[3-manifold]] can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}}, and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].

Thurston's [[hyperbolization theorem]] implies that [[Haken manifold]]s satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.

[[Grigori Perelman]] sketched a proof of the full geometrization conjecture in 2003 using [[Ricci flow]] with [[Surgery theory|surgery]].
There are now four different manuscripts (see below) with details of the proof. The Poincaré conjecture and the [[spherical space form conjecture]] are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

==The conjecture==
A 3-manifold is called '''closed''' if it is compact and has no boundary.

Every closed 3-manifold has a [[prime decomposition (3-manifold)|prime decomposition]]: this means it is the [[connected sum]] of prime three-manifolds (this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds). This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.

Here is a statement of Thurston's conjecture:

:Every oriented prime closed [[3-manifold]] can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are [[Seifert manifold]]s or [[atoroidal]] called the [[JSJ decomposition]], which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an [[Anosov map]] of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)

For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the [[Orientable double cover|oriented double cover]]. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure so this gives a minor extra complication.

In 2 dimensions the analogous statement says that every surface (without boundary) has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first.

==The eight Thurston geometries==
A '''model geometry''' is a simply connected smooth manifold ''X'' together with a transitive action of a [[Lie group]] ''G'' on ''X'' with compact stabilizers.

A model geometry is called '''maximal''' if ''G'' is maximal among groups acting smoothly and transitively on ''X'' with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.

A '''geometric structure''' on a manifold ''M'' is a diffeomorphism from ''M'' to ''X''/Γ for some model geometry ''X'', where Γ is a discrete subgroup of ''G'' acting freely on ''X''. If a given manifold admits a geometric structure, then it admits one whose model is maximal.

A 3-dimensional model geometry ''X'' is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on ''X''. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called '''Thurston geometries'''. (There are also uncountably many model geometries without compact quotients.)

There is some connection with the [[Bianchi classification|Bianchi groups]]: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However '''S'''2 × '''R''' cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.

===Spherical geometry S3===
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group O(4, '''R'''), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite fundamental group. Examples include the 3-sphere, the [[Poincaré homology sphere]], [[Lens space]]s. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type IX]]. Manifolds with this geometry are all compact, orientable, and have the structure of a [[Seifert fiber space]] (often in several ways). The complete list of such manifolds is given in the article on [[Spherical 3-manifold]]s. Under Ricci flow manifolds with this geometry collapse to a point in finite time.

===Euclidean geometry ''E''3 ===
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group '''R'''3 × O(3, '''R'''), with 2 components. Examples are the 3-torus, and more generally the [[mapping torus]] of a finite order automorphism of the 2-torus; see [[torus bundle]]. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi groups of type I or VII0]]. Finite volume manifolds with this geometry are all compact, and have the structure of a [[Seifert fiber space]] (sometimes in two ways). The complete list of such manifolds is given in the article on [[Seifert fiber space]]s. Under Ricci flow manifolds with Euclidean geometry remain invariant.

===Hyperbolic geometry H3===
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group O+(1, 3, '''R'''), with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the [[Weeks manifold]]. Other examples are given by the [[Seifert–Weber space]], or "sufficiently complicated" [[Dehn surgery|Dehn surgeries]] on links, or most [[Haken manifold]]s. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, [[atoroidal]], and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type V]]. Under Ricci flow manifolds with hyperbolic geometry expand.

===The geometry of S2 × R===
The point stabilizer is O(2, '''R''') × '''Z'''/2'''Z''', and the group ''G'' is O(3, '''R''') × '''R''' × '''Z'''/2'''Z''', with 4 components. The four finite volume manifolds with this geometry are: '''S'''2 × '''S'''1, the mapping torus of the antipode map of '''S'''2, the connected sum of two copies of 3 dimensional projective space, and the product of '''S'''1 with two-dimensional projective space. The first two are mapping tori of the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible. The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a [[Seifert fiber space]] (often in several ways). Under normalized Ricci flow manifolds with this geometry converge to a 1-dimensional manifold.

===The geometry of H2 × R===
The point stabilizer is O(2, '''R''') × '''Z'''/2'''Z''', and the group ''G'' is O+(1, 2, '''R''') × '''R''' × '''Z'''/2'''Z''', with 4 components. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry have the structure of a [[Seifert fiber space]] if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.<ref>Ronald Fintushel, Local S1 actions on 3-manifolds, Pacific J. o. M. 66 No1 (1976) 111-118, [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102818222&page=record http://projecteuclid.org/...]</ref>) The classification of such (oriented) manifolds is given in the article on [[Seifert fiber space]]s. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type III]]. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.

===The geometry of the universal cover of SL(2, R)===
<math>{\tilde{\rm{SL}}}(2, \mathbf{R})</math> is the universal cover of [[SL2(R)|SL(2, '''R''')]], which fibers over '''H'''2. The point stabilizer is O(2, '''R'''). The group ''G'' has 2 components. Its identity component has the structure <math>(\mathbf{R}\times\tilde{\rm{SL}}_2 (\mathbf{R}))/\mathbf{Z}</math>. Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the [[homology sphere|Brieskorn homology spheres]] (excepting the 3-sphere and the [[Poincare dodecahedral space]]). This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type VIII]]. Finite volume manifolds with this geometry are orientable and have the structure of a [[Seifert fiber space]]. The classification of such manifolds is given in the article on [[Seifert fiber space]]s. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.

===Nil geometry===
{{see also|Nilmanifold}}

This fibers over ''E''2, and is the geometry of the [[Heisenberg group]]. The point stabilizer is O(2, '''R'''). The group ''G'' has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O(2, '''R''') of isometries of a circle. Compact manifolds with this geometry include the mapping torus of a [[Dehn twist]] of a 2-torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi group of type II]]. Finite volume manifolds with this geometry are compact and orientable and have the structure of a [[Seifert fiber space]]. The classification of such manifolds is given in the article on [[Seifert fiber space]]s. Under normalized Ricci flow compact manifolds with this geometry converge to '''R'''2 with the flat metric.

===Sol geometry===
{{see also|Solvmanifold}}

This geometry fibers over the line with fiber the plane, and is the geometry of the identity component of the group ''G''. The point stabilizer is the dihedral group of order 8. The group ''G'' has 8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component has a normal subgroup '''R'''2 with quotient '''R''', where '''R''' acts on '''R'''2 with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the [[Bianchi classification|Bianchi group of type VI0]] and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with sol geometry are compact. The compact manifolds with sol geometry are either the [[mapping torus]] of an [[Anosov map]] of the 2-torus (an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as <math>\left( {\begin{array}{*{20}c}
2 & 1 \\
1 & 1 \\
\end{array}} \right)</math>), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the sol manifolds could in principle be classified in terms of the units and ideal classes of this order, though the details do not seem to be written down anywhere.
Under normalized Ricci flow compact manifolds with this geometry converge (rather slowly) to '''R'''1.

==Uniqueness==
A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (However a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if ''M'' is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π1(''M''):
*If π1(''M'') is finite then the geometric structure on ''M'' is spherical, and ''M'' is compact.
*If π1(''M'') is virtually cyclic but not finite then the geometric structure on ''M'' is '''S'''2×'''R''', and ''M'' is compact.
*If π1(''M'') is virtually abelian but not virtually cyclic then the geometric structure on ''M'' is Euclidean, and ''M'' is compact.
*If π1(''M'') is virtually nilpotent but not virtually abelian then the geometric structure on ''M'' is nil geometry, and ''M'' is compact.
*If π1(''M'') is virtually solvable but not virtually nilpotent then the geometric structure on ''M'' is sol geometry, and ''M'' is compact.
*If π1(''M'') has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on ''M'' is either '''H'''2×'''R''' or the universal cover of SL(2, '''R'''). The manifold ''M'' may be either compact or non-compact. If it is compact, then the 2 geometries can be distinguished by whether or not π1(''M'') has a finite [[Index of a subgroup|index]] subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
*If π1(''M'') has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on ''M'' is hyperbolic, and ''M'' may be either compact or non-compact.

Infinite volume manifolds can have many different types of geometric structure: for example, '''R'''3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.

There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:
*Taking connected sums with several copies of '''S'''3 does not change a manifold.
*The connected sum of two projective 3-spaces has a '''S'''2×'''R''' geometry, and is also the connected sum of two pieces with '''S'''3 geometry.
*The product of a surface negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.

It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.

==History==
The [[Fields Medal]] was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for [[Haken manifold]]s.

The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for [[Richard Hamilton (professor)|Richard Hamilton]] to develop his [[Ricci flow]]. In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive [[Ricci curvature]], the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by [[Ricci flow with surgery]]. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries '''S'''3 and '''S'''2 × '''R''', while what is left at large times should have a [[thick-thin decomposition]] into a "thick" piece with hyperbolic geometry and a "thin" [[graph manifold]].

In 2003 [[Grigori Perelman]] sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above. The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7.4 in the preprint 'Ricci Flow with surgery on three-manifolds'. This theorem was stated by Perelman without proof. There are now several different proofs of Perelman's Theorem 7.4, or variants of it which are sufficient to prove geometrization. There is the paper of Shioya and Yamaguchi <ref>T. Shioya and T. Yamaguchi, 'Volume collapsed three-manifolds with a lower curvature bound,' Math. Ann. 333 (2005), no. 1, 131-155.</ref> that uses Perelman's stability theorem <ref>V. Kapovitch, 'Perelman's Stability Theorem', in Surveys of Differential Geometry, Metric and Comparison Geometry, vol. XI, International press, (2007), 103-136. There is a preprint on http://arxiv.org/abs/math/0703002v3</ref> and a fibration theorem for Alexandrov spaces.<ref>T. Yamaguchi. A convergence theorem in the geometry of Alexandrov spaces. In Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), volume 1 of Semin. Congr., pages 601-642. Soc. math. France, Paris, 1996.</ref> This method, with full details leading to the proof of Geometrization, can be found in the exposition by [[Bruce Kleiner|B. Kleiner]] and J. Lott in 'Notes on Perelman's papers' in the journal Geometry & Topology.<ref>B. Kleiner and J. Lott, 'Notes on Perelman's papers', Geometry & Topology, volume 12, pp. 2587-2855, 2008. There is a preprint at http://arxiv.org/abs/math/0605667v3</ref>

A second route to Geometrization is the method of [[Laurent Bessières|Bessières]] ''et al.'',<ref>{{cite arxiv |first=L. |last=Bessieres |first2=G. |last2=Besson |first3=M. |last3=Boileau |first4=S. |last4=Maillot |first5=J. |last5=Porti |title=Weak collapsing and geometrization of aspherical 3-manifolds |eprint=0706.2065 |class=math.GT |year=2007 }}</ref><ref>{{cite journal |first=L. |last=Bessieres |first2=G. |last2=Besson |first3=M. |last3=Boileau |first4=S. |last4=Maillot |first5=J. |last5=Porti |title=Collapsing irreducible 3-manifolds with nontrivial fundamental group |journal=[[Inventiones Mathematicae|Invent. Math.]] |volume=179 |issue=2 |year=2010 |pages=435–460 |doi=10.1007/s00222-009-0222-6 |ref=harv }}</ref> which uses Thurston's hyperbolization theorem for Haken manifolds <ref>J.-P. Otal, 'Thurston's hyperbolization of Haken manifolds,'Surveys in differential geometry, Vol. III Cambridge, MA, 77-194, Int. Press, Boston, MA, 1998.</ref> and Gromov's norm for 3-manifolds.<ref>{{aut|[[Mikhail Gromov (mathematician)|M. Gromov]]}}. Volume and bounded cohomology. Inst. Hautes Etudes Sci. Publ. Math., (56):5-99 (1983), 1982.</ref> A book by the same authors with complete details of their version of the proof has been published by the European Mathematical Society.<ref>L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. Available at http://www-fourier.ujf-grenoble.fr/~lbessier/book.pdf</ref>

Also containing proofs of Perelman's Theorem 7.4, there is a paper of Morgan and Tian,<ref>{{cite paper |first=J. |last=Morgan |first2=G. |last2=Tian |title=Completion of the Proof of the Geometrization Conjecture |id={{ArXiv|math|0809.4040}} |year=2008}}</ref> another paper of Kleiner and Lott,<ref>{{cite paper |first=B. |last=Kleiner |first2=J. |last2=Lott |title=Locally collapsed 3-manifolds |arxiv=1005.5106 |year=2010 }}</ref> and a paper by Cao and Ge.<ref>{{cite paper |first=J. |last=Cao |first2=J. |last2=Ge |title=A proof of Perelman's collapsing theorem for 3-manifolds |arxiv=0908.3229 }}</ref>

==Notes==
{{reflist|2}}

==References==
*L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. [http://www-fourier.ujf-grenoble.fr/~lbessier/book.pdf]
*M. Boileau [http://www.crm.es/Publications/Quaderns/Quadern25-1.pdf Geometrization of 3-manifolds with symmetries]
*F. Bonahon ''Geometric structures on 3-manifolds'' Handbook of Geometric Topology (2002) Elsevier.
* Allen Hatcher: [http://www.math.cornell.edu/~hatcher/3M/3M.pdf ''Notes on Basic 3-Manifold Topology''] 2000
*J. Isenberg, M. Jackson, ''Ricci flow of locally homogeneous geometries on a Riemannian manifold'', J. Diff. Geom. 35 (1992) no. 3 723-741.
* G. Perelman, [http://arxiv.org/abs/math.DG/0211159 ''The entropy formula for the Ricci flow and its geometric applications''], 2002
* G. Perelman,[http://arxiv.org/abs/math.DG/0303109 ''Ricci flow with surgery on three-manifolds''], 2003
* G. Perelman, [http://arxiv.org/abs/math.DG/0307245 ''Finite extinction time for the solutions to the Ricci flow on certain three-manifolds''], 2003
*Bruce Kleiner and John Lott, [http://arxiv.org/abs/math.DG/0605667 ''Notes on Perelman's Papers''] (May 2006) (fills in the details of Perelman's proof of the geometrization conjecture).
* {{cite journal
|title=A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelman theory of the Ricci flow
|last=Cao
|first=Huai-Dong
|coauthors=Zhu, Xi-Ping
|journal=Asian Journal of Mathematics
|volume=10
|issue=2
|date=June 2006
|pages=165–498
|url=http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492-Abstract.php
|format=[[PDF]]
|accessdate=2006-07-31
|ref=harv
|doi=10.4310/AJM.2006.v10.n2.a2}} Revised version (December 2006): [http://arxiv.org/abs/math.DG/0612069 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture]
* [[John Morgan (mathematician)|John W. Morgan]]. [http://www.ams.org/bull/2005-42-01/S0273-0979-04-01045-6/S0273-0979-04-01045-6.pdf ''Recent progress on the Poincaré conjecture and the classification of 3-manifolds.''] Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57-78 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture)
*{{Cite book
| title = Ricci Flow and Geometrization of 3-Manifolds
| year = 2010
| series = University Lecture Series
| url = http://www.ams.org/bookstore-getitem/item=ulect-53
| last1 = Morgan | first1 = John W.
| last2 = Fong | first2 = Frederick Tsz-Ho
| isbn = 978-0-8218-4963-7
| accessdate = 2010-09-26 }}
*[[G. Peter Scott|Scott, Peter]] [http://www.math.lsa.umich.edu/~pscott/8geoms.pdf ''The geometries of 3-manifolds.''] ([http://www.math.lsa.umich.edu/~pscott/errata8geoms.pdf errata]) Bull. London Math. Soc. 15 (1983), no. 5, 401-487.
*{{Cite journal | last1=Thurston | first1=William P. | author1-link=William Thurston | title=Three-dimensional manifolds, Kleinian groups and hyperbolic geometry | doi=10.1090/S0273-0979-1982-15003-0 | mr=648524 | year=1982 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=6 | issue=3 | pages=357–381 | ref=harv | postscript={{inconsistent citations}}}} This gives the original statement of the conjecture.
* William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5 (in depth explanation of the eight geometries and the proof that there are only eight)
* William Thurston. [http://www.msri.org/publications/books/gt3m/ The Geometry and Topology of Three-Manifolds], 1980 Princeton lecture notes on geometric structures on 3-manifolds.

==External links==
* [http://athome.harvard.edu/threemanifolds/ The Geometry of 3-Manifolds(video)] A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.

{{DEFAULTSORT:Geometrization Conjecture}}
[[Category:Geometric topology]]
[[Category:Riemannian geometry]]
[[Category:3-manifolds]]
[[Category:Conjectures]]

{{Link GA|de}}

Newton–Cotes formulas

2014-01-27T15:48:22Z

Jitse Niesen: Reverted edits by 194.210.227.168 (talk) to last version by Jitse Niesen

{{Disputed|date=June 2010}}

In finance, a '''forward rate agreement''' ('''FRA''') is a [[forward contract]], an over-the-counter contract between parties that determines the rate of interest, or the currency [[exchange rate]], to be paid or received on an obligation beginning at a future start date. The contract will determine the rates to be used along with the termination date and notional value.<ref>http://www.investopedia.com/terms/f/fra.asp</ref> On this type of agreement, it is only the differential that is paid on the notional amount of the contract. It is paid on the ''effective date''. The reference rate is fixed one or two days before the effective date, dependent on the market convention for the particular currency. FRAs are [[derivative (finance)#OTC_and_exchange-traded|over-the counter derivatives]]. FRAs are very similar to [[Swap (finance)|swap]]s except that in a FRA a payment is only made once at maturity. Instruments such as [[interest rate swap]] could be viewed as a chain of FRAs.

Many banks and large corporations will use FRAs to hedge future interest or exchange rate exposure. The buyer hedges against the risk of rising interest rates, while the seller hedges against the risk of falling interest rates. Other parties that use Forward Rate Agreements are speculators purely looking to make bets on future directional changes in interest rates.{{Citation needed|date=September 2009}} The development swaps in the 1980s provided organisations with an alternative to FRAs for hedging and speculating.

In other words, a forward rate agreement (FRA) is a tailor-made, over-the-counter financial futures contract on short-term deposits. A FRA transaction is a contract between two parties to exchange payments on a deposit, called the ''Notional amount'', to be determined on the basis of a short-term interest rate,
referred to as the ''Reference rate'', over a predetermined time period at a future date. FRA transactions are entered as a hedge against interest rate changes. The buyer of the contract locks in the interest rate in an effort to protect against an interest rate increase, while the seller protects against a possible interest rate decline. At maturity, no funds exchange hands; rather, the
difference between the contracted interest rate and the market rate is exchanged. The buyer of the contract is paid if the reference rate is above the contracted rate, and the buyer pays to the seller if the reference rate is below the contracted rate. A company that seeks to hedge against a possible increase in interest rates would purchase FRAs, whereas a company that seeks an interest hedge against a possible decline of the rates would sell FRAs.

==Payoff formula==
The netted payment made at the effective date is as follows

<math>
\mbox{Payment} = \mbox{Notional Amount} * \left( \frac{(\mbox{Reference Rate}-\mbox{Fixed Rate}) * \alpha }{ 1 + \mbox{Reference Rate} * \alpha } \right)
</math>

* The Fixed Rate is the rate at which the contract is agreed.
* The Reference Rate is typically [[Euribor]] or [[LIBOR]].
* <math>\alpha </math> is the ''day count fraction'', i.e. the portion of a year over which the rates are calculated, using the [[day count convention]] used in the money markets in the underlying currency. For EUR and USD this is generally the number of days divided by 360, for GBP it is the number of days divided by 365 days.
* The Fixed Rate and Reference Rate are rates that should accrue over a period starting on the effective date, and then paid at the end of the period (termination date). However, as the payment is already known at the beginning of the period, it is also paid at the beginning. This is why the discount factor is used in the denominator.

==FRAs Notation==
FRA Descriptive Notation and Interpretation
{|class="wikitable"
!|Notation
!|Effective Date from now
!|Termination Date from now
!|Underlying Rate
|-
|1 x 4
|1 month
|4 months
|4-1 = 3 months LIBOR
|-
|1 x 7
|1 month
|7 months
|7-1 = 6 months LIBOR
|-
|3 x 6
|3 months
|6 months
|6-3 = 3 months LIBOR
|-
|3 x 9
|3 months
|9 months
|9-3 = 6 months LIBOR
|-
|6 x 12
|6 months
|12 months
|12-6 = 6 months LIBOR
|-
|12 x 18
|12 months
|18 months
|18-12 = 6 months LIBOR
|}

How to interpret a quote for FRA?

[US$ 3x9 - 3.25/3.50%p.a ] - means deposit interest starting 3 months from now for 6 month is 3.25% and borrowing interest rate
starting 3 months from now for 6 month is 3.50% (see also [[bid–offer spread]]). Entering an "payer FRA" means paying the fixed rate (3.50% p.a.) and receiving a floating 6-month rate, while entering a "receiver FRA" means paying the same floating rate and receiving a fixed rate (3.25% p.a.).

==References==
{{reflist}}

==See also==
*[[Forward rate]]
*[[Derivative (finance)]]
*[[List of finance topics]]
*[[Wikinvest:Forward Rate Agreement|Forward Rate Agreements on Wikinvest]]

{{Finance}}

{{DEFAULTSORT:Forward Rate Agreement}}
[[Category:Derivatives (finance)]]

Rastrigin function

2014-01-23T14:34:46Z

Jitse Niesen: remove {{mathanalysis-stub}}

{{confusing|date=June 2010}}

'''First-break picking ''' detecting or picking the onset arrivals of refracted signals from all the signals received by receiver arrays and produced by a particular source signal generation. It is also called first arrival picking or first break detection. First-break picking can be done automatically, manually or as a combination of both. With the development of computer science and the size of seismic surveys, automatic picking is often preferred.<ref name=Yilmaz/>

==Significance==
'''First-break picks''' associated with the [[refracted]] arrival times are used in an [[Inversion (geology)|inversion scheme]] to study the near-surface low-[[velocity]] zone and subsequent determination of static corrections.
Static correction is a correction applied to geophysical data, especially seismic data, to compensate for the effect of near-surface irregularities, differences in the elevation of shots and [[geophone]]s, or any application to correct the positions of source and receivers.

==History of First Break Picking==
Gelchinsky and Shtivelman<ref name=Gelchinsky/>(1983) used [[correlation]] properties of signals and applied a statistical criterion for the estimation of first arrivals time.

Coppens<ref name=Coppens/>(1985) calculated the ratio of energy of [[seismogram]] of the two windows and used that to differentiate in signal and noise.

Michael D. McCormark ''et al.''<ref name=Michael/>(1993) introduced a [[backpropagation]] neural network (BNN) method. The [[Neural network]] which edits [[seismic]] data or pick first breaks was trained by users, who were just selecting and presenting to the network examples of trace edits or refraction picks. The network then changes internal weights iteratively until it can reproduce the examples accurately provided by the users.

Fabio Boschetti ''et al.''<ref name=Fabio/>(1996) introduce a [[fractal]]-based [[algorithm]] which detects the presence of a signal by analyzing the variation in fractal dimension along the trace. This method works when signal-to-noise ratio is small, but it is considerably slow.

A direct correlation method was introduced by Joseph ''et al.''<ref name=Joseph/>(1999) which was developed for use in highly time-resolved, low-noise signals acquired in the laboratory. In this method, the greatest value of Pearson's correlation coefficient between segments of observed [[waveform]]s near the [[pulse onset]] and at an appropriate reference serves as the time determination criterion.

Zuolin Chen, ''et al.''<ref name=Zuolin/>(2005) introduced a multi-window algorithm to detect the first break. In this method, three moving windows were used and the averages of absolute amplitudes in each window need to be calculated, then ratios based on the averages of the windows provide standards to differentiate signals from unwanted noise.

Wong ''et al.''<ref name=Wong/>(2009) introduced STA/LTA ratio method. This method is similar as Coppens’<ref name=Coppens/> algorithm. The difference is to do the ratio of two averages of energy between a short-term window and a long-term window, which is denoted as STA/LTA (short-term average/long-term average), instead of calculating the ratio of energy of seismogram of the two windows in Coppens’ algorithm.

==Methods of Automatic First Break Picking==
===STA/LTA ratio Method<ref name=Wong/>===
This method is similar as Coppens’ (1985) algorithm. The difference is to do the ratio of two averages of energy between a short-term window and a long-term window, which is denoted as STA/LTA (short-term average/long-term average), instead of calculating the ratio of energy of seismogram of the two windows in Coppens’ algorithm. The numerical [[derivative]] of the ratio can be defined as,
: <math>\begin{align}
& d_i=r_{i+1}-r_i , i=1,2,...(n-1) \\
\end{align}</math>
where ''ri+1'' is the STA/LTA ratio at time index ''i+1'', and ''ri'' is the STA/LTA ratio at time index ''i''. For noise-free seismograms, the maximum value of the numerical derivative of the STA/LTA ratio is close to the time of the first arrival.

Wong et al. (2009) modified the algorithm of the energy ratio method, where they named the method as modified energy ratio. In this method, they define the energy ratio as,
: <math>\begin{align}
& er=\sum_{j=i-ne}^i
x^2_j/\sum_{j=i}^{i+ne}
x^2_j \\
\end{align}</math>
where ''xi'' is the times series representing a seismogram with the time index ''i=1, 2 … N.'' and the number of points in an energy window is ''ne''. Then, the modified energy ratio is defined as
: <math>\begin{align}
& er3_i=(abs(x_i)^*er)^3 \\
\end{align}</math>
The peak of the modified energy ratio ''er3i'' is very closed to the time of first arrivals on noise-free seismograms.

=== Multi-Window Method<ref name=Zuolin/>===
This method needs to calculate the averages of absolute [[amplitude]]s from a seismic trace by using three moving time windows before and after each time point (sample).

When the instantaneous absolute amplitude exceeds an automatically adjusted threshold, ratios based on the averages of the windows over previous time samples provide standards to differentiate signals from unwanted noise.

The multi-window automatic P phase picker operates in the time-domain. It includes procedures to define: time windows, standards, corresponding thresholds and [[waveform]] correction.

1. The averages of absolute amplitudes within BTA (Before Term Average), ATA (After Term Average) and DTA (Delayed Term Average) windows are respectively defined as follows:
: <math>\begin{align}
&\overline{BTA(t)}=\sum_{i=i}^{m}\frac{|u(t-i)|}{m}\\
&\overline{ATA(t)}=\sum_{i=i}^{n}\frac{|u(t+j)|}{n}\\
&\overline{DTA(t)}=\sum_{k=i}^{q}\frac{|u(t+j+d)|}{q}\\
&R_2(t)=\frac{\overline{ATA(t)}}{\overline{BTA(t)}}\\
&R_3(t)=\frac{\overline{DTA(t)}}{\overline{BTA(t)}}\\
\end{align}</math>
Standards ''R2(t)'' and ''R3(t)'' are used for the discrimination of high-amplitude short-duration and long-duration noise.

2.Thresholds is defined as
: <math>\begin{align}
& H_1(t)=E_m(t-p)+\alpha E_{sd}(t-p) \\
\end{align}</math>
where ''Em'' is mean and ''Esd'' is [[standard deviation]]; ''p'' is the number of shifted samples; ''α'' is the coefficient to adjust the height of the first threshold and is taken to be 3. From this equation it is obvious that ''H1(t)'' is automatically adjusted with the variance of the background noise.

3. ''H1(t)'' is defined larger than most pre-existing noise levels, and the instantaneous absolute amplitude at the trigger time point is higher than ''H1(t)'',according to the configuration of the first arrival of an event the real onset time must be earlier than the trigger time point. A waveform correction should be used to compensate this belated onset time. For an impulsive first arrival, the height of the absolute amplitude and the representative [[gradient]] at the trigger point can be used to accomplish the correction.

==Available Code==
Potash SU is a package including [[Seismic Unix]] style codes developed by Balazs Nemeth, it provides a [[subroutine]] called simple window-based first break picker, the figure shows the seismic images before and after the application of subroutine.
[[File:Wagc=1.PNG|thumb|Right one is before first break picking, left one is after first break picking]]

==Future Trend of the Topic==
Methods of Picking: the automatic first-break picking has been playing a more important role in processing the seismic data, and directly influences the authenticity of the seismic sections. Because of the increasing of [[seismic survey]] size, more efficient and fast first break picking method will be needed, and with the development of computer science, the method which can be paralleled will be preferred.

Application of First Break: Traditionally the [[geologist]]s use the first break to do the static correction. First break signal can also be used as the observation data to do the history matching.

== Notes ==

{{reflist|refs=
<ref name=Coppens>Coppens F., (1985). ''First arrival picking on common-offset trace collections for automatic estimation of static corrections.'' Geophysical Prospecting, 33, 1212-1231.</ref>
<ref name=Fabio>Fabio Boschetti, Mike D. Dentith, and Ron D. List, (1996). ''A fractal-based algorithm for detecting first arrivals on seismic traces.'' Geophysics, Vol.61, No.4, P. 1095-1102.</ref>
<ref name=Gelchinsky>Gelchinsky B. and Shtivelman V., (1983). ''Automatic picking of first arrivals and parameterization of traveltime curves.'' Geophysical Prospecting, 31, 915-928.</ref>
<ref name=Joseph>Joseph B. Molyneux and Douglas R. Schmitt, (1999). ''First-break timing: Arrival onset times by direct correlation.'' Geophysics, Vol.64, No.5, P. 1492-1501.</ref>
<ref name=Michael>Michael D. McCormack et al, (1993). ''First-break refraction event picking and seismic data trace editing using neural networks.'' Geophysics, Vol.58, No.1, P. 67-78.</ref>
<ref name=Yilmaz>ÖZ, YILMAZ, (2000). ''Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data.'' Volume Ⅰ: Society of Exploration Geophysicists.</ref>
<ref name=Wong>Wong J., Han L., Bancroft J.C., Stewart R.R., (2009). ''Automatic time-picking of first arrivals on noisy microseismic data.[http://www.crewes.org/ForOurSponsors/ConferenceAbstracts/2009/CSEG/Wong_CSEG_2009.pdf]''</ref>
<ref name=Zuolin>Zuolin Chen and Robert Stewart, (2005). ''Multi-window algorithm for detecting seismic first arrivals.[http://www.crewes.org/ForOurSponsors/ConferenceAbstracts/2005/CSEG/Chen_CSEG_2005-1.pdf]''</ref>
}}

[[Category:Acoustics]]
[[Category:Seismology]]

Interpolation

2014-01-22T15:28:15Z

Jitse Niesen: Undid revision 591836790 by 36.70.29.148 (talk) - what is "nomor"?

{{Use British English|date=November 2012}}
{{sprotect2}}
{{Infobox scientist
| name = Sir Isaac Newton
| image = GodfreyKneller-IsaacNewton-1689.jpg
| alt = Portrait of man in black with shoulder-length, wavy brown hair, a large sharp nose, and a distracted gaze
| caption = [[Godfrey Kneller]]'s 1689 portrait of Isaac Newton (age 46).
| birth_date = 25 December 1642 {{small|<nowiki>[</nowiki>[[Old Style and New Style dates|NS]]: {{Birth date|1643|01|04|df=y}}<nowiki>]</nowiki>}}{{lower|0.1em|<ref name="OSNS"/>}}
| birth_place = {{nowrap|[[Woolsthorpe-by-Colsterworth]],}} [[Lincolnshire]], [[Kingdom of England|England]]
| death_date = {{nowrap|20 March 1727 (aged 84)}} {{small|<nowiki>[</nowiki>[[Old Style and New Style dates|OS]]: {{death date|1726|03|20|df=y}}  [[Old Style and New Style dates|NS]]: {{death date|1727|03|31|df=y}}<nowiki>]</nowiki>}}{{lower|0.1em|<ref name="OSNS"/>}}
| death_place = [[Kensington]], [[Middlesex]], [[England]], [[Kingdom of Great Britain|Great Britain]]
| resting_place = [[Westminster Abbey]]
| residence = England
| nationality = [[Kingdom of England|English]] {{small|(later [[Kingdom of Great Britain|British]])}}
| fields =
{{unbulleted list
| {{hlist |[[Physics]] |[[Natural philosophy]]}}
| {{hlist |[[Mathematics]] |[[Astronomy]]}}
| {{hlist |[[Alchemy]] |[[Christian theology]]}}
| {{hlist |[[Economics]] }}
}}
| workplaces = {{unbulleted list |[[University of Cambridge]] |[[Royal Society]] |[[Royal Mint]]}}
| alma_mater = [[Trinity College, Cambridge]]
| academic_advisors =
{{unbulleted list
| [[Isaac Barrow]]<ref>Mordechai Feingold, [http://www.oxforddnb.com/view/article/1541 Barrow, Isaac (1630–1677)], ''Oxford Dictionary of National Biography'', [[Oxford University Press]], September 2004; online edn, May 2007. Retrieved 24 February 2009; explained further in Mordechai Feingold's "[http://www.jstor.org/stable/236236 Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation]" in ''Isis'', Vol. 84, No. 2 (June 1993), pp. 310–338.</ref>
| [[Benjamin Pulleyn]]<ref>''[http://www.chlt.org/sandbox/lhl/dsb/page.50.a.php "Newton, Isaac"] in the ''Dictionary of Scientific Biography'', n.4.</ref><ref name="The Newton Handbook"/>
}}
| notable_students = {{unbulleted list |[[Roger Cotes]] |[[William Whiston]]}}
| known_for =
{{unbulleted list
|[[Newtonian mechanics]] |[[Universal gravitation]] |[[Infinitesimal calculus]]
|{{hlist |[[Optics]] |[[Binomial series]]}}
|{{hlist |''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' |[[Newton's method]]}}
}}
| influences =
{{unbulleted list
| [[Johannes Kepler]]
| [[Henry More]]<ref name="More" />
| [[Polish Brethren]]<ref name="heretic"/>
| [[Robert Boyle]]<ref>{{cite book |title=Isaac Newton |last=Stokes |first=Mitch |year=2010 |publisher=Thomas Nelson |isbn=1595553037 |page=97 |url=http://books.google.gr/books?id=zpsoSXCeg5gC&pg=PA97&lpg=PA97&dq=#v=onepage&q=%22Boyle%20influenced%20Newton%22&f=false |accessdate=17 October 2012}}</ref>
}}
| influenced = {{unbulleted list |[[Nicolas Fatio de Duillier]] |[[John Keill]]}}
| signature = Isaac Newton signature.svg
| signature_alt = Is. Newton

}}
{{Isaac Newton sidebar}}

'''[[Knight Bachelor|Sir]] Isaac Newton''' {{small|[[President of the Royal Society|PRS]] [[Member of Parliament|MP]]}} ({{IPAc-en|ˈ|n|j|uː|t|ən}};<ref>[http://www.collinsdictionary.com/dictionary/english/newton "Newton"] entry in ''[[Collins English Dictionary]]'', HarperCollins Publishers, 1998.</ref> 25 December 1642{{spaced ndash}}20 March 1727<ref name="OSNS"/>) was an English [[Physics|physicist]] and [[Methematics|mathematician]] who is widely regarded as one of the most influential scientists of all time and as a key figure in the [[scientific revolution]]. His book ''[[Philosophiæ Naturalis Principia Mathematica]]'' ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations for most of [[classical mechanics]]. Newton also made seminal contributions to [[optics]] and shares credit with [[Gottfried Leibniz]] for the invention of the [[infinitesimal calculus]].

Newton's ''Principia'' formulated the [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|universal gravitation]] that dominated scientists' view of the physical universe for the next three centuries. It also demonstrated that the [[Kinematics|motion of objects]] on the Earth and that of [[celestial mechanics|celestial]] bodies could be described by the same principles. By deriving [[Johannes Kepler|Kepler's]] [[Kepler's laws of planetary motion|laws of planetary motion]] from his mathematical description of gravity, Newton removed the last doubts about the validity of the [[heliocentrism|heliocentric]] model of the cosmos.

Newton built the first practical [[reflecting telescope]] and developed a theory of colour based on the observation that a [[Dispersive prism|prism]] decomposes white light into the many colours of the [[visible spectrum]]. He also formulated [[Newton's law of cooling|an empirical law of cooling]] and studied the [[speed of sound]]. In addition to his work on the calculus, as a mathematician Newton contributed to the study of [[power series]], generalised the [[binomial theorem]] to non-integer exponents, and developed [[Newton's method]] for approximating the [[Zero of a function|roots of a function]].

Newton was a fellow of [[Trinity College, Cambridge|Trinity College]] and the second [[Lucasian Professor of Mathematics]] at the [[University of Cambridge]]. He was a devout but unorthodox Christian and, unusually for a member of the Cambridge faculty, he refused to take [[holy orders]] in the [[Church of England]], perhaps because he privately rejected the doctrine of the [[Trinity]]. In addition to his work on the mathematical sciences, Newton also dedicated much of his time to the study of [[alchemy]] and [[Chronology of the Bible|biblical chronology]], but most of his work in those areas remained unpublished until long after his death. In his later life, Newton became president of the [[Royal Society]]. He also served the British government as Warden and Master of the [[Royal Mint]].

== Life ==

=== Early life ===
{{Main|Early life of Isaac Newton}}

Isaac Newton was born (according to the [[Julian calendar]] in use in England at the time) on Christmas Day, 25 December 1642 ([[Old Style and New Style dates|NS]] 4 January 1643<ref name="OSNS"/>), at [[Woolsthorpe Manor]] in [[Woolsthorpe-by-Colsterworth]], a [[hamlet (place)|hamlet]] in the county of [[Lincolnshire]]. He was born three months after the death of his father, a prosperous farmer also named Isaac Newton. Born [[premature birth|prematurely]], he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a [[quart]] mug (≈ 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young Isaac disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house over them." Her mother had three children from her second marriage.<ref>{{harvnb|Westfall|1980|p=55}}</ref> Although it was claimed that he was once engaged,<ref>This claim was made Dr. Stukeley in 1727, in a letter about Newton written to Dr. Richard Mead. Charles Hutton, who in the late 18th century collected oral traditions about earlier scientists, declares that there "do not appear to be any sufficient reason for his never marrying, if he had an inclination so to do. It is much more likely that he had a constitutional indifference to the state, and even to the sex in general." Charles Hutton "A Mathematical and Philosophical Dictionary" (1795/6) II p.100.</ref> Newton never married.

[[File:Sir Isaac Newton by Sir Godfrey Kneller, Bt.jpg|thumb|upright|Newton in a 1702 portrait by [[Godfrey Kneller]]]]
[[File:Bolton-newton.jpg|thumb|upright|Isaac Newton (''Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889'')]]

From the age of about twelve until he was seventeen, Newton was educated at [[The King's School, Grantham]]. He was removed from school, and by October 1659, he was to be found at [[Woolsthorpe-by-Colsterworth]], where his mother, widowed by now for a second time, attempted to make a farmer of him. He hated farming.<ref>Westfall 1994, pp 16–19</ref> Henry Stokes, master at the King's School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student.<ref>White 1997, p. 22</ref>

In June 1661, he was admitted to [[Trinity College, Cambridge]] as a [[sizar]] – a sort of work-study role.<ref>Michael White, ''Isaac Newton'' (1999) [http://books.google.com/books?id=l2C3NV38tM0C&pg=PA24&dq=storer+intitle:isaac+intitle:newton&lr=&num=30&as_brr=0&as_pt=ALLTYPES#PPA46,M1 page 46]</ref> At that time, the college's teachings were based on those of [[Aristotle]], whom Newton supplemented with modern philosophers, such as [[René Descartes|Descartes]], and [[astronomers]] such as [[Nicolaus Copernicus|Copernicus]], [[Galileo Galilei|Galileo]], and [[Johannes Kepler|Kepler]]. In 1665, he discovered the generalised [[binomial theorem]] and began to develop a mathematical theory that later became [[infinitesimal calculus]]. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the [[Great Plague of London|Great Plague]]. Although he had been undistinguished as a Cambridge student,<ref>ed. Michael Hoskins (1997). Cambridge Illustrated History of Astronomy, p. 159. [[Cambridge University Press]]</ref> Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus,<ref>{{cite web |last=Newton |first=Isaac |title=Waste Book |url=http://cudl.lib.cam.ac.uk/view/MS-ADD-04004/|publisher=Cambridge University Digital Library|accessdate=10 January 2012}}</ref> [[optics]] and the [[law of gravitation]]. In 1667, he returned to Cambridge as a fellow of Trinity.<ref>{{acad|id=RY644J|name=Newton, Isaac}}</ref> Fellows were required to become ordained priests, something Newton desired to avoid due to his unorthodox views. Luckily for Newton, there was no specific deadline for ordination, and it could be postponed indefinitely. The problem became more severe later when Newton was elected for the prestigious [[Lucasian Chair]]. For such a significant appointment, ordaining normally could not be dodged. Nevertheless, Newton managed to avoid it by means of a special permission from [[Charles II of England|Charles II]] (see "Middle years" section below).

=== Middle years ===

==== Mathematics ====
Newton's work has been said "to distinctly advance every branch of mathematics then studied".<ref>W W Rouse Ball (1908), "A short account of the history of mathematics", at page 319.</ref> His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton's mathematical papers.<ref>D T Whiteside (ed.), ''The Mathematical Papers of Isaac Newton'' (Volume 1), (Cambridge University Press, 1967), part 7 "The October 1666 Tract on Fluxions", [http://books.google.com/books?id=1ZcYsNBptfYC&pg=PA400 at page 400, in 2008 reprint].</ref> The author of the manuscript ''[[De analysi per aequationes numero terminorum infinitas]]'', sent by [[Isaac Barrow]] to [[John Collins (mathematician)|John Collins]] in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as:<ref>D Gjertsen (1986), "The Newton handbook", (London (Routledge & Kegan Paul) 1986), at page 149.</ref> {{quote|Mr Newton, a fellow of our College, and very young ... but of an extraordinary genius and proficiency in these things. }}
Newton later became involved in a dispute with [[Gottfried Wilhelm Leibniz|Leibniz]] over priority in the development of infinitesimal calculus (the [[Leibniz–Newton calculus controversy]]). Most modern historians believe that Newton and Leibniz developed [[infinitesimal calculus]] independently, although with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in 1684. (Leibniz's notation and "differential Method", nowadays recognised as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) Such a suggestion, however, fails to notice the content of calculus which critics of Newton's time and modern times have pointed out in [[Philosophiæ Naturalis Principia Mathematica#Book 1, De motu corporum|Book 1]] of Newton's ''Principia'' itself (published 1687) and in its forerunner manuscripts, such as ''[[De motu corporum in gyrum]]'' ("On the motion of bodies in orbit"), of 1684. The ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' is not written in the language of calculus either as we know it or as Newton's (later) 'dot' notation would write it. But his work extensively uses an infinitesimal calculus in geometric form, based on limiting values of the ratios of vanishing small quantities: in the ''Principia'' itself Newton gave demonstration of this under the name of 'the method of first and last ratios'<ref>Newton, 'Principia', 1729 English translation, [http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA41 at page 41].</ref> and explained why he put his expositions in this form,<ref>Newton, 'Principia', 1729 English translation, [http://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA54 at page 54].</ref> remarking also that 'hereby the same thing is performed as by the method of indivisibles'.

Because of this, the ''Principia'' has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times<ref>Clifford Truesdell, ''Essays in the History of Mechanics'' (Berlin, 1968), at p.99.</ref> and "lequel est presque tout de ce calcul" ('nearly all of it is of this calculus') in Newton's time.<ref>In the preface to the Marquis de L'Hospital's ''Analyse des Infiniment Petits'' (Paris, 1696).</ref> His use of methods involving "one or more orders of the infinitesimally small" is present in his ''De motu corporum in gyrum'' of 1684<ref>Starting with [[De motu corporum in gyrum#Contents|De motu corporum in gyrum]], see also [http://books.google.com/books?id=uvMGAAAAcAAJ&pg=RA1-PA2 (Latin) Theorem 1].</ref> and in his papers on motion "during the two decades preceding 1684".<ref>D T Whiteside (1970), "The Mathematical principles underlying Newton's Principia Mathematica" in ''Journal for the History of Astronomy'', vol.1, pages 116–138, especially at pages 119–120.</ref>

Newton had been reluctant to publish his calculus because he feared controversy and criticism.<ref>Stewart 2009, p.107</ref> He was close to the Swiss mathematician [[Nicolas Fatio de Duillier]]. In 1691, Duillier started to write a new version of Newton's ''Principia'', and corresponded with Leibniz.<ref>Westfall 1980, pp 538–539</ref> In 1693 the relationship between Duillier and Newton deteriorated, and the book was never completed.

Starting in 1699, other members of the [[Royal Society]] (of which Newton was a member) accused Leibniz of [[plagiarism]], and the dispute broke out in full force in 1711. The Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later found that Newton himself wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter's death in 1716.<ref>Ball 1908, p. 356ff</ref>

Newton is generally credited with the [[Catalon series|generalised binomial theorem]], valid for any exponent. He discovered [[Newton's identities]], [[Newton's method]], classified cubic plane curves ([[polynomials]] of degree three in two [[variable (mathematics)|variables]]), made substantial contributions to the theory of [[finite differences]], and was the first to use fractional indices and to employ [[coordinate geometry]] to derive solutions to [[Diophantine equations]]. He approximated partial sums of the [[harmonic series (mathematics)|harmonic series]] by [[logarithms]] (a precursor to [[Euler's summation formula]]), and was the first to use [[power series]] with confidence and to revert power series. Newton's work on infinite series was inspired by [[Simon Stevin]]'s decimals.<ref>{{citation | last1 = Błaszczyk | first1 = Piotr |last2 = Katz | first2 = Mikhail |author2-link = Mikhail Katz | last3 = Sherry | first3 = David | arxiv = 1202.4153 |doi = 10.1007/s10699-012-9285-8 | issue = |journal = [[Foundations of Science]] | pages = | title = Ten misconceptions from the history of analysis and their debunking | volume = |year = 2012}}</ref>

He was appointed [[Lucasian Professor of Mathematics]] in 1669 on Barrow's recommendation. In that day, any fellow of Cambridge or Oxford was required to become an ordained [[Anglicanism|Anglican]] priest. However, the terms of the Lucasian professorship required that the holder ''not'' be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the ordination requirement, and [[Charles II of England|Charles II]], whose permission was needed, accepted this argument. Thus a conflict between Newton's religious views and Anglican orthodoxy was averted.<ref>White 1997, p. 151</ref>

==== Optics ====
In 1666, Newton observed that the spectrum of colours exiting a [[Triangular prism (optics)|prism]] is oblong, even when the light ray entering the prism is circular, which is to say, the prism refracts different colours by different angles.<ref name="Whittaker" /> This led him to conclude that colour is a property intrinsic to light—a point which had been debated.

[[File:NewtonsTelescopeReplica.jpg|thumb|Replica of Newton's second [[Reflecting telescope]] that he presented to the [[Royal Society]] in 1672<ref>{{cite book|url=http://books.google.com/?id=KAWwzHlDVksC&dq=history+of+the+telescope&printsec=frontcover |title=''The History of the Telescope'' By Henry C. King, Page 74 |publisher=Google Books |accessdate=16 January 2010|isbn=978-0-486-43265-6|author1=King, Henry C|year=2003}}</ref>]]
From 1670 to 1672, Newton lectured on optics.<ref>{{cite web |last=Newton |first=Isaac |title=Hydrostatics, Optics, Sound and Heat|url=http://cudl.lib.cam.ac.uk/view/MS-ADD-03970/|publisher=Cambridge University Digital Library|accessdate=10 January 2012}}</ref> During this period he investigated the [[refraction]] of light, demonstrating that the multicoloured spectrum produced by a prism could be recomposed into white light by a [[lens (optics)|lens]] and a second prism.<ref>Ball 1908, p. 324</ref> Modern scholarship has revealed that Newton's analysis and resynthesis of white light owes a debt to [[Corpuscularianism|corpuscular]] alchemy.<ref>[[William R. Newman]], "Newton's Early Optical Theory and its Debt to Chymistry," in Danielle Jacquart and Michel Hochmann, eds., ''Lumière et vision dans les sciences et dans les arts'' (Geneva: Droz, 2010), pp. 283-307. A free access online version of this article can be found at [http://webapp1.dlib.indiana.edu/newton/html/Newton_optics-alchemy_Jacquart_paper.pdf the ''Chymistry of Isaac Newton'' project]</ref>

He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as [[Early life of Isaac Newton#Newton's theory of colour|Newton's theory of colour]].<ref>Ball 1908, p. 325</ref>

[[File:Dispersive Prism Illustration.jpg|thumb|250px|left|Illustration of a [[dispersive prism]] decomposing white light into the colours of the spectrum, as discovered by Newton]]
From this work, he concluded that the lens of any [[refracting telescope]] would suffer from the [[dispersion (optics)|dispersion]] of light into colours ([[chromatic aberration]]). As a proof of the concept, he constructed a telescope using a mirror as the [[objective (optics)|objective]] to bypass that problem.<ref name="The Early Period (1608–1672)"/><ref name="White 1997, p170"/> Building the design, the first known functional reflecting telescope, today known as a [[Newtonian telescope]],<ref name="White 1997, p170"/> involved solving the problem of a suitable mirror material and shaping technique. Newton ground his own mirrors out of a custom composition of highly reflective [[speculum metal]], using [[Newton's rings]] to judge the [[quality (philosophy)|quality]] of the optics for his telescopes. In late 1668<ref name="books.google.com"/> he was able to produce this first ''reflecting telescope''. In 1671, the Royal Society asked for a demonstration of his reflecting telescope.<ref>White 1997, p168</ref> Their interest encouraged him to publish his notes ''On Colour'', which he later expanded into his ''[[Opticks]]''. When [[Robert Hooke]] criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. Newton and Hooke had brief exchanges in 1679–80, when Hooke, appointed to manage the Royal Society's correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal Society transactions,<ref name="hooke1679nov24"/> which had the effect of stimulating Newton to work out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see [[Newton's law of universal gravitation#History|Newton's law of universal gravitation – History]] and ''[[De motu corporum in gyrum]]''). But the two men remained generally on poor terms until Hooke's death.<ref>Iliffe, Robert (2007) Newton. A very short introduction, Oxford University Press 2007</ref>

[[File:Newton-letter-to-briggs 03.jpg|thumb|right|Facsimile of a 1682 letter from Isaac Newton to Dr William Briggs, commenting on Briggs' "A New Theory of Vision"]]

Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films (Opticks Bk.II, Props. 12), but still retained his theory of 'fits' that disposed corpuscles to be reflected or transmitted (Props.13). Later physicists instead favoured a purely wavelike explanation of light to account for the [[interference (wave propagation)|interference]] patterns, and the general phenomenon of [[diffraction]]. Today's [[quantum mechanics]], [[photons]] and the idea of [[wave–particle duality]] bear only a minor resemblance to Newton's understanding of light.

In his ''Hypothesis of Light'' of 1675, Newton [[wikt:posit|posited]] the existence of the [[luminiferous aether|ether]] to transmit forces between particles. The contact with the [[theosophist]] [[Henry More]], revived his interest in alchemy. He replaced the ether with occult forces based on [[Hermeticism|Hermetic]] ideas of attraction and repulsion between particles. [[John Maynard Keynes]], who acquired many of Newton's writings on alchemy, stated that "Newton was not the first of the age of reason: He was the last of the magicians."<ref name="The Collected Writings of John Maynard Keynes Volume X"/> Newton's interest in alchemy cannot be isolated from his contributions to science.<ref name="More"/> This was at a time when there was no clear distinction between alchemy and science. Had he not relied on the [[occult]] idea of [[action at a distance (physics)|action at a distance]], across a vacuum, he might not have developed his theory of gravity. (See also [[Isaac Newton's occult studies]].)

In 1704, Newton published ''[[Opticks]]'', in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?"<ref name="Newton's Alchemy and His Theory of Matter"/> Newton also constructed a primitive form of a frictional [[electrostatic generator]], using a glass globe (Opticks, 8th Query).

In an article entitled "Newton, prisms, and the 'opticks' of tunable lasers"<ref name="OPN1"/> it is indicated that Newton in his book ''[[Opticks]]'' was the first to show a diagram using a prism as a beam expander. In the same book he describes, via diagrams, the use of multiple-prism arrays. Some 278 years after Newton's discussion, [[beam expander#Multiple-prism beam expanders|multiple-prism beam expanders]] became central to the development of [[laser linewidth|narrow-linewidth]] [[tunable laser]]s. Also, the use of these prismatic beam expanders led to the [[multiple-prism dispersion theory]].<ref name=OPN1/>

Subsequent to Newton, much has been amended. Young and Fresnel combined Newton's particle theory with Huygen's wave theory to show that colour is the visible manifestation of light's wavelength. Science also slowly came to realise the difference between perception of colour and mathematisable optics. The German poet, [[Johann Wolfgang von Goethe|Goethe]], could not shake the Newtonian foundation but "one hole Goethe did find in Newton's armour, ... Newton had committed himself to the doctrine that refraction without colour was impossible. He therefore thought that the object-glasses of telescopes must for ever remain imperfect, achromatism and refraction being incompatible. This inference was proved by [[John Dollond |
Dollond]] to be wrong."<ref>Tyndall, John. (1880). ''Popular Science Monthly'' Volume 17, July. [http://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_17/July_1880/Goethe's_Farbenlehre:_Theory_of_Colors_II]</ref>

==== Mechanics and gravitation ====
[[File:NewtonsPrincipia.jpg|thumb|Newton's own copy of his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', with hand-written corrections for the second edition]]
{{further2|[[Writing of Principia Mathematica]]}}

In 1679, Newton returned to his work on ([[celestial mechanics|celestial]]) mechanics, i.e., gravitation and its effect on the orbits of [[planet]]s, with reference to [[Kepler's laws]] of planetary motion. This followed stimulation by a brief exchange of letters in 1679–80 with Hooke, who had been appointed to manage the Royal Society's correspondence, and who opened a correspondence intended to elicit contributions from Newton to Royal Society transactions.<ref name=hooke1679nov24 /> Newton's reawakening interest in astronomical matters received further stimulus by the appearance of a comet in the winter of 1680–1681, on which he corresponded with [[John Flamsteed]].<ref>{{harvnb|Westfall|1980|p=391–2}}</ref> After the exchanges with Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see [[Newton's law of universal gravitation#History|Newton's law of universal gravitation – History]] and De motu corporum in gyrum). Newton communicated his results to [[Edmond Halley]] and to the Royal Society in ''[[De motu corporum in gyrum]]'', a tract written on about 9 sheets which was copied into the Royal Society's Register Book in December 1684.<ref>D T Whiteside (ed.), 'Mathematical Papers of Isaac Newton', vol.6, 1684–1691, Cambridge University Press 1974, at page 30.</ref> This tract contained the nucleus that Newton developed and expanded to form the ''Principia''.

The ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' was published on 5 July 1687 with encouragement and financial help from [[Edmond Halley]]. In this work, Newton stated the [[Isaac Newton#Laws of motion|three universal laws of motion]] that enabled many of the advances of the [[Industrial Revolution]] which soon followed and were not to be improved upon for more than 200 years, and are still the underpinnings of the non-relativistic technologies of the modern world. He used the Latin word ''gravitas'' (weight) for the effect that would become known as [[gravity]], and defined the law of [[Newton's law of universal gravitation|universal gravitation]].

In the same work, Newton presented a calculus-like method of geometrical analysis by 'first and last ratios', gave the first analytical determination (based on [[Boyle's law]]) of the speed of sound in air, inferred the oblateness of the spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the [[Lunar theory#Newton|irregularities in the motion of the moon]], provided a theory for the determination of the orbits of comets, and much more.

Newton made clear his [[heliocentric]] view of the solar system – developed in a somewhat modern way, because already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the solar system.<ref>See Curtis Wilson, "The Newtonian achievement in astronomy", pages 233–274 in R Taton & C Wilson (eds) (1989) ''The General History of Astronomy'', Volume, 2A', [http://books.google.com/books?id=rkQKU-wfPYMC&pg=PA233 at page 233].</ref> For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this centre of gravity "either is at rest or moves uniformly forward in a right line" (Newton adopted the "at rest" alternative in view of common consent that the centre, wherever it was, was at rest).<ref>Text quotations are from 1729 translation of Newton's ''Principia'', Book 3 (1729 vol.2) [http://books.google.com/books?id=6EqxPav3vIsC&pg=PA233 at pages 232–233].</ref>

Newton's postulate of an invisible [[action at a distance (physics)|force able to act over vast distances]] led to him being criticised for introducing "[[occult]] agencies" into science.<ref>Edelglass et al., ''Matter and Mind'', ISBN 0-940262-45-2. p. 54</ref> Later, in the second edition of the ''Principia'' (1713), Newton firmly rejected such criticisms in a concluding [[General Scholium]], writing that it was enough that the phenomena implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous expression ''"hypotheses non fingo"<ref>On the meaning and origins of this expression, see Kirsten Walsh, [https://blogs.otago.ac.nz/emxphi/2010/10/does-newton-feign-an-hypothesis/ Does Newton feign an hypothesis?], [https://blogs.otago.ac.nz/emxphi/ Early Modern Experimental Philosophy], 18 October 2010.</ref>'').

With the ''Principia'', Newton became internationally recognised.<ref>Westfall 1980. Chapter 11.</ref> He acquired a circle of admirers, including the [[Switzerland|Swiss]]-born mathematician [[Nicolas Fatio de Duillier]], with whom he formed an intense relationship. This abruptly ended in 1693, and at the same time Newton suffered a [[nervous breakdown]].<ref>{{harvnb|Westfall| 1980|p= 493–497}} on the friendship with Fatio, pp 531–540 on Newton's breakdown.</ref>

=== Classification of cubics ===
Besides the work of Newton and others on calculus, the first important demonstration of the power of analytic geometry was Newton's classification of cubic curves in the Euclidean plane in the late 1600s. He divided them into four types, satisfying different equations, and in 1717 [[James Stirling (mathematician)|Stirling]], probably with Newton's help, proved that every cubic was one of these four. Newton also claimed that the four types could be obtained by plane projection from one of them, and this was proved in 1731.<ref>''Conics and Cubics,'' Robert Bix, Springer Undergraduate Texts in Mathematics, 2nd edition, 2006, Springer Verlag.</ref>

=== Later life ===
[[File:Newton 25.jpg|thumb|upright|left|Isaac Newton in old age in 1712, portrait by [[Sir James Thornhill]]]]
{{Main|Later life of Isaac Newton}}
In the 1690s, Newton wrote a number of [[religious tracts]] dealing with the literal and symbolic interpretation of the Bible. A manuscript Newton sent to [[John Locke]] in which he disputed the fidelity of [[1 John 5:7]] and its fidelity to the original manuscripts of the New Testament, remained unpublished until 1785.<ref>[http://www.libraries.psu.edu/tas/locke/mss/c1690.html John Locke Manuscripts -- Chronological Listing: 1690]</ref><ref>John C. Attig, [http://www.libraries.psu.edu/tas/locke/bib/ch5c.html#01160 John Locke Bibliography — Chapter 5, Religion, 1751-1900]</ref>

Even though a number of authors have claimed that the work might have been an indication that Newton disputed the belief in [[Trinity]]; others assure that Newton did question the passage but never denied Trinity as such. His biographer, scientist [[Sir David Brewster]], who compiled his manuscripts for over 20 years, wrote about the controversy in well-known book''Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton'', where he explains that Newton questioned the veracity of those passages, but he never denied the doctrine of Trinity as such. Brewster states that Newton was never known as an [[Arianism|Arian]] during his lifetime, it was first [[William Whiston]] (an Arian) who argued that ''"Sir Isaac Newton was so hearty for the Baptists, as well as for the Eusebians or Arians, that he sometimes suspected these two were the two witnesses in the Revelations,"'' while other like [[Hopton Haynes]] (a Mint employee and [[Humanitarian]]), "mentioned to [[Richard Baron]], that Newton held the same doctrine as himself".<ref>David Brewster. "Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton:" p.268.</ref>

Later works{{spaced ndash}}''[[The Chronology of Ancient Kingdoms Amended]]'' (1728) and ''Observations Upon the Prophecies of Daniel and the Apocalypse of St. John'' (1733){{spaced ndash}}were published after his death. He also devoted a great deal of time to [[alchemy]] (see above).

Newton was also a member of the [[Parliament of England]] for [[Cambridge University (UK Parliament constituency)|Cambridge University]] in 1689–90 and 1701–2, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed.<ref>White 1997, p. 232</ref><ref>{{cite book |first1=E.R. |last1=Edwards |first2=Geoffrey |last2=Jaggar |chapter=Newton, Isaac (1642-1727) |title=The History of Parliament: the House of Commons 1660-1690 |editor-first=B.D. |editor-last=Henning |year=1983 |url=http://www.historyofparliamentonline.org/volume/1660-1690/member/newton-isaac-1642-1727 }}</ref><ref>{{cite book |first=D.W. |last=Hayton |chapter=Newton, Isaac (1642-1727) |title=The History of Parliament: the House of Commons 1690-1715 |editor1-first=D. |editor1-last=Hayton |editor2-first=E. |editor2-last=Cruickshanks |editor3-first=S. |editor3-last=Handley |year=2002 |url=http://www.historyofparliamentonline.org/volume/1690-1715/member/newton-isaac-1642-1727 }}</ref>

Newton moved to London to take up the post of warden of the [[Royal Mint]] in 1696, a position that he had obtained through the patronage of [[Charles Montagu, 1st Earl of Halifax]], then [[Chancellor of the Exchequer]]. He took charge of England's great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy [[comptroller]] of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known [[Master of the Mint]] upon the death of [[Thomas Neale]] in 1699, a position Newton held for the last 30 years of his life.<ref name="Mint">{{cite episode |series=Science Friday |title=Isaac Newton: Physicist And ... Crime Fighter? |airdate=5 June 2009 |transcripturl=http://www.npr.org/templates/transcript/transcript.php?storyId=105012144 |episodelink=http://www.npr.org/templates/story/story.php?storyId=105012144 |network=NPR}}</ref><ref>{{cite book |title=Newton and the counterfeiter : the unknown detective career of the world's greatest scientist |author=Thomas Levenson |publisher=Houghton Mifflin Harcourt |year=2009 |isbn=978-0-15-101278-7 |oclc=276340857}}</ref> These appointments were intended as [[sinecure]]s, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish [[debasement|clippers]] and counterfeiters. As Master of the Mint in 1717 in the "[[Anne, Queen of Great Britain|Law of Queen Anne]]" Newton moved the [[Pound Sterling]] ''de facto'' from the [[silver standard]] to the [[gold standard]] by setting the bimetallic relationship between gold coins and the silver penny in favour of gold. This caused silver sterling coin to be melted and shipped out of Britain. Newton was made President of the [[Royal Society]] in 1703 and an associate of the French [[French Academy of Sciences|Académie des Sciences]]. In his position at the Royal Society, Newton made an enemy of [[John Flamsteed]], the [[Astronomer Royal]], by prematurely publishing Flamsteed's ''Historia Coelestis Britannica'', which Newton had used in his studies.<ref>White 1997, p.317</ref>
[[File:ENG COA Newton.svg|thumb|upright|Personal coat of arms of Sir Isaac Newton<ref name="Coat of arms of Isaac Newton"/>]]

In April 1705, Queen Anne [[Knight Bachelor|knighted]] Newton during a royal visit to Trinity College, Cambridge. The knighthood is likely to have been motivated by political considerations connected with the [[Parliamentary elections|Parliamentary election]] in May 1705, rather than any recognition of Newton's scientific work or services as Master of the Mint.<ref>"The Queen's 'great Assistance' to Newton's election was his knighting, an honor bestowed not for his contributions to science, nor for his service at the Mint, but for the greater glory of party politics in the election of 1705." Westfall 1994 p.245</ref> Newton was the second scientist to be knighted, after Sir [[Francis Bacon]].

Towards the end of his life, Newton took up residence at [[Cranbury Park]], near [[Winchester]] with his niece and her husband, until his death in 1727.<ref name = "Yonge6">{{cite web|last=[[Charlotte M. Yonge|Yonge]] |first=Charlotte M. |title=Cranbury and Brambridge|url=http://www.online-literature.com/charlotte-yonge/john-keble/6/ |year= 1898|work=[[John Keble]]'s Parishes – Chapter 6 |publisher=www.online-literature.com|accessdate=23 September 2009}}</ref> His half-niece, [[Catherine Barton Conduitt]],<ref>{{harvnb|Westfall| 1980|p=44}}</ref> served as his hostess in social affairs at his house on [[Jermyn Street]] in London; he was her "very loving Uncle,"<ref>{{harvnb|Westfall| 1980|p=595}}</ref> according to his letter to her when she was recovering from [[smallpox]].

Newton died in his sleep in London on 20 March 1727 ([[Old Style and New Style dates|OS]] 20 March 1726; [[Old Style and New Style dates|NS]] 31 March 1727)<ref name="OSNS"/> and was buried in [[Westminster Abbey]]. [[Voltaire]] was present at his funeral and praised the British for honoring a scientist of [[heretical]] religious beliefs with burial there. A bachelor, he had divested much of his estate to relatives during his last years, and died [[intestacy|intestate]]. After his death, Newton's hair was examined and found to contain [[mercury (element)|mercury]], probably resulting from his alchemical pursuits. [[Mercury poisoning]] could explain Newton's eccentricity in late life.<ref name="Newton, Isaac (1642–1727)"/>

=== After death ===

==== Fame ====
The mathematician [[Joseph Louis Lagrange|Joseph-Louis Lagrange]] often said that Newton was the greatest genius who ever lived, and once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to establish."<ref>Fred L. Wilson, ''History of Science: Newton'' citing: Delambre, M. "Notice sur la vie et les ouvrages de M. le comte J. L. Lagrange," ''Oeuvres de Lagrange'' I. Paris, 1867, p. xx.</ref> English poet [[Alexander Pope]] was moved by Newton's accomplishments to write the famous [[epitaph]]:
<blockquote>
Nature and nature's laws lay hid in night; 
God said "Let Newton be" and all was light.</blockquote>

Newton himself had been rather more modest of his own achievements, famously writing in a letter to [[Robert Hooke]] in February 1676:
<blockquote>If I have seen further it is by [[standing on the shoulders of giants]].<ref>Letter from Isaac Newton to Robert Hooke, 5 February 1676, as transcribed in Jean-Pierre Maury (1992) ''Newton: Understanding the Cosmos'', New Horizons</ref></blockquote>

Two writers think that the above quote, written at a time when Newton and Hooke were in dispute over optical discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather than{{spaced ndash}}or in addition to{{spaced ndash}}a statement of modesty.<ref>John Gribbin (2002) ''Science: A History 1543–2001'', p 164.</ref><ref>White 1997, p187.</ref> On the other hand, the widely known proverb about [[standing on the shoulders of giants]] published among others by 17th-century poet [[George Herbert]] (a former orator of the University of Cambridge and fellow of Trinity College) in his ''Jacula Prudentum'' (1651), had as its main point that "a dwarf on a giant's shoulders sees farther of the two", and so its effect as an analogy would place Newton himself rather than Hooke as the 'dwarf'.

In a later memoir, Newton wrote:
<blockquote>I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.<ref>Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) by Sir David Brewster (Volume II. Ch. 27)</ref></blockquote>

In 1816 a tooth said to have belonged to Isaac Newton was sold for £730<ref>{{cite journal|date=16 January 1881|title=Silly relic-worship|journal=The New York Times|pages=10|url=http://query.nytimes.com/gst/abstract.html?res=9E02E4DE1730EE3ABC4E52DFB766838A699FDE|accessdate=2009-07-12}}</ref> ({{smallcaps|us}}$3,633) in London to an aristocrat who passed to have it set in a ring.<ref name=Guiness>{{cite book|title=Guinness World Records 2002|url=http://www.amazon.com/Guinness-World-Records-Antonia-Cunningham/dp/product-description/0553583786|accessdate=2009-07-12}}</ref> The ''[[Guinness World Records]] 2002'' classified it as the most valuable tooth, which would value approximately £25,000 ({{smallcaps|us}}$35,700) in late 2001's terms.<ref name=Guiness/> Who has bought it and to whom it currently pertains are mysteries.

[[Albert Einstein]] kept a picture of Newton on his study wall alongside ones of [[Michael Faraday]] and [[James Clerk Maxwell]].<ref>"Einstein's Heroes: Imagining the World through the Language of Mathematics", by Robyn Arianrhod UQP, reviewed by Jane Gleeson-White, 10 November 2003, The Sydney Morning Herald</ref> Newton remains influential to today's scientists, as demonstrated by a 2005 survey of members of Britain's [[Royal Society]] (formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Einstein. Royal Society scientists deemed Newton to have made the greater overall contribution.<ref name="royalsoc.ac.uk"/> In 1999, an opinion poll of 100 of today's leading physicists voted Einstein the "greatest physicist ever;" with Newton the runner-up, while a parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton.<ref>{{cite news|url=http://news.bbc.co.uk/2/hi/science/nature/541840.stm |title=Opinion poll. Einstein voted "greatest physicist ever" by leading physicists; Newton runner-up|publisher=BBC News |date=29 November 1999 |accessdate=17 January 2012}}</ref>

==== Commemorations ====
[[File:Isaac Newton statue.jpg|thumb|upright|Newton statue on display at the [[Oxford University Museum of Natural History]]]]

Newton's monument (1731) can be seen in [[Westminster Abbey]], at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor [[Michael Rysbrack]] (1694–1770) in white and grey marble with design by the architect [[William Kent]]. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts [[putti]] using instruments such as a telescope and prism.<ref name="wmabbey"/> The Latin inscription on the base translates as:
<blockquote>Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20 March 1726/7. — Translation from G.L. Smyth, ''The Monuments and Genii of St. Paul's Cathedral, and of Westminster Abbey'' (1826), ii, 703–4.<ref name="wmabbey"/>
</blockquote>

From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 [[banknotes of the pound sterling|banknotes]] issued by the [[Bank of England]] (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of the notes holding a book and accompanied by a telescope, a prism and a map of the [[Solar System]].<ref name="bankofengland"/>

[[File:Newton by Eduardo Paolozzi 2003-03-10.jpg|thumb|left|[[Eduardo Paolozzi]]'s ''Newton, after William Blake'' (1995), outside the British Library]]
A statue of Isaac Newton, looking at an apple at his feet, can be seen at the [[Oxford University Museum of Natural History]]. A large bronze statue, ''Newton, after William Blake'', by [[Eduardo Paolozzi]], dated 1995 and inspired by [[William Blake|Blake]]'s [[Newton (Blake)|etching]], dominates the piazza of the [[British Library]] in London.

=== In popular culture ===
{{Main|Isaac Newton in popular culture}}

== Personal life ==
Newton never married. Although it is impossible to verify, it is commonly believed that he died a [[Virginity|virgin]], as has been commented on by such figures as mathematician [[Charles Hutton]],<ref>{{cite book |title=A Philosophical and Mathematical Dictionary Containing... Memoirs of the Lives and Writings of the Most Eminent Authors, Volume 2 |last=Hutton |first=Charles |authorlink=Charles Hutton |year=1815 |page=100 |url=http://books.google.ca/books?id=_xk2AAAAQAAJ&pg=PA100&lpg=PA100&dq=Charles+Hutton+Isaac+Newton+constitutional+indifference&source=bl&ots=gxI1T-5UzL&sig=NJHnmCqkPwNalnOSrUXZZgkfODs&hl=en#v=onepage&q=Charles%20Hutton%20Isaac%20Newton%20constitutional%20indifference&f=false |accessdate=11 September 2012}}</ref> economist [[John Maynard Keynes]],<ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/Extras/Keynes_Newton.html |title=Newton: the Man |author=John Maynard Keynes |publisher=[[University of St Andrews]] School of Mathematics and Statistics |accessdate=11 September 2012}}</ref> and physicist [[Carl Sagan]].<ref>{{cite book |title=Cosmos |last=Carl |first=Sagan |authorlink=Carl Sagan |year=1980 |publisher=Random House |location=New York |isbn=0394502949 |page= |url=http://books.google.ca/books?id=_-XhL6_xsVkC&pg=PA55&lpg=PA55&dq=Isaac+Newton+virgin&source=bl&ots=pfxDt6lG8I&sig=u4GtOW8G0jCFdrppKL2o0j9ZAKU&hl=en&sa=X&ei=jrJJULeTIYnDigLs14Fo&ved=0CEMQ6AEwAzge#v=onepage&q=Isaac%20Newton%20virgin&f=false |accessdate=11 September 2012}}</ref>

French writer and philosopher [[Voltaire]], who was in London at the time of Newton's funeral, claimed to have verified the fact, writing that "I have had that confirmed by the doctor and the surgeon who were with him when he died".<ref>''Letters on England'', 14, pp. 68-70, as referenced in the footnote for the quote in p. 6 of James Gleick's biography, ''Isaac Newton''</ref> In 1733, Voltaire publicly stated that Newton "had neither passion nor weakness; he never went near any woman".<ref>{{cite book |title=The Newton Handbook |last=Gjertsen |first=Derek |year=1986 |publisher=Taylor & Francis |isbn=0710202792 |page=105 |url=http://books.google.ca/books?id=cqIOAAAAQAAJ&pg=PA105&lpg=PA105&dq=Isaac+Newton+virgin&source=bl&ots=Sf2QL1yV2J&sig=0m7VW3Ca0_jKFl-k-P8FNAATuaY&hl=en#v=onepage&q=Isaac%20Newton%20virgin&f=false |accessdate=11 September 2012}}</ref><ref>{{cite book |title=Newton: The Making of Genius |last=Fara |first=Patricia |authorlink=Patricia Fara |year=2011 |publisher=Pan Macmillan |isbn=1447204530 |page=}}</ref>

Newton did have a close friendship with the Swiss mathematician [[Nicolas Fatio de Duillier]], whom he met in London around 1690.<ref>{{cite web |url=http://web.clas.ufl.edu/users/ufhatch/pages/13-NDFE/newton/05-newton-timeline-m.htm |title=Newton Timeline |author=Professor Robert A. Hatch, University of Florida |accessdate=13 August 2012}}</ref> Their friendship came to an unexplained end in 1693. Some of their correspondence has survived.<ref>{{cite web |url=http://janus.lib.cam.ac.uk/db/node.xsp?id=CV%2FPers%2FDuillier%2C%20Nicholas%20Fatio%20de%20%281664-1753%29%20mathematician%20and%20natural%20philosopher |title=Duillier, Nicholas Fatio de (1664-1753) mathematician and natural philosopher |publisher=Janus database |accessdate=22 March 2013}}</ref><ref>{{cite web |url=http://www.oac.cdlib.org/search?style=oac4;Institution=UCLA::Clark%20%28William%20Andrews%29%20Memorial%20Library;idT=4859632 |title=Collection Guide: Fatio de Duillier, Nicolas [Letters to Isaac Newton] |publisher=Online Archive of California |accessdate=22 March 2013}}</ref>

Newton was one of many people who lost heavily when the [[South Sea Company]] collapsed. Their most significant trade was slaves and according to his niece, he lost around £20,000.

== Religious views ==
{{Main|Isaac Newton's religious views}}
[[File:Isaac Newton grave in Westminster Abbey.jpg|thumb|upright|Newton's tomb in [[Westminster Abbey]]]]Although born into an [[Anglicanism|Anglican]] family, by his thirties Newton held a Christian faith that, had it been made public, would not have been considered orthodox by mainstream [[Christianity]];<ref name="Newton - 1">[[Richard S. Westfall]] - [[Indiana University]] {{cite book | url =http://galileo.rice.edu/Catalog/NewFiles/newton.html | title = The Galileo Project | publisher = ([[Rice University]]) | accessdate = 2008-07-05}}</ref> in recent times he has been described as a [[heresy|heretic]].<ref name="heretic">{{cite journal |last=Snobelen |first=Stephen D. |authorlink=Stephen Snobelen | title=Isaac Newton, heretic : the strategies of a Nicodemite |journal=British Journal for the History of Science |volume=32 | pages=381–419 | year =1999 | url =http://www.isaac-newton.org/heretic.pdf |format=PDF|doi=10.1017/S0007087499003751 |issue=4 }}</ref>

In Newton's eyes, worshipping Christ as God was [[idolatry]], to him the fundamental sin.<ref name="westfall2"/> Historian [[Stephen Snobelen|Stephen D. Snobelen]] says of Newton, "Isaac Newton was a [[heresy|heretic]]. But ... he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical. He hid his faith so well that scholars are still unravelling his personal beliefs."<ref name="heretic"/> Snobelen concludes that Newton was at least a [[Socinian]] sympathiser (he owned and had thoroughly read at least eight Socinian books), possibly an [[Arianism|Arian]] and almost certainly an [[anti-trinitarian]].<ref name="heretic"/> During and after the [[English Civil War]] (1642–1651), a period notable for its religious intolerance, there are few public expressions of Newton's radical views, most notably his refusal to receive holy orders and his refusal, on his death bed, to receive the [[Anointing of the Sick]] sacrament when it was offered to him.{{citation needed|date=January 2014}}

In a minority view, T.C. Pfizenmaier argues that Newton held the [[Eastern Orthodox]] view on the Trinity.<ref name="Was Isaac Newton an Arian?"/> However, this type of view 'has lost support of late with the availability of Newton's theological papers',<ref>{{cite journal|last=Snobelen |first=Stephen D. |title=Isaac Newton, heretic: the strategies of a Nicodemite |journal=British Journal for the History of Science |volume=32|issue=4 |pages=381–419 [383] |year=1999|url=http://www.isaac-newton.org/heretic.pdf |format=PDF|doi=10.1017/S0007087499003751 }}</ref> and now most scholars identify Newton as an [[Nontrinitarianism|Antitrinitarian]] [[monotheist]].<ref name="heretic"/><ref name="dulles"/>

Although the laws of motion and universal gravitation became Newton's best-known discoveries, he warned against using them to view the Universe as a mere machine, as if akin to a great clock. He said, "Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done."<ref name="tiner"/>

Along with his scientific fame, Newton's studies of the Bible and of the early [[Church Fathers]] were also noteworthy. Newton wrote works on [[textual criticism]], most notably ''[[An Historical Account of Two Notable Corruptions of Scripture]]''. He placed the crucifixion of [[Jesus Christ]] at 3 April, AD 33, which agrees with one traditionally accepted date.<ref>John P. Meier, ''[[John P. Meier#A Marginal Jew: Rethinking the Historical Jesus|A Marginal Jew]]'', v. 1, pp. 382–402 after narrowing the years to 30 or 33, provisionally judges 30 most likely.</ref> He also tried unsuccessfully to find [[Bible code|hidden messages within the Bible]].

He believed in a rationally [[immanent]] world, but he rejected the [[hylozoism]] implicit in [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Baruch Spinoza]]. The ordered and dynamically informed Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed that in writing the ''Principia'' "I had an eye upon such Principles as might work with considering men for the belief of a Deity".<ref>Newton to [[Richard Bentley]] 10 December 1692, in Turnbull et al. (1959–77), vol 3, p. 233.</ref> He saw evidence of design in the system of the world: "Such a wonderful uniformity in the planetary system must be allowed the effect of choice". But Newton insisted that divine intervention would eventually be required to reform the system, due to the slow growth of instabilities.<ref>Opticks, 2nd Ed 1706. Query 31.</ref> For this, Leibniz lampooned him: "God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion."<ref>H. G. Alexander (ed) ''The Leibniz-Clarke correspondence'', [[Manchester University Press]], 1998, p. 11.</ref> Newton's position was vigorously defended by his follower [[Samuel Clarke]] in a [[Leibniz-Clarke correspondence|famous correspondence]]. A century later, [[Pierre-Simon Laplace]]'s work "[[Celestial Mechanics]]" had a natural explanation for why the planet orbits don't require periodic divine intervention.<ref>{{cite journal |author=Neil Degrasse Tyson |work=Natural History Magazine |date=November 2005 |title=The Perimeter of Ignorance |url=http://www.haydenplanetarium.org/tyson/read/2005/11/01/the-perimeter-of-ignorance}}</ref>

=== Effect on religious thought ===
Newton and [[Robert Boyle]]'s approach to the [[mechanical philosophy]] was promoted by [[rationalist]] pamphleteers as a viable alternative to the [[pantheism|pantheists]] and [[enthusiasm|enthusiasts]], and was accepted hesitantly by orthodox preachers as well as dissident preachers like the [[latitudinarian]]s.<ref name="The Newtonians and the English Revolution: 1689–1720"/> The clarity and simplicity of science was seen as a way to combat the emotional and [[metaphysics|metaphysical]] superlatives of both [[superstition|superstitious]] enthusiasm and the threat of [[atheism]],<ref name="Science and Religion in Seventeenth-Century England"/> and at the same time, the second wave of English [[deism|deists]] used Newton's discoveries to demonstrate the possibility of a "Natural Religion".
[[File:Newton-WilliamBlake.jpg|thumb|left|''[[Newton (Blake)|Newton]]'', by [[William Blake]]; here, Newton is depicted critically as a "divine geometer". This copy of the work is currently held by the [[Tate Collection]].<ref>{{cite web| url = http://www.blakearchive.org/exist/blake/archive/copyinfo.xq?copyid=but306.1| title = Newton, object 1 (Butlin 306) "Newton"| date = September 25, 2013|publisher = [[William Blake Archive]]}}</ref> ]]

The attacks made against pre-[[Age of Enlightenment|Enlightenment]] "magical thinking", and the [[Christian mysticism|mystical elements of Christianity]], were given their foundation with Boyle's mechanical conception of the Universe. Newton gave Boyle's ideas their completion through [[mathematical proof]]s and, perhaps more importantly, was very successful in popularising them.<ref name="Enlightenment and Religion: Rational Dissent in eighteenth-century Britain"/> Newton refashioned the world governed by an interventionist God into a world crafted by a God that designs along rational and universal principles.<ref name="The Faith of Reason: The Idea of Progress in the French Enlightenment"/> These principles were available for all people to discover, allowed people to pursue their own aims fruitfully in this life, not [[afterlife|the next]], and to perfect themselves with their own rational powers.<ref name="A Discourse on Disenchantment: Reflections on Politics and Technology"/>

Newton saw God as the master creator whose existence could not be denied in the face of the grandeur of all creation.<ref>Principia, Book III; cited in; Newton's Philosophy of Nature: Selections from his writings, p. 42, ed. H.S. Thayer, Hafner Library of Classics, NY, 1953.</ref><ref>A Short Scheme of the True Religion, manuscript quoted in Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton by Sir David Brewster, Edinburgh, 1850; cited in; ibid, p. 65.</ref><ref>Webb, R.K. ed. Knud Haakonssen. "The emergence of Rational Dissent." Enlightenment and Religion: Rational Dissent in eighteenth-century Britain. Cambridge University Press, Cambridge: 1996. p19.</ref> His spokesman, Clarke, rejected Leibniz' [[theodicy]] which cleared God from the responsibility for ''[[Problem of evil|l'origine du mal]]'' by making God removed from participation in his creation, since as Clarke pointed out, such a deity would be a king in name only, and but one step away from atheism.<ref>H. G. Alexander (ed) ''The Leibniz-Clarke correspondence'', Manchester University Press, 1998, p. 14.</ref> But the unforeseen [[Christian theology|theological]] consequence of the success of Newton's system over the next century was to reinforce the [[deism|deist]] position advocated by Leibniz.<ref>Westfall, 1958 p201.</ref>
The understanding of the world was now brought down to the level of simple human reason, and humans, as Odo Marquard argued, became responsible for the correction and elimination of evil.<ref>Marquard, Odo. "Burdened and Disemburdened Man and the Flight into Unindictability," in Farewell to Matters of Principle. Robert M. Wallace trans. London: Oxford UP, 1989.</ref>

=== End of the world ===
{{See also|Isaac Newton's occult studies|eschatology}}
In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible, he estimated that the world would end no earlier than 2060. In predicting this he said, "This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."<ref name="Papers Show Isaac Newton's Religious Side, Predict Date of Apocalypse"/>

=== Alchemy ===

Speculative fiction author [[Fritz Leiber]] said of Newton, "Everyone knows Newton as the great scientist. Few remember that he spent half his life muddling with alchemy, looking for the philosopher's stone. That was the pebble by the seashore he really wanted to find." <ref>Heinlein, ed., p. 208.</ref>

== Enlightenment philosophers ==
[[Age of Enlightenment|Enlightenment]] philosophers chose a short history of scientific predecessors – Galileo, Boyle, and Newton principally – as the guides and guarantors of their applications of the singular concept of [[Nature]] and [[Natural law|Natural Law]] to every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it could be discarded.<ref>Cassels, Alan. Ideology and International Relations in the Modern World. p2.</ref>

It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one of the seeds for Enlightenment ideology.<ref>"Although it was just one of the many factors in the Enlightment, the success of Newtonian physics in providing a mathematical description of an ordered world clearly played a big part in the flowering of this movement in the eighteenth century" John Gribbin (2002) ''Science: A History 1543–2001'', p 241</ref> Locke and [[Voltaire]] applied concepts of Natural Law to political systems advocating intrinsic rights; the [[physiocrat]]s and [[Adam Smith]] applied Natural conceptions of [[psychology]] and self-interest to economic systems; and [[sociology|sociologists]] criticised the current [[social order]] for trying to fit history into Natural models of [[progress (history)|progress]]. [[Monboddo]] and [[Samuel Clarke]] resisted elements of Newton's work, but eventually rationalised it to conform with their strong religious views of nature.

== Royal Mint ==
As Warden, and afterwards Master, of the [[Royal Mint]], Newton estimated that 20 percent of the coins taken in during The [[Great Recoinage of 1696]] were [[counterfeit]]. Counterfeiting was [[High treason in the United Kingdom|high treason]], punishable by the felon's being [[hanged, drawn and quartered]]. Despite this, convicting the most flagrant criminals could be extremely difficult. However, Newton proved to be equal to the task.<ref>White 1997, p. 259</ref> Disguised as a habitué of bars and taverns, he gathered much of that evidence himself.<ref>White 1997, p. 267</ref> For all the barriers placed to prosecution, and separating the branches of government, [[English law]] still had ancient and formidable customs of authority. Newton had himself made a [[justice of the peace]] in all the home counties—there is a draft of a letter regarding this matter stuck into Newton's personal first edition of his ''Philosophiæ Naturalis Principia Mathematica'' which he must have been amending at the time.<ref>{{cite web|last=Newton|first=Isaac|title=Philosophiæ Naturalis Principia Mathematica|url=http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/|publisher=Cambridge University Digital Library|accessdate=10 January 2012|pages=265–266}}</ref> Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June 1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.<ref>Westfall 2007, p.73</ref>

One of Newton's cases as the King's attorney was against [[William Chaloner]].<ref name="White 1997, p 269"/> Chaloner's schemes included setting up phony conspiracies of Catholics and then turning in the hapless conspirators whom he had entrapped. Chaloner made himself rich enough to posture as a gentleman. Petitioning Parliament, Chaloner accused the Mint of providing tools to counterfeiters (a charge also made by others). He proposed that he be allowed to inspect the Mint's processes in order to improve them. He petitioned Parliament to adopt his plans for a coinage that could not be counterfeited, while at the same time striking false coins.<ref>Westfall 1994, p 229</ref> Newton put Chaloner on trial for counterfeiting and had him sent to Newgate Prison in September 1697. But Chaloner had friends in high places, who helped him secure an acquittal and his release.<ref name="White 1997, p 269"/> Newton put him on trial a second time with conclusive evidence. Chaloner was convicted of high treason and hanged, drawn and quartered on 23 March 1699 at [[Tyburn, London|Tyburn gallows]].<ref>{{harvnb|Westfall| 1980|p=571–5}}</ref>

As a result of a report written by Newton on 21 September 1717 to the Lords Commissioners of His Majesty's Treasury the bimetallic relationship between gold coins and silver coins was changed by Royal proclamation on 22 December 1717, forbidding the exchange of gold guineas for more than 21 silver shillings.<ref>[http://www.pierre-marteau.com/editions/1701-25-mint-reports/report-1717-09-25.html On the Value of Gold and Silver in European Currencies and the Consequences on the World-wide Gold- and Silver-Trade], Sir Isaac Newton, 21 September 1717.</ref><ref>By The King, A Proclamation Declaring the Rates at which Gold shall be current in Payments [http://www.archive.org/details/numismaticser1v05royauoft reproduced in the numismatic chronicle and journal of the Royal Numismatic Society, Vol V., April 1842 - January 1843]</ref> This inadvertently resulted in a silver shortage as silver coins were used to pay for imports, while exports were paid for in gold, effectively moving Britain from the silver standard to its first gold standard. It is a matter of debate as whether he intended to do this or not.<ref>http://www.jstor.org/discover/10.2307/3020836?uid=3738032&uid=2129&uid=2&uid=70&uid=4&sid=21101668938967</ref> It has been argued that Newton conceived of his work at the Mint as a continuation of his alchemical work.<ref>http://gtresearchnews.gatech.edu/newsrelease/newton.htm</ref>

== Laws of motion ==
{{Classical mechanics|cTopic=Scientists}}
{{Main|Newton's laws of motion}}

In the ''Principia'', Newton gives the famous three laws of motion, stated here in modern form.

''Newton's First Law'' (also known as the Law of [[Inertia]]) states that an object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. The meaning of this law is the existence of reference frames (called inertial frames) where objects not acted upon by forces move in uniform motion (in particular, they may be at rest).

''Newton's Second Law'' states that an applied force, <math>\mathbf{F}</math>, on an object equals the rate of change of its momentum, <math>\mathbf{p}</math>, with time. Mathematically, this is expressed as
:<math> \mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{\mathrm{d}}t} = \frac{\mathrm{d} (m\mathbf{v})}{\mathrm{\mathrm{d}}t}.</math>

Since the law applies only to systems of constant mass,<ref name=Halliday>{{cite book|last=Halliday|coauthors=Resnick|title=Physics|volume=1|pages=199|quote=It is important to note that we ''cannot'' derive a general expression for Newton's second law for variable mass systems by treating the mass in '''F''' = ''d'''''P'''/''dt'' = ''d''(''M'''''v''') as a ''variable''. [...] We ''can'' use '''F''' = ''d'''''P'''/''dt'' to analyze variable mass systems ''only'' if we apply it to an ''entire system of constant mass'' having parts among which there is an interchange of mass.|isbn=0-471-03710-9}} [Emphasis as in the original]</ref> <math>m</math> can be brought out of the derivative operator. By substitution using the definition of [[acceleration]], the equation can be written in the iconic form

:<math> \mathbf{F} = m \mathbf{a}.</math>

The first and second laws represent a break with the physics of [[Aristotle]], in which it was believed that a force was necessary in order to maintain motion. They state that a force is only needed in order to ''change'' an object's state of motion. The SI unit of force is the [[newton (unit)|newton]], named in Newton's honour.

''Newton's Third Law'' states that for every action there is an equal and opposite reaction. This means that any force exerted onto an object has a counterpart force that is exerted in the opposite direction back onto the first object. A common example is of two [[ice skater]]s pushing against each other and sliding apart in opposite directions. Another example is the [[recoil]] of a [[firearm]], in which the force propelling the [[bullet]] is exerted equally back onto the gun and is felt by the shooter. Since the objects in question do not necessarily have the same mass, the resulting acceleration of the two objects can be different (as in the case of firearm recoil).

Unlike Aristotle's, Newton's physics is meant to be universal. For example, the second law applies both to a planet and to a falling stone.

The [[Euclidean vector|vector]] nature of the second law addresses the geometrical relationship between the direction of the force and the manner in which the object's momentum changes. Before Newton, it had typically been assumed that a planet orbiting the Sun would need a forward force to keep it moving. Newton showed instead that all that was needed was an inward attraction from the Sun. Even many decades after the publication of the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', this counterintuitive idea was not universally accepted, and many scientists preferred [[Descartes]]' theory of vortices.<ref>Ball 1908, p. 337</ref>

== Apple incident ==
{{Multiple image|direction=vertical|align=right|image1=Supposedly_Isaac_Newton's_apple_tree.jpg|image2=Newton's tree, Botanic Gardens, Cambridge.JPG|image3=Newtons apple.jpg|width=150|caption3=Reputed descendants of Newton's [[apple tree]], (from top to bottom) at [[Trinity College, Cambridge]], the [[Cambridge University Botanic Garden]], and the [[Instituto Balseiro]] library garden}}
Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree.<ref>White 1997, p. 86</ref> Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity in any single moment,<ref name="Berkun2010"/> acquaintances of Newton (such as [[William Stukeley]], whose manuscript account of 1752 has been made available by the Royal Society)<ref name="Newton's apple: The real story"/> do in fact confirm the incident, though not the cartoon version that the apple actually hit Newton's head. Stukeley recorded in his ''Memoirs of Sir Isaac Newton's Life'' a conversation with Newton in Kensington on 15 April 1726:<ref name="Hamblyn (2011)"/>

{{Quotation|... We went into the garden, & drank tea under the shade of some appletrees, only he, & myself. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "why should that apple always descend perpendicularly to the ground," thought he to him self: occasion'd by the fall of an apple, as he sat in a comtemplative mood: "why should it not go sideways, or upwards? but constantly to the earths centre? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths centre, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the centre. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple."}}

John Conduitt, Newton's assistant at the Royal Mint and husband of Newton's niece, also described the event when he wrote about Newton's life:<ref name="Keynes Ms. 130.4:Conduitt's account of Newton's life at Cambridge"/>

{{Quotation|In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon said he to himself & if so, that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition.}}

In similar terms, [[Voltaire]] wrote in his ''Essay on Epic Poetry'' (1727), "Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree."

It is known from his notebooks that Newton was grappling in the late 1660s with the idea that terrestrial gravity extends, in an inverse-square proportion, to the Moon; however it took him two decades to develop the full-fledged theory.<ref>I. Bernard Cohen and George E. Smith, eds. ''The Cambridge Companion to Newton'' (2002) p. 6</ref> The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the force holding the Moon to its orbit. Newton showed that if the force decreased as the inverse square of the distance, one could indeed calculate the Moon's orbital period, and get good agreement. He guessed the same force was responsible for other orbital motions, and hence named it "universal gravitation".

Various trees are claimed to be "the" apple tree which Newton describes. The King's School, Grantham, claims that the tree was purchased by the school, uprooted and transported to the headmaster's garden some years later. The staff of the [now] [[National Trust for Places of Historic Interest or Natural Beauty|National Trust]]-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is the one described by Newton. A descendant of the original tree<ref>Alberto A. Martinez ''Science Secrets: The Truth about Darwin's Finches, Einstein's Wife, and Other Myths'', page 69 (University of Pittsburgh Press, 2011). ISBN 978-0-8229-4407-2</ref> can be seen growing outside the main gate of Trinity College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at Brogdale<ref name="Brogdale — Home of the National Fruit Collection"/> can supply grafts from their tree, which appears identical to [[Flower of Kent]], a coarse-fleshed cooking variety.<ref name="From the National Fruit Collection: Isaac Newton's Tree"/>

== Writings ==
{{See also|Writing of Principia Mathematica}}

* ''[[De analysi per aequationes numero terminorum infinitas]]'' (1669, published 1711)
* ''[[Method of Fluxions]]'' (1671)
*''Of Natures Obvious Laws & Processes in Vegetation'' (unpublished, c. 1671–75)<ref>[http://webapp1.dlib.indiana.edu/newton/index.jsp Newton's alchemical works] transcribed and online at [[Indiana University (Bloomington)|Indiana University]]. Retrieved 11 January 2007.</ref>
* ''[[De motu corporum in gyrum]]'' (1684)
* ''[[Philosophiæ Naturalis Principia Mathematica]]'' (1687)
* ''[[Opticks]]'' (1704)
* ''[http://www.pierre-marteau.com/editions/1701-25-mint-reports.html Reports as Master of the Mint]'' (1701–25)
* ''[[Arithmetica Universalis]]'' (1707)
* ''The System of the World'', ''Optical Lectures'', ''[[The Chronology of Ancient Kingdoms|The Chronology of Ancient Kingdoms, (Amended)]]'' and ''De mundi systemate'' (published posthumously in 1728)
* ''Observations on Daniel and The Apocalypse of St. John'' (1733)
* ''[[An Historical Account of Two Notable Corruptions of Scripture]]'' (1754)

== See also ==
{{Wikipedia books|Isaac Newton}}
{|
|-
| valign=top |
*[[Ismaël Bullialdus]]
*[[De Motu (Berkeley's essay)]]
* ''[[Elements of the Philosophy of Newton]]''
*[[Newton series|Finite difference: Newton_series]]
*[[Gauss–Newton algorithm]]
*[[History of calculus]]
*[[History of the telescope]]
*[[Leibniz–Newton calculus controversy]]
*[[List of multiple discoveries#17th century|List of multiple discoveries: 17th century]]
| {{in5|11}} || valign=top |
*[[Retrospective diagnoses of autism]]
*[[List of things named after Isaac Newton]]
|}

== References ==
{{reflist|colwidth=30em|refs=
<ref name="A Discourse on Disenchantment: Reflections on Politics and Technology">{{cite book|last=Germain |first=Gilbert G. |title=A Discourse on Disenchantment: Reflections on Politics and Technology |page=28 |isbn=0-7914-1319-5}}</ref>

<ref name="Berkun2010">{{cite book|author=[[Scott Berkun]]|title=The Myths of Innovation|url=http://books.google.com/books?id=kPCgnc70MSgC&pg=PA4|accessdate=7 September 2011|date=27 August 2010|publisher=O'Reilly Media, Inc.|isbn=978-1-4493-8962-8|page=4}}</ref>

<ref name="Brogdale — Home of the National Fruit Collection">{{cite web|url=http://www.brogdale.org/ |title=Brogdale — Home of the National Fruit Collection |publisher=Brogdale.org |accessdate=20 December 2008}}</ref>

<ref name="Coat of arms of Isaac Newton">{{cite web|author=Gerard Michon |url=http://www.numericana.com/arms/index.htm#newton |title=Coat of arms of Isaac Newton |publisher=Numericana.com |accessdate=16 January 2010}}</ref>

<ref name="Enlightenment and Religion: Rational Dissent in eighteenth-century Britain">{{cite book|last=Haakonssen |first=Knud |editor=Martin Fitzpatrick ed. |chapter=The Enlightenment, politics and providence: some Scottish and English comparisons |title=Enlightenment and Religion: Rational Dissent in eighteenth-century Britain |publisher=Cambridge University Press |location=Cambridge |page=64 |isbn=0-521-56060-8}}</ref>

<ref name="From the National Fruit Collection: Isaac Newton's Tree">{{cite web|url=http://www.brogdale.org.uk/image1.php?varietyid=1089 |title=From the National Fruit Collection: Isaac Newton's Tree|accessdate=10 January 2009}}</ref>

<ref name="Hamblyn (2011)">{{cite book |last=Hamblyn |first=Richard |year=2011 |title=The Art of Science |publisher=[[Pan Macmillan]] |isbn=978-1-4472-0415-2 |contribution=[http://books.google.co.uk/books?id=1xKFSqsDj0MC&pg=PT57 Newtonian Apples: William Stukeley]}}</ref>

<ref name="Keynes Ms. 130.4:Conduitt's account of Newton's life at Cambridge">{{cite web|url=http://www.newtonproject.sussex.ac.uk/view/texts/normalized/THEM00167 |last=Conduitt|first=John |title=Keynes Ms. 130.4:Conduitt's account of Newton's life at Cambridge|work=Newtonproject|publisher=Imperial College London|accessdate=30 August 2006}}</ref>

<ref name="More">{{cite book|last=Westfall |first=Richard S. |origyear=1980 |year=1983 |title=Never at Rest: A Biography of Isaac Newton |publisher=Cambridge University Press |location=Cambridge|isbn=978-0-521-27435-7 |pages=530–1}}</ref>

<ref name="Newton's Alchemy and His Theory of Matter">{{cite journal |last=Dobbs |first=J.T. |date=December 1982 |title=Newton's Alchemy and His Theory of Matter |journal=Isis |volume=73 |issue=4 |page=523 |doi=10.1086/353114 }} quoting ''Opticks''</ref>

<ref name="Newton's apple: The real story">{{cite journal| publisher=New Scientist | url=http://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.php | title=Newton's apple: The real story | date=18 January 2010 | accessdate=10 May 2010 | postscript=}}</ref>

<ref name="Newton, Isaac (1642–1727)">{{cite web|url=http://scienceworld.wolfram.com/biography/Newton.html |title=Newton, Isaac (1642–1727) |work=Eric Weisstein's World of Biography |accessdate=30 August 2006}}</ref>

<ref name="OPN1">{{cite journal |last=[[F. J. Duarte|Duarte]] |year=2000 |title=Newton, prisms, and the 'opticks' of tunable lasers |journal=Optics and Photonics News |volume=11 |issue=5 |pages=24–25 |doi=10.1364/OPN.11.5.000024|bibcode = 2000OptPN..11...24D |url=http://www.opticsjournal.com/F.J.DuarteOPN%282000%29.pdf |first1=F. J.}}</ref>

<ref name="OSNS">During Newton's lifetime, two calendars were in use in Europe: the [[Julian Calendar|Julian]] ("[[Old Style and New Style dates|Old Style]]") calendar in [[Protestantism|protestant]] and [[Eastern Orthodox Church|Orthodox]] regions, including Britain; and the [[Gregorian Calendar|Gregorian]] ("[[Old Style and New Style dates|New Style]]") calendar in Roman Catholic Europe. At Newton's birth, Gregorian dates were ten days ahead of Julian dates: thus his birth is recorded as taking place on 25 December 1642 Old Style, but can be converted to a New Style (modern) date of 4 January 1643. By the time of his death, the difference between the calendars had increased to eleven days: moreover, he died in the period after the start of the New Style year on 1 January, but before that of the Old Style new year on 25 March. His death occurred on 20 March 1726 according to the Old Style calendar, but the year is usually adjusted to 1727. A full conversion to New Style gives the date 31 March 1727.</ref>

<ref name="Papers Show Isaac Newton's Religious Side, Predict Date of Apocalypse">{{cite news|title=Papers Show Isaac Newton's Religious Side, Predict Date of Apocalypse |agency=Associated Press |date=19 June 2007 |url=http://www.christianpost.com/article/20070619/28049_Papers_Show_Isaac_Newton%27s_Religious_Side%2C_Predict_Date_of_Apocalypse.htm |archiveurl=http://web.archive.org/web/20070813033620/http://www.christianpost.com/article/20070619/28049_Papers_Show_Isaac_Newton%27s_Religious_Side,_Predict_Date_of_Apocalypse.htm |archivedate=13 August 2007|accessdate=1 August 2007}}</ref>

<ref name="Science and Religion in Seventeenth-Century England">{{cite book|last=Westfall |first=Richard S. |year=1958 |title=Science and Religion in Seventeenth-Century England |publisher=Yale University Press |location=New Haven |page=200 |isbn=0-208-00843-8}}</ref>



<ref name="The Collected Writings of John Maynard Keynes Volume X">{{cite book|last=Keynes |first=John Maynard |year=1972 |chapter=Newton, The Man |title=The Collected Writings of John Maynard Keynes Volume X |publisher=MacMillan St. Martin's Press |pages=363–4}}</ref>

<ref name="The Early Period (1608–1672)">{{cite web|url=http://etoile.berkeley.edu/~jrg/TelescopeHistory/Early_Period.html|title=The Early Period (1608–1672)|accessdate=3 February 2009|publisher=James R. Graham's Home Page}}</ref>

<ref name="The Faith of Reason: The Idea of Progress in the French Enlightenment">{{cite book|last=Frankel |first=Charles |year=1948 |title=The Faith of Reason: The Idea of Progress in the French Enlightenment |publisher=King's Crown Press |location=New York |page=1}}</ref>

<ref name="The Newton Handbook">{{cite book|author=Gjersten, Derek |title=The Newton Handbook |year=1986 |location=London |publisher=Routledge & Kegan Paul}}</ref>

<ref name="The Newtonians and the English Revolution: 1689–1720">{{cite book|last=Jacob |first=Margaret C. |year=1976 |title=The Newtonians and the English Revolution: 1689–1720 |publisher=Cornell University Press |pages=37, 44 |isbn=0-85527-066-7}}</ref>

<ref name="Was Isaac Newton an Arian?">{{cite journal|last=Pfizenmaier |first=T.C. |year=1997 |title=Was Isaac Newton an Arian? |journal=Journal of the History of Ideas |volume=58 |issue=1 |pages=57–80}}</ref>

<ref name="White 1997, p 269">White 1997, p 269</ref>

<ref name="White 1997, p170">White 1997, p170</ref>

<ref name="bankofengland">{{cite web|url=http://www.bankofengland.co.uk/banknotes/denom_guide/nonflash/1-SeriesD-Revised.htm|title=Withdrawn banknotes reference guide|publisher=Bank of England|accessdate=27 August 2009}}</ref>

<ref name="books.google.com">{{cite book|url=http://books.google.com/?id=32IDpTdthm4C&pg=PA67&lpg=PA67&dq=newton+reflecting+telescope++1668+letter+1669&q=newton%20reflecting%20telescope%20%201668%20letter%201669 |title='''Isaac Newton: adventurer in thought''', by Alfred Rupert Hall, page 67 |publisher=Google Books |accessdate=16 January 2010|isbn=978-0-521-56669-8|author1=Hall, Alfred Rupert|year=1996}}</ref>

<ref name="dulles">Avery Cardinal Dulles. [http://www.firstthings.com/print.php?type=article&year=2008&month=08&title_link=the-deist-minimum--28 The Deist Minimum]. January 2005.</ref>

<ref name="hooke1679nov24">See 'Correspondence of Isaac Newton, vol.2, 1676–1687' ed. H W Turnbull, Cambridge University Press 1960; at page 297, document No. 235, letter from Hooke to Newton dated 24 November 1679.</ref>

<ref name="royalsoc.ac.uk">{{cite web|title=Newton beats Einstein in polls of Royal Society scientists and the public |work=The Royal Society |url=http://royalsociety.org/News.aspx?id=1324&terms=Newton+beats+Einstein+in+polls+of+scientists+and+the+public}}</ref>

<ref name="tiner">{{cite book|last=Tiner |first=J.H. |year=1975 |title=Isaac Newton: Inventor, Scientist and Teacher |publisher=Mott Media |location=Milford, Michigan, U.S. |isbn=0-915134-95-0}}</ref>

<ref name="westfall2">{{Page needed|date=September 2010}}{{cite book| last =Westfall | first =Richard S. | title =The Life of Isaac Newton | publisher=Cambridge University Press | year =1994 | location = Cambridge | isbn =0-521-47737-9}}</ref>

<ref name="wmabbey">{{cite web|url=http://www.westminster-abbey.org/our-history/people/sir-isaac-newton|title=Famous People & the Abbey: Sir Isaac Newton|publisher=Westminster Abbey|accessdate=13 November 2009}}</ref>

<ref name="Whittaker">[[E. T. Whittaker|Whittaker, E. T.]], ''A History of the Theories of Aether and Electricity''. Dublin University Press, 1910.</ref>
}}

== Bibliography ==
{{refbegin|30em}}
* {{cite book|last=Ball|first=W.W. Rouse|title=A Short Account of the History of Mathematics|location=New York|publisher=Dover|year=1908|isbn=0-486-20630-0}}
* {{cite book|last=Christianson|first=Gale|title=In the Presence of the Creator: Isaac Newton & His Times|location=New York|publisher=Free Press|year=1984|isbn=0-02-905190-8}} This well documented work provides, in particular, valuable information regarding Newton's knowledge of [[Patristics]]
* {{cite journal|last=Craig |first=John |title=Isaac Newton – Crime Investigator |journal=Nature |year=1958 |volume=182|issue=4629 |pages=149–152 |doi=10.1038/182149a0 |bibcode = 1958Natur.182..149C }}
* {{cite journal|last=Craig |first=John |title=Isaac Newton and the Counterfeiters |journal=Notes and Records of the Royal Society of London |volume=18|issue=2 |year=1963 |pages=136–145 |doi=10.1098/rsnr.1963.0017 }}
* {{cite book|last=Levenson|first=Thomas|title= Newton and the Counterfeiter: The Unknown Detective Career of the World's Greatest Scientist|publisher=Mariner Books|year=2010|isbn=978-0-547-33604-6 }}
* {{cite book|last=Stewart|first=James|title= Calculus: Concepts and Contexts|publisher=Cengage Learning|year=2009|isbn=978-0-495-55742-5}}
* {{cite book|authorlink=Richard S. Westfall |last=Westfall |first=Richard S. |title=Never at Rest |publisher=Cambridge University Press |year=1980 |isbn=0-521-27435-4|ref=harv }}
*{{cite book|last=Westfall|first=Richard S.|title=Isaac Newton|publisher=Cambridge University Press|year=2007|isbn=978-0-19-921355-9}}
*{{cite book|last=Westfall|first=Richard S.|title=The Life of Isaac Newton|publisher=Cambridge University Press|year=1994|isbn=0-521-47737-9}}
*{{cite book|authorlink=Michael White (author) |title=Isaac Newton: The Last Sorcerer |first=Michael |last=White |publisher=Fourth Estate Limited |year=1997 |isbn=1-85702-416-8}}
{{refend}}

== Further reading ==
{{refbegin|colwidth=30em}}
* {{cite book|last=Andrade |first=E. N. De C. |title=Isaac Newton |publisher=Chanticleer Press |location=New York |year=1950 |isbn=0-8414-3014-4}}
*Bardi, Jason Socrates. ''The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time.'' 2006. 277 pp. [http://www.amazon.com/dp/1560259922 excerpt and text search]
* {{cite book| author=Bechler, Zev | title = Newton's Physics and the Conceptual Structure of the Scientific Revolution | year = 1991 | publisher=Springer | isbn = 0-7923-1054-3}}.
* Berlinski, David. ''Newton's Gift: How Sir Isaac Newton Unlocked the System of the World.'' (2000). 256 pages. [http://www.amazon.com/dp/0743217764 excerpt and text search] ISBN 0-684-84392-7
* Buchwald, Jed Z. and Cohen, I. Bernard, eds. ''Isaac Newton's Natural Philosophy.'' [[MIT Press]], 2001. 354 pages. [http://www.amazon.com/dp/0262524252 excerpt and text search]
*{{cite journal|author=Casini, P|title=Newton's Principia and the Philosophers of the Enlightenment|journal=Notes and Records of the Royal Society of London|year=1988|volume=42|issue=1|pages=35–52|issn=0035–9149|doi=10.1098/rsnr.1988.0006|jstor=531368}}
* {{cite book| author=Christianson, Gale E | title = Isaac Newton and the Scientific Revolution | publisher=[[Oxford University Press]] | year = 1996 | isbn = 0-19-530070-X}} See [http://www.amazon.com/dp/019530070X this site] for excerpt and text search.
* {{cite book|last=Christianson |first=Gale |title=In the Presence of the Creator: Isaac Newton & His Times |location=New York |publisher=Free Press |year=1984 |isbn=0-02-905190-8 }}
* Cohen, I. Bernard and Smith, George E., ed. ''The Cambridge Companion to Newton.'' (2002). 500 pp. focuses on philosophical issues only; [http://www.amazon.com/dp/0521656966 excerpt and text search]; [http://www.questia.com/read/105054986 complete edition online]
* {{cite book|author=Cohen, I. B |title=The Newtonian Revolution |year=1980 |location= Cambridge |publisher=Cambridge University Press |isbn=0-521-22964-2}}
* {{cite book|author=Craig, John |title=Newton at the Mint |year=1946 |publisher=Cambridge University Press |location=Cambridge, England }}
* {{cite book|author=Dampier, William C |coauthors=Dampier, M. |title=Readings in the Literature of Science |publisher=Harper & Row |location=New York |year=1959 |isbn=0-486-42805-2 }}
* {{cite book|author=[[Richard de Villamil|de Villamil, Richard]] |title=Newton, the Man |publisher=G.D. Knox |location=London |year=1931}} – Preface by Albert Einstein. Reprinted by Johnson Reprint Corporation, New York (1972).
* {{cite book|author=Dobbs, B. J. T |title=The Foundations of Newton's Alchemy or "The Hunting of the Greene Lyon" |year=1975 |publisher=Cambridge University Press | place = Cambridge }}
* {{cite book|author=Gjertsen, Derek |title=The Newton Handbook |publisher=Routledge & Kegan Paul |location=London |year=1986 |isbn=0-7102-0279-2 }}
*{{cite book|author=Gleick, James|title=Isaac Newton|publisher=Alfred A. Knopf|year=2003 |isbn=0-375-42233-1}}
*{{cite journal|author=Halley, E |title=Review of Newton's Principia |year=1687 |journal=Philosophical Transactions |volume=186 |pages=291–297}}
*[[Stephen Hawking|Hawking, Stephen]], ed. ''On the Shoulders of Giants''. ISBN 0-7624-1348-4 Places selections from Newton's ''Principia'' in the context of selected writings by Copernicus, Kepler, Galileo and Einstein
*{{cite book|last=Herivel, J. W.|title=The Background to Newton's Principia. A Study of Newton's Dynamical Researches in the Years 1664–84|publisher=Clarendon Press|location=Oxford|year=1965 }}
* {{cite book|author=[[John Maynard Keynes|Keynes, John Maynard]] |title=Essays in Biography |publisher=W. W. Norton & Co |year=1963 |isbn=0-393-00189-X}} Keynes took a close interest in Newton and owned many of Newton's private papers.
* {{cite book|author=Koyré, A |title=Newtonian Studies |location=Chicago |publisher=University of Chicago Press |year=1965 }}
* Newton, Isaac. ''Papers and Letters in Natural Philosophy'', edited by [[I. Bernard Cohen]]. [[Harvard University Press]], 1958,1978. ISBN 0-674-46853-8.
* Newton, Isaac (1642–1727). ''The Principia'': a new Translation, Guide by I. Bernard Cohen ISBN 0-520-08817-4 University of California (1999)
* {{cite book|author=Pemberton, H |title=A View of Sir Isaac Newton's Philosophy |publisher=S. Palmer |location=London |year=1728 }}
* {{cite book|last=Shamos, Morris H. |title=Great Experiments in Physics |location=New York |publisher=Henry Holt and Company, Inc. |year=1959 |isbn=0-486-25346-5 }}
* Shapley, Harlow, S. Rapport, and H. Wright. ''A Treasury of Science''; "Newtonia" pp. 147–9; "Discoveries" pp. 150–4. Harper & Bros., New York, (1946).
* {{cite book| author=Simmons, J | title = The Giant Book of Scientists – The 100 Greatest Minds of all Time | location = Sydney | publisher=The Book Company | year = 1996}}
* {{cite book|last=Stukeley |first=W. |title=Memoirs of Sir Isaac Newton's Life |publisher=Taylor and Francis | place = London |year=1936}} (edited by A. H. White; originally published in 1752)
* {{cite book|author=Westfall, R. S |title=Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century |location=London |publisher=Macdonald |year=1971 |isbn=0-444-19611-0 }}
{{refend}}

'''Religion'''
{{refbegin|colwidth=30em}}
* Dobbs, Betty Jo Tetter. ''The Janus Faces of Genius: The Role of Alchemy in Newton's Thought.'' (1991), links the alchemy to Arianism
* Force, James E., and Richard H. Popkin, eds. ''Newton and Religion: Context, Nature, and Influence.'' (1999), 342pp . Pp. xvii + 325. 13 papers by scholars using newly opened manuscripts
* Ramati, Ayval. "The Hidden Truth of Creation: Newton's Method of Fluxions" ''British Journal for the History of Science'' 34: 417–438. [http://www.jstor.org/stable/4028372 in JSTOR], argues that his calculus had a theological basis
*Snobelen, Stephen "'God of Gods, and Lord of Lords': The Theology of Isaac Newton's General Scholium to the Principia," ''Osiris,'' 2nd Series, Vol. 16, (2001), pp. 169–208 [http://www.jstor.org/stable/301985 in JSTOR]
* {{cite journal | last1 = Snobelen | first1 = Stephen D. | title = Isaac Newton, Heretic: The Strategies of a Nicodemite | jstor = 4027945 | journal=British Journal for the History of Science | volume = 32 | pages = 381–419 | doi=10.1017/S0007087499003751 | year = 1999 | issue = 4}}
* {{cite journal | last1 = Pfizenmaier | first1 = Thomas C. |date=January 1997 | title = Was Isaac Newton an Arian? | jstor = 3653988 | journal=Journal of the History of Ideas | volume = 58 | issue = 1| pages = 57–80 }}
* Wiles, Maurice. ''Archetypal Heresy. Arianism through the Centuries.'' (1996) 214 pages, with chapter 4 on 18th century England; pp. 77–93 on Newton, [http://books.google.com/books?id=DGksMzk37hMC&printsec=frontcover&dq=%22Arianism+through+the+Centuries%22 excerpt and text search].
{{refend}}

'''Primary sources'''
{{refbegin|colwidth=30em}}
* Newton, Isaac. ''The Principia: Mathematical Principles of Natural Philosophy.'' [[University of California Press]], (1999). 974 pp.
** Brackenridge, J. Bruce. ''The Key to Newton's Dynamics: The Kepler Problem and the Principia: Containing an English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's Mathematical Principles of Natural Philosophy.'' University of California Press, 1996. 299 pp.
* Newton, Isaac. ''The Optical Papers of Isaac Newton. Vol. 1: The Optical Lectures, 1670–1672.'' Cambridge U. Press, 1984. 627 pp.
** Newton, Isaac. ''Opticks'' (4th ed. 1730) [http://books.google.com/books?id=GnAFAAAAQAAJ&dq=newton+opticks&pg=PP1&ots=Nnl345oqo_&sig=0mBTaXUI_K6w-JDEu_RvVq5TNqc&prev=http://www.google.com/search%3Fq%3Dnewton%2Bopticks%26rls%3Dcom.microsoft:en-us:IE-SearchBox%26ie%3DUTF-8%26oe%3DUTF-8%26sourceid%3Die7%26rlz%3D1I7GGLJ&sa=X&oi=print&ct=title&cad=one-book-with-thumbnail online edition]
** Newton, I. (1952). Opticks, or A Treatise of the Reflections, Refractions, Inflections & Colours of Light. New York: Dover Publications.
* Newton, I. ''Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World,'' tr. A. Motte, rev. [[Florian Cajori]]. Berkeley: University of California Press. (1934).
* {{cite book|author=[[Whiteside, D. T.|Whiteside, D. T]] |title=The Mathematical Papers of Isaac Newton |location=Cambridge |publisher=Cambridge University Press |year=1967–82 |isbn=0-521-07740-0}} – 8 volumes.
*Newton, Isaac. ''The correspondence of Isaac Newton,'' ed. H. W. Turnbull and others, 7 vols. (1959–77).
* ''Newton's Philosophy of Nature: Selections from His Writings'' edited by H. S. Thayer, (1953), [http://www.questia.com/read/5876270 online edition].
* Isaac Newton, Sir; J Edleston; [[Roger Cotes]], [http://books.google.com/books?as_brr=1&id=OVPJ6c9_kKgC&vid=OCLC14437781&dq=%22isaac+newton%22&jtp=I ''Correspondence of Sir Isaac Newton and Professor Cotes, including letters of other eminent men''], London, John W. Parker, West Strand; Cambridge, John Deighton, 1850 (Google Books).
* Maclaurin, C. (1748). An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books. London: A. Millar and J. Nourse.
* Newton, I. (1958). Isaac Newton's Papers and Letters on Natural Philosophy and Related Documents, eds. I. B. Cohen and R. E. Schofield. Cambridge: Harvard University Press.
* Newton, I. (1962). The Unpublished Scientific Papers of Isaac Newton: A Selection from the Portsmouth Collection in the University Library, Cambridge, ed. A. R. Hall and M. B. Hall. Cambridge: Cambridge University Press.
*Newton, I. (1975). Isaac Newton's 'Theory of the Moon's Motion' (1702). London: Dawson.
{{refend}}

== External links ==
{{Spoken Wikipedia|Isaac_Newton.ogg|2008-07-30}}
{{Sister project links|s=Author:Isaac Newton|wikt=no|n=no|b=Introduction to Astrophysics/Historical Context/Isaac Newton}}
*{{librivox author|Isaac+Newton}}
*[http://scholar.google.com.au/citations?user=xJaxiEEAAAAJ&hl=en Newton's Scholar Google profile]
*[http://scienceworld.wolfram.com/biography/Newton.html ScienceWorld biography] by [[Eric Weisstein]]
*[http://www.chlt.org/sandbox/lhl/dsb/page.50.a.php Dictionary of Scientific Biography]
*[http://www.newtonproject.sussex.ac.uk/prism.php?id=1 "The Newton Project"]
*[http://www.isaacnewton.ca/ "The Newton Project – Canada"]
*[http://web.archive.org/web/20080629021908/http://www.skepticreport.com/predictions/newton.htm "Rebuttal of Newton's astrology"] (via [[archive.org]])
*[http://www.galilean-library.org/snobelen.html "Newton's Religious Views Reconsidered"]
*[http://www.pierre-marteau.com/editions/1701-25-mint-reports.html "Newton's Royal Mint Reports"]
*[http://www.pbs.org/wgbh/nova/newton/ "Newton's Dark Secrets"] - [[Nova (TV series)|NOVA]] TV programme
*from ''The [[Stanford Encyclopedia of Philosophy]]:''
**[http://plato.stanford.edu/entries/newton/ "Isaac Newton"], by George Smith
**[http://plato.stanford.edu/entries/newton-principia/ "Newton's ''Philosophiae Naturalis Principia Mathematica''"], by George Smith
**[http://plato.stanford.edu/entries/newton-philosophy/ "Newton's Philosophy"], by Andrew Janiak
**[http://plato.stanford.edu/entries/newton-stm/ "Newton's views on space, time, and motion"], by Robert Rynasiewicz
*[http://www.tqnyc.org/NYC051308/index.htm "Newton's Castle"] - educational material
*[http://www.dlib.indiana.edu/collections/newton "The Chymistry of Isaac Newton"], research on his alchemical writings
*[http://www.fmalive.com/ "FMA Live!"] - program for teaching Newton's laws to kids
*[http://www.adherents.com/people/pn/Isaac_Newton.html Newton's religious position]
*[http://hss.fullerton.edu/philosophy/GeneralScholium.htm The "General Scholium" to Newton's ''Principia'']
*Kandaswamy, Anand M. [http://www.math.rutgers.edu/courses/436/Honors02/newton.html "''The Newton/Leibniz Conflict in Context''"]
*[http://www.phaser.com/modules/historic/newton/index.html Newton's First ODE] – A study by on how Newton approximated the solutions of a first-order ODE using infinite series
*{{MacTutor Biography|id=Newton}}
*{{MathGenealogy|id=74313}}
*[http://www.ltrc.mcmaster.ca/newton/ "The Mind of Isaac Newton"] - images, audio, animations and interactive segments
*[http://www.enlighteningscience.sussex.ac.uk/home Enlightening Science] Videos on Newton's biography, optics, physics, reception, and on his views on science and religion
*[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Newton.html Newton biography (University of St Andrews)]
* Chisholm, Hugh, ed. (1911). "[[s:1911 Encyclopædia Britannica/Newton, Sir Isaac|Newton, Sir Isaac]]". ''[[Encyclopædia Britannica]]'' (11th ed.). Cambridge University Press.
*and see at [[s:Author:Isaac Newton]] for the following works about him:
**"Newton, Sir Isaac" in A Short Biographical Dictionary of English Literature by John William Cousin, London: J. M. Dent & Sons, 1910.
**"Newton, Isaac," in Dictionary of National Biography, London: Smith, Elder, & Co., (1885–1900)
**Memoirs of Sir Isaac Newton's life by William Stukeley, 1752
* {{NRA | id=P21155}}
* {{npg name|id=03286|name=Sir Isaac Newton}}

'''Writings by Newton'''
*[http://www.newtonproject.sussex.ac.uk/prism.php?id=43 Newton's works – full texts, at the Newton Project]
* [http://web.nli.org.il/sites/NLI/English/collections/Humanities/Pages/newton.aspx The Newton Manuscripts at the National Library of Israel - the collection of all his religious writings]
*{{gutenberg author|id=Isaac_Newton|name=Isaac Newton}}
*[http://rack1.ul.cs.cmu.edu/is/newton/ "Newton's ''Principia''"] – read and search
*[http://www.earlymoderntexts.com/ ''Descartes, Space, and Body'' and ''A New Theory of Light and Colour''], modernised readable versions by Jonathan Bennett
*[http://www.archive.org/stream/opticksoratreat00newtgoog#page/n6/mode/2up ''Opticks, or a Treatise of the Reflections, Refractions, Inflexions and Colours of Light''], full text on [[archive.org]]
*[http://cudl.lib.cam.ac.uk/collections/newton "Newton Papers"] - Cambridge Digital Library
*See Wikisource at [[s:Author:Isaac Newton]] for the following works by him:
** ''Philosophiae Naturalis Principia Mathematica''
** ''Opticks: or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light''
** ''Observations upon the Prophecies of Daniel and the Apocalypse of St. John''
** ''New Theory About Light and Colour''
** ''An Historical Account of Two Notable Corruptions of Scripture''

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{{Persondata
|NAME=Newton, Isaac
|ALTERNATIVE NAMES=Newton, Sir Isaac
|SHORT DESCRIPTION=English mathematician, physicist, and astronomer
|DATE OF BIRTH=4 January 1643 Gregorian calendar
|PLACE OF BIRTH=[[Woolsthorpe-by-Colsterworth]], Lincolnshire, England
|DATE OF DEATH=31 March 1727 Gregorian calendar
|PLACE OF DEATH=[[Kensington]], London, England
}}
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Numerical methods for ordinary differential equations

2014-01-20T16:35:43Z

Jitse Niesen: revert - \mathcal is a command which produces a curly O

The '''ultraproduct''' is a [[mathematics|mathematical]] construction that appears mainly in [[abstract algebra]] and in [[model theory]], a branch of [[mathematical logic]]. An ultraproduct is a quotient of the direct product of a family of [[structure (mathematical logic)|structures]]. All factors need to have the same [[signature (logic)|signature]]. The '''ultrapower''' is the special case of this construction in which all factors are equal.

For example, ultrapowers can be used to construct new [[field (mathematics)|field]]s from given ones. The [[hyperreal numbers]], an ultrapower of the [[real numbers]], are a special case of this.

Some striking applications of ultraproducts include very elegant proofs of the [[compactness theorem]] and the [[completeness theorem]], [[H. Jerome Keisler|Keisler]]'s ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson-Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of [[non-standard analysis]], which was pioneered (as an application of the compactness theorem) by [[Abraham Robinson]].

==Definition==
The general method for getting ultraproducts uses an index set ''I'', a [[structure (mathematical logic)|structure]] ''M''''i'' for each element ''i'' of ''I'' (all of the same [[signature (logic)|signature]]), and an [[ultrafilter]] ''U'' on ''I''. The usual choice is for ''I'' to be infinite and ''U'' to contain all [[cofinite]] subsets of ''I''. Otherwise the ultrafilter is [[Principal ultrafilter|principal]], and the ultraproduct is isomorphic to one of the factors.

Algebraic operations on the [[Cartesian product]]

:<math>\prod_{i \in I} M_i </math>

are defined in the usual way (for example, for a binary function +, (''a'' + ''b'') ''i'' = ''a''''i'' + ''b''''i'' ), and an [[equivalence relation]] is defined by ''a'' ~ ''b'' if and only if

:<math>\left\{ i \in I: a_i = b_i \right\}\in U,</math>

and the '''ultraproduct''' is the [[quotient set]] with respect to ~. The ultraproduct is therefore sometimes denoted by

:<math>\prod_{i\in I}M_i / U . </math>

One may define a finitely additive [[Measure (mathematics)|measure]] ''m'' on the index set ''I'' by saying ''m''(''A'') = 1 if ''A'' ∈ ''U'' and = 0 otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal [[almost everywhere]] on the index set. The ultraproduct is the set of equivalence classes thus generated.

Other [[relation (mathematics)|relation]]s can be extended the same way:

:<math>R([a^1],\dots,[a^n]) \iff \left\{ i \in I: R^{M_i}(a^1_i,\dots,a^n_i) \right\}\in U,</math>

where [''a''] denotes the equivalence class of ''a'' with respect to ~.

In particular, if every ''M''''i'' is an [[ordered field]], then so is the ultraproduct.

An '''ultrapower''' is an ultraproduct for which all the factors ''M''''i'' are equal:

: <math>M^\kappa/U=\prod_{\alpha<\kappa}M/U.\,</math>

More generally, the construction above can be carried out whenever ''U'' is a [[filter (mathematics)|filter]] on ''I''; the resulting model <math>\prod_{i\in I}M_i / U</math> is then called a '''reduced product'''.

== Examples ==

The [[hyperreal numbers]] are the ultraproduct of one copy of the [[real numbers]] for every natural number, with regard to an ultrafilter over the natural numbers containing all cofinite sets. Their order is the extension of the order of the real numbers. For example, the sequence ''ω'' given by ''ω''''i'' = ''i'' defines an equivalence class representing a hyperreal number that is greater than any real number.

Analogously, one can define [[nonstandard integer]]s, [[nonstandard complex numbers]], etc., by taking the ultraproduct of copies of the corresponding structures.

As an example of the carrying over of relations into the ultraproduct, consider the sequence ''ψ'' defined by ''ψ''''i'' = 2''i''. Because ''ψ''''i'' > ''ω''''i'' = ''i'' for all ''i'', it follows that the equivalence class of ''ψ''''i'' = 2''i'' is greater than the equivalence class of ''ω''''i'' = ''i'', so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let ''χ''''i'' = ''i'' for ''i'' not equal to 7, but ''χ''7 = 8. The set of indices on which ''ω'' and ''χ'' agree is a member of any ultrafilter (because ''ω'' and ''χ'' agree almost everywhere), so ''ω'' and ''χ'' belong to the same equivalence class.

In the theory of [[large cardinal]]s, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter ''U''. Properties of this ultrafilter ''U'' have a strong influence on (higher order) properties of the ultraproduct; for example, if ''U'' is σ-complete, then the ultraproduct will again be well-founded. (See [[measurable cardinal]] for the prototypical example.)

==Łoś's theorem==

Łoś's theorem, also called ''the fundamental theorem of ultraproducts'', is due to [[Jerzy Łoś]] (the surname is pronounced {{IPA-pl|ˈwɔɕ|}}, approximately "wash"). It states that any [[first-order predicate calculus|first-order]] formula is true in the ultraproduct if and only if the set of indices ''i'' such that the formula is true in ''M''''i'' is a member of ''U''. More precisely:

Let σ be a signature, <math> U </math> an ultrafilter over a set <math> I </math>, and for each <math> i \in I </math> let <math> M_{i} </math> be a σ-structure. Let <math> M </math> be the ultraproduct of the <math> M_{i} </math> with respect to <math>U</math>, that is, <math> M = \prod_{ i\in I }M_i/U.</math> Then, for each <math> a^{1}, \ldots, a^{n} \in \prod M_{i} </math>, where <math> a^{k} = (a^{k}_{i})_{i \in I} </math>, and for every σ-formula <math>\phi</math>,

:<math> M \models \phi[[a^1], \ldots, [a^n]] \iff \{ i \in I : M_{i} \models \phi[a^1_{i}, \ldots, a^n_{i} ] \} \in U.</math>

The theorem is proved by induction on the complexity of the formula <math>\phi</math>. The fact that <math>U</math> is an ultrafilter (and not just a filter) is used in the negation clause, and the [[axiom of choice]] is needed at the existential quantifier step. As an application, one obtains the [[transfer principle|transfer theorem]] for [[hyperreal number|hyperreal fields]].

===Examples===
Let ''R'' be a unary relation in the structure ''M'', and form the ultrapower of ''M''. Then the set <math>S=\{x \in M|R x\}</math> has an analog ''*S'' in the ultrapower, and first-order formulas involving S are also valid for ''*S''. For example, let ''M'' be the reals, and let ''Rx'' hold if ''x'' is a rational number. Then in ''M'' we can say that for any pair of rationals ''x'' and ''y'', there exists another number ''z'' such that ''z'' is not rational, and ''x'' < ''z'' < ''y''. Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that ''*S'' has the same property. That is, we can define a notion of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals.

Consider, however, the [[Archimedean property]] of the reals, which states that there is no real number ''x'' such that ''x'' > 1, ''x'' > 1 +1 , ''x'' > 1 + 1 + 1, ... for every inequality in the infinite list. Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic. In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number ''ω'' above.

==Ultralimit==
:''For the ultraproduct of a sequence of metric spaces, see [[Ultralimit]].''
In [[model theory]] and [[set theory]], an '''ultralimit''' or '''limiting ultrapower''' is a [[direct limit]] of a sequence of ultrapowers.

Beginning with a structure, ''A''0, and an ultrafilter, ''D''0, form an ultrapower, ''A''1. Then repeat the process to form ''A''2, and so forth. For each ''n'' there is a canonical diagonal embedding <math>A_n\to A_{n+1}</math>. At limit stages, such as ''A''ω, form the direct limit of earlier stages. One may continue into the transfinite.

==References==
* {{ cite book | last=Bell | first=John Lane | coauthors=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 edition | origyear=1969 | publisher=[[Dover Publications]] | isbn=0-486-44979-3 }}
* {{ cite book | last=Burris | first =Stanley N. | coauthors=Sankappanavar, H.P. | title=A Course in Universal Algebra | origyear=1981 | year=2000 | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html | edition=Millennium edition}}

[[Category:Model theory]]
[[Category:Universal algebra]]
[[Category:Non-standard analysis]]

Backward Euler method

2014-01-18T16:57:10Z

Jitse Niesen: local error is O(h^2), see talk

{| border="1" cellpadding="5" cellspacing="0" align="right" style="width:220px; text-align:justify"
! style="background:#ffdead;" | [[Feynman diagrams]]
|-
| align="center" | '''Annihilation''' [[Image:Electron-positron-annihilation.svg|220px]]
|-
| align="center" | '''Scattering''' [[Image:Electron-positron-scattering.svg|220px]]
|-
|Note: Time moves forward from the left side of the diagram to the right. '''The arrows are simply markers of particle motion, and are ''not'' the same as the arrows conventionally written into Feynman diagrams'''.
|}

In [[quantum electrodynamics]], '''Bhabha scattering''' is the [[electron]]-[[positron]] [[scattering]] process:
::<math>e^+ e^- \rightarrow e^+ e^-</math>

There are two leading-order [[Feynman diagram]]s contributing to this interaction: an annihilation process and a scattering process. Bhabha scattering is named after the Indian physicist [[Homi J. Bhabha]].

The Bhabha scattering rate is used as a [[luminosity]] monitor in electron-positron colliders.

==Differential cross section==
To [[leading-order|leading order]], the spin-averaged [[differential cross section]] for this process is

::<math>\frac{\mathrm{d} \sigma}{\mathrm{d} (\cos\theta)} = \frac{\pi \alpha^2}{s} \left( u^2 \left( \frac{1}{s} + \frac{1}{t} \right)^2 + \left( \frac{t}{s} \right)^2 + \left( \frac{s}{t} \right)^2 \right) \,</math>

where ''s'',''t'', and ''u'' are the [[Mandelstam variables]], <math>\alpha</math> is the [[fine-structure constant]], and <math>\theta</math> is the scattering angle.

This cross section is calculated neglecting the electron mass relative to the collision energy and including only the contribution from photon exchange. This is a valid approximation at collision energies small compared to the mass scale of the [[Z boson]], about 91 GeV; at higher energies the contribution from Z boson exchange also becomes important.

===Mandelstam variables===
In this article, the [[Mandelstam variables]] are defined by
::{|
|align="right"|<math>s= \,</math>
|align="right"|<math>(k+p)^2= \,</math>
|align="right"|<math>(k'+p')^2 \approx \,</math>
|align="right"|<math>2 k \cdot p \approx\,</math>
|align="right"|<math> 2 k' \cdot p' \,</math>
|rowspan="3"|        [[Image:Mandelstam01.png]]
|-
|align="right"|<math>t= \,</math>
|align="right"|<math>(k-k')^2= \,</math>
|align="right"|<math>(p-p')^2\approx \,</math>
|<math> -2 k \cdot k' \approx \,</math>
|<math> -2 p \cdot p' \,</math>
|-
|align="right"|<math>u= \,</math>
|align="right"|<math>(k-p')^2= \,</math>
|align="right"|<math>(p-k')^2\approx \,</math>
|<math> -2 k \cdot p' \approx \,</math>
|<math> -2 k' \cdot p \,</math>
|}

where the approximations are for the high-energy (relativistic) limit.

==Deriving unpolarized cross section==

===Matrix elements===
Both the scattering and annihilation diagrams contribute to the transition matrix element. By letting ''k'' and ''k' '' represent the four-momentum of the positron, while letting ''p'' and ''p' '' represent the four-momentum of the electron, and by using [[Feynman rules]] one can show the following diagrams give these matrix elements:
:{| border="0" cellpadding="5" cellspacing="0"
|
|align="center" | [[Image:Feynman-scattering.svg|160px]]
|align="center" | [[Image:Feynman-annihilation.svg|160px]]
|Where we use: <math>\gamma^\mu \,</math> are the [[Gamma matrices]], <math>u, \ \mathrm{and} \ \bar{u}\,</math> are the four-component spinors for fermions, while <math>v, \ \mathrm{and} \ \bar{v}\,</math> are the four-component spinors for anti-fermions (see [[Dirac equation#Four spinor|Four spinors]]).
|-
|
|align="center" | (scattering)
|align="center" | (annihilation)
|
|-
|<math>\mathcal{M} = \,</math>
|<math>-e^2 \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \frac{1}{(k-k')^2} \left( \bar{u}_{p'} \gamma_\mu u_p \right) </math>
|<math>+e^2 \left( \bar{v}_{k} \gamma^\nu u_p \right) \frac{1}{(k+p)^2} \left( \bar{u}_{p'} \gamma_\nu v_{k'} \right) </math>
|
|}

Notice that there is a relative sign difference between the two diagrams.

===Square of matrix element===
To calculate the unpolarized [[Cross section (physics)|cross section]], one must ''average'' over the spins of the incoming particles (''s''e- and ''s''e+ possible values) and ''sum'' over the spins of the outgoing particles. That is,
::{|
|<math>\overline{|\mathcal{M}|^2} \,</math>
|<math> = \frac{1}{(2s_{e-} + 1)(2 s_{e+} + 1)} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
|-
|
|<math>= \frac{1}{4} \sum_{s=1}^2 \sum_{s'=1}^2 \sum_{r=1}^2 \sum_{r'=1}^2 |\mathcal{M}|^2 \,</math>
|}

First, calculate <math>|\mathcal{M}|^2 \,</math>:
::{| cellpadding=4
|<math>|\mathcal{M}|^2 \,</math>=
|<math> e^4 \left| \frac{(\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right|^2 \,</math>
| align="center" | (scattering)
|-
|
|<math>{}- e^4 \left( \frac{ (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right)^* \left( \frac{ (\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'}) }{(k+p)^2} \right) \,</math>
| align="center" | (interference)
|-
|
|<math>{}- e^4 \left( \frac{ (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right) \left( \frac{ (\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'}) }{(k+p)^2} \right)^* \,</math>
| align="center" | (interference)
|-
|
|<math>{}+ e^4 \left| \frac{(\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'} )}{(k+p)^2} \right|^2 \,</math>
| align="center" | (annihilation)
|}

===Scattering term (t-channel)===

====Magnitude squared of M====
::{|
|<math>|\mathcal{M}|^2 \,</math>
|<math>= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p) \Big)^* \Big( (\bar{v}_{k} \gamma^\nu v_{k'})( \bar{u}_{p'} \gamma_\nu u_p) \Big) \,</math>
|     <math>(1) \,</math>
|-
|
|<math>= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )^* ( \bar{u}_{p'} \gamma_\mu u_p)^* \Big) \Big( (\bar{v}_{k} \gamma^\nu v_{k'})( \bar{u}_{p'} \gamma_\nu u_p) \Big) \,</math>
|     <math>(2) \,</math>
|-
|
|align="center" | (complex conjugate will flip order)
|
|-
|
|<math>= \frac{e^4}{(k-k')^4} \Big( \left(\bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \Big) \Big( \left( \bar{v}_{k} \gamma^\nu v_{k'} \right) \left( \bar{u}_{p'} \gamma_\nu u_p \right) \Big) \,</math>
|     <math>(3) \,</math>
|-
|
|align="center" | (move terms that depend on same momentum to be next to each other)
|
|-
|
|<math>= \frac{e^4}{(k-k')^4} \left( \bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{v}_{k} \gamma^\nu v_{k'} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \left( \bar{u}_{p'} \gamma_\nu u_p \right) \,</math>
|     <math>(4) \,</math>
|}

====Sum over spins====
Next, we'd like to sum over spins of all four particles. Let ''s'' and ''s' '' be the spin of the electron and ''r'' and ''r' '' be the spin of the positron.
::{|
|<math> \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
|<math>= \frac{e^4}{(k-k')^4}\left(\sum_{r'} \bar{v}_{k'} \gamma^\mu (\sum_{r}v_{k} \bar{v}_{k}) \gamma^\nu v_{k'} \right) \left(\sum_{s} \bar{u}_{p} \gamma_\mu (\sum_{s'}{u_{p'} \bar{u}_{p'}}) \gamma_\nu u_p \right) \,</math>
|     <math>(5) \,</math>
|-
|
|<math>= \frac{e^4}{(k-k')^4}\operatorname{Tr}\left( \Big(\sum_{r'} v_{k'} \bar{v}_{k'} \Big) \gamma^\mu \Big(\sum_{r}v_{k} \bar{v}_{k} \Big) \gamma^\nu \right) \operatorname{Tr} \left( \Big(\sum_{s} u_p \bar{u}_{p} \Big) \gamma_\mu \Big( \sum_{s'}{u_{p'} \bar{u}_{p'}} \Big) \gamma_\nu \right) \,</math>
|     <math>(6) \,</math>
|-
|
|align="center" | (now use [[Bhabha scattering#Completeness relations|Completeness relations]])
|
|-
|
|<math>=\frac{e^4}{(k-k')^4}\operatorname{Tr}\left( (k\!\!\!/' - m) \gamma^\mu (k\!\!\!/ - m) \gamma^\nu \right) \cdot \operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,</math>
|     <math>(7) \,</math>
|-
|
|align="center" | (now use [[Bhabha scattering#Trace identities|Trace identities]])
|
|-
|
|<math>=\frac{e^4}{(k-k')^4}\left(4 \left( {k'}^\mu k^\nu - (k' \cdot k)\eta^{\mu\nu} + k'^\nu k^\mu \right) + 4 m^2 \eta^{\mu\nu} \right) \left( 4 \left( {p'}_\mu p_\nu - (p' \cdot p)\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \right) \,</math>
|     <math>(8) \,</math>
|-
|
|<math>=\frac{32{e^4}}{(k-k')^4}\left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') -m^2 p' \cdot p - m^2 k' \cdot k + 2m^4 \right) \,</math>
|     <math>(9) \,</math>
|}

Now that is the exact form, in the case of electrons one is usually interested in energy scales that far exceed the electron mass. Neglecting the electron mass yields the simplified form:
::{|
|<math>\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
|<math> = \frac{32e^4}{4(k-k')^4} \left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') \right) \,</math>
|-
|
|(use the [[Bhabha scattering#Mandelstam variables|Mandelstam variables]] in this relativistic limit)
|-
|
|<math>=\frac{8e^4}{t^2} \left(\tfrac{1}{2} s \tfrac{1}{2}s + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,</math>
|-
|
|<math>= 2 e^4 \frac{s^2 +u^2}{t^2} \,</math>
|}

===Annihilation term (s-channel)===
The process for finding the annihilation term is similar to the above. Since the two diagrams are related by [[crossing symmetry]], and the initial and final state particles are the same, it is sufficient to permute the momenta, yielding
::{|
|<math>\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
|<math> = \frac{32e^4}{4(k+p)^4} \left( (k \cdot k') (p \cdot p') + (k' \cdot p) (k \cdot p') \right) \,</math>
|-
|
|<math>=\frac{8e^4}{s^2} \left(\tfrac{1}{2} t \tfrac{1}{2}t + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,</math>
|-
|
|<math>= 2 e^4 \frac{t^2 +u^2}{s^2} \,</math>
|}
(This is proportional to
<math>(1 + \cos^2\theta)</math>
where <math>\theta</math> is the scattering angle in the center-of-mass frame.)

===Solution===
Evaluating the interference term along the same lines and adding the three terms yields the final result
::<math>\frac{\overline{|\mathcal{M}|^2}}{2e^4} = \frac{u^2 + s^2}{t^2} + \frac{2 u^2}{st} + \frac{u^2 + t^2}{s^2} \,</math>

==Simplifying steps==

===Completeness relations===
The completeness relations for the [[Dirac_spinor#Four-spinor_for_particles|four-spinors]] ''u'' and ''v'' are
::<math>\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,</math>
::<math>\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,</math>
:where
::<math>p\!\!\!/ = \gamma^\mu p_\mu \,</math>      (see [[Feynman slash notation#With four-momentum|Feynman slash notation]])
::<math>\bar{u} = u^{\dagger} \gamma^0 \,</math>

===Trace identities===
:''Main article:'' [[Gamma matrix#Identities|Trace identities]]
To simplify the trace of the [[Gamma matrices|Dirac gamma matrices]], one must use trace identities. Three used in this article are:
#The Trace of any product of an ''odd number'' of <math>\gamma_\mu \,</math>'s is zero
#<math>\operatorname{Tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}</math>
#<math>\operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) = 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) \,</math>

Using these two one finds that, for example,
::{|
|<math>\operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,</math>
|<math> = \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + \operatorname{Tr}\left(m \gamma_\mu p\!\!\!/ \gamma_\nu \right) \,</math>
|-
|
|         <math>+ \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu m \gamma_\nu \right) + \operatorname{Tr}\left(m^2 \gamma_\mu \gamma_\nu \right) \,</math>
|-
|
|align="center"|(the two middle terms are zero because of (1))
|-
|
|<math> = \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + m^2 \operatorname{Tr}\left(\gamma_\mu \gamma_\nu \right) \,</math>
|-
|
|align="center"|(use identity (2) for the term on the right)
|-
|
|<math>= {p'}^{\rho} p^\sigma \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) + m^2 \cdot 4\eta_{\mu\nu} \,</math>
|-
|
|align="center"|(now use identity (3) for the term on the left)
|-
|
|<math>= {p'}^{\rho} p^\sigma 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) + 4 m^2 \eta_{\mu\nu} \,</math>
|-
|
|<math>=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,</math>
|}

==Uses==
Bhabha scattering has been used as a [[luminosity]] monitor in a number of e+e- collider physics experiments. The accurate measurement of luminosity is necessary for accurate measurements of cross sections.
* Small-angle Bhabha scattering was used to measure the luminosity of the 1993 run of the [[Stanford Large Detector]] (SLD), with a relative uncertainty of less than 0.5%.<ref>[http://adsabs.harvard.edu/abs/1995PhDT.......160W A Study of Small Angle Radiative Bhabha Scattering and Measurement of the Luminosity at SLD]</ref>
* Electron-positron colliders operating in the region of the low-lying hadronic resonances (about 1 GeV to 10 GeV), such as the [[Beijing Electron Synchrotron]] (BES) and the [[Belle experiment|Belle]] and [[BaBar experiment|BaBar]] "B-factory" experiments, use large-angle Bhabha scattering as a luminosity monitor. To achieve the desired precision at the 0.1% level, the experimental measurements must be compared to a theoretical calculation including next-to-leading-order [[renormalization|radiative corrections]].[http://arxiv.org/abs/hep-ph/0003268] The high-precision measurement of the total hadronic cross section at these low energies is a crucial input into the theoretical calculation of the [[anomalous magnetic dipole moment]] of the [[muon]], which is used to constrain [[supersymmetry]] and other models of physics [[beyond the Standard Model]].

==References==
<references/>

*{{cite book | author=Halzen, Francis; Martin, Alan | title=Quarks & Leptons: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}
*{{cite book | author=Peskin, Michael E.; Schroeder, Daniel V. | title=An Introduction to Quantum Field Theory | publisher=Perseus Publishing | year=1994 | isbn=0-201-50397-2}}
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=270563 Bhabha scattering on arxiv.org]
{{QED}}

[[Category:Quantum field theory]]
[[Category:Quantum electrodynamics]]

Euler method

2014-01-14T14:33:57Z

Jitse Niesen: Reverted edits by 117.18.231.60 (talk) to last version by 186.55.143.100

'''Upper and lower probabilities''' are representations of [[imprecise probability]]. Whereas [[probability theory]] uses a single number, the [[probability]], to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.

Because [[frequentist statistics]] disallows [[metaprobability|metaprobabilities]],{{cn|date=May 2012}} frequentists have had to propose new solutions. [[Cedric Smith (statistician)|Cedric Smith]] and [[Arthur P. Dempster|Arthur Dempster]] each developed a theory of upper and lower probabilities. [[Glenn Shafer]] developed Dempster's theory further, and it is now known as [[Dempster–Shafer theory]]: see also Choquet(1953).
More precisely, in the work of these authors one considers in a [[power set]], <math>P(S)\,\!</math>, a ''mass'' function <math>m : P(S)\rightarrow R</math> satisfying the conditions

:<math>m(\varnothing) = 0 \,\,\,\,\,\,\! ; \,\,\,\,\,\, \sum_{A \in P(X)} m(A) = 1. \,\!</math>

In turn, a mass is associated with two non-additive continuous measures called '''belief''' and '''plausibility''' defined as follows:

:<math>\operatorname{bel}(A) = \sum_{B \mid B \subseteq A} m(B)\,\,\,\,;\,\,\,\,
\operatorname{pl}(A) = \sum_{B \mid B \cap A \ne \varnothing} m(B)</math>

In the case where <math>S</math> is infinite there can be <math>\operatorname{bel}</math> such that there is no associated mass function. See p. 36 of Halpern (2003). Probability measures are a special case of belief functions in which the mass function assigns positive mass to singletons of the event space only.

A different notion of upper and lower probabilities is obtained by the ''lower and upper envelopes'' obtained from a class ''C'' of probability distributions by setting

:<math>\operatorname{env_1}(A) = \inf_{p \in C} p(A)\,\,\,\,;\,\,\,\,
\operatorname{env_2}(A) = \sup_{p \in C} p(A)</math>

The upper and lower probabilities are also related with [[probabilistic logic]]: see Gerla (1994).

Observe also that a [[necessity measure]] can be seen as a lower probability and a [[possibility measure]] can be seen as an upper probability.

==See also==
*[[Possibility theory]]
*[[Fuzzy measure theory]]
*[[Interval finite element]]

==References==
*G. Gerla, Inferences in Probability Logic, ''Artificial Intelligence'' 70(1–2):33–52, 1994.
*J.Y. Halpern 2003 ''Reasoning about Uncertainty'' MIT Press
*J. Y. Halpern and R. Fagin, Two views of belief: Belief as generalized probability and belief as evidence. ''Artificial Intelligence'', 54:275–317, 1992.
*P. J. Huber, ''Robust Statistics''. Wiley, New York, 1980.
*Saffiotti, A., A Belief-Function Logic, in ''Procs of the 10h AAAI Conference'', San Jose, CA 642–647, 1992.
*Choquet, G., Theory of Capacities, ''Annales de l'Institut Fourier'' 5, 131–295, 1953.
*Shafer, G., ''A Mathematical Theory of Evidence'', (Princeton University Press, Princeton), 1976.
*P. Walley and T. L. Fine, Towards a frequentist theory of upper and lower probability. ''Annals of Statistics'', 10(3):741–761, 1982.

[[Category:Exotic probabilities]]
[[Category:Probability bounds analysis]]

Differential equation

2014-01-14T08:29:47Z

Jitse Niesen: remove {{incoherent}} and {{technical}} as it is unclear what they are refering to; see talk

In the history of [[science]], '''''vis viva''''' (from the [[Latin]] for ''living force'') is an [[obsolete scientific theory]] that served as an elementary and limited early formulation of the principle of [[conservation of energy]]. It was the first [known] description of what we now call [[kinetic energy]] or of energy related to sensible motions.

Proposed by [[Gottfried Leibniz]] over the period 1676–1689, the theory was controversial as it seemed to oppose the theory of [[conservation of momentum]] advocated by Sir [[Isaac Newton]] and [[René Descartes]]. The two theories are now understood to be complementary.

The theory was eventually absorbed into the modern theory of [[energy]] though the term still survives in the context of [[celestial mechanics]] through the [[Vis-viva equation|''vis viva'' equation]].

==History==

Although [[ancient]] [[philosopher]]s as far back as [[Thales|Thales of Miletus]] had inklings of the law of conservation of energy{{Citation needed|date=January 2011}}, it was the [[Germany|German]] [[Gottfried Wilhelm Leibniz]] during 1676–1689 who first attempted a mathematical formulation. Leibniz noticed that in many mechanical systems (of several [[mass]]es, ''mi'' each with [[velocity]] ''vi'') the quantity:

:<math>\sum_{i} m_i v_i^2</math>

was conserved. He called this quantity the ''vis viva'' or ''living force'' of the system. The principle, it is now realised, represents an accurate statement of the conservation of [[kinetic energy]] in [[elastic collision]]s, and is a consequence of the [[conservation of momentum]]. However, many [[physicist]]s at the time were unaware of this connection and, instead, were influenced by the prestige of Sir [[Isaac Newton]] in [[England]] and of [[René Descartes]] in [[France]], both of whom had set great store by the [[conservation of momentum]] as a guiding principle. Thus the [[momentum]]:

:<math>\,\!\sum_{i} m_i \mathbf{v}_i</math>

was held by the rival camp to be the conserved ''vis viva''. It was largely [[engineer]]s such as [[John Smeaton]], [[Peter Ewart]], [[Karl Holtzmann]], [[Gustave-Adolphe Hirn]] and [[Marc Séguin]] who objected that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle. The principle was also championed by some [[chemist]]s such as [[William Hyde Wollaston]].

The French mathematician [[Émilie du Châtelet]], who had a sound grasp of Newtonian mechanics, developed Leibniz's concept and, combining it with the observations of [[Willem 's Gravesande]], showed that ''vis viva'' was dependent on the square of the velocities.

Members of the academic establishment such as [[John Playfair]] were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the [[second law of thermodynamics]] but in the 18th and 19th centuries, the fate of the lost energy was still unknown. Gradually it came to be suspected that the [[heat]] inevitably generated by motion was another form of ''vis viva''. In 1783, [[Antoine Lavoisier]] and [[Pierre-Simon Laplace]] reviewed the two competing theories of ''vis viva'' and [[caloric theory]].{{ref|LL}} [[Benjamin Thompson|Count Rumford]]'s 1798 observations of heat generation during the [[Boring (manufacturing)|boring]] of [[cannon]]s added more weight to the view that mechanical motion could be converted into heat. ''Vis viva'' now started to be known as ''energy'', after the term was first used in that sense by [[Thomas Young (scientist)|Thomas Young]] in 1807.

[[File:Bernoulli-vis-viva-with-0.5-multiplier-1736 (1741).gif|thumb|An excerpt from [[Daniel Bernoulli|D.Bernoulli's]] article, published in 1741,<ref>{{cite journal|author=Bernoulli D.|title=De legibus quibusdam mechanicis…|url=http://gidropraktikum.narod.ru/jet.htm|journal=Commentarii Academiae scientiarum imperialis Petropolitanae|year=1741 (1736)|volume=8|pages=99–127}}</ref> with the definition of vis viva with 1/2 multiplier.]]
The recalibration of ''vis viva'' to include the coefficient of a half, namely:

:<math>\frac {1} {2}\sum_{i} m_i v_i^2</math>

was largely the result of the work of [[Gaspard-Gustave Coriolis]] and [[Jean-Victor Poncelet]] over the period 1819–1839, although present-day definition can occasionally be found earlier (e.g., in [[Daniel Bernoulli]] texts).

The former called the ''quantité de travail'' (quantity of work) and the latter, ''travail mécanique'' (mechanical work) and both championed its use in engineering calculation.

==See also==
*[[Conservation of energy#History|Conservation of energy: Historical development]]
*[[Vis-viva equation]]

==References==
<references />
* George E. Smith, [http://ptonline.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTOAD000059000010000031000001&idtype=cvips "The Vis Viva Dispute: A Controversy at the Dawn of Dynamics"], ''Physics Today'' '''59''' (October 2006) Issue 10 pp 31–36. [http://ptonline.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTOAD000059000010000031000001&idtype=cvips ] (see also [http://www.physicstoday.org/vol-59/iss-12/p16.html erratum])

{{Gottfried Wilhelm Leibniz}}

{{DEFAULTSORT:Vis Viva}}
[[Category:Natural philosophy]]
[[Category:Obsolete scientific theories]]
[[Category:Mechanics]]
[[Category:Thermodynamics]]
[[Category:Gottfried Leibniz]]
[[Category:History of thermodynamics]]

Talk:Linear programming

2014-01-06T13:23:20Z

Jitse Niesen: reply

== Sacs Michael Kors Maroc 58 ==

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Hamiltonian matrix

2013-09-06T08:11:18Z

Jitse Niesen: add citation for alternative definition

In [[mathematics]], a '''shift matrix''' is a [[binary matrix]] with ones only on the [[superdiagonal]] or [[subdiagonal]], and zeroes elsewhere. A shift matrix ''U'' with ones on the superdiagonal is an '''upper shift matrix'''.
The alternative subdiagonal matrix ''L'' is unsurprisingly known as a '''lower shift matrix'''. The ''(i,j)'':th component of ''U'' and ''L'' are
:<math> U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]] symbol.

For example, the ''5×5'' shift matrices are
::<math>
U_5=\begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix} \quad
L_5=\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix}.
</math>

Clearly, the [[matrix transpose|transpose]] of a lower shift matrix is an upper shift matrix and vice versa.

Premultiplying a matrix ''A'' by a lower shift matrix results in the elements of ''A'' being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left.
Similar operations involving an upper shift matrix result in the opposite shift.

Clearly all shift matrices are [[nilpotent]]; an ''n'' by ''n'' shift matrix ''S'' becomes the [[null matrix]] when raised to the power of its dimension ''n''.

==Properties==
Let ''L'' and ''U'' be the ''n'' by ''n'' lower and upper shift matrices, respectively. The following properties hold for both ''U'' and ''L''.
Let us therefore only list the properties for ''U'':
* [[determinant|det]](''U'') = 0
* [[Trace (linear algebra)|trace]](''U'') = 0
* [[Rank (linear algebra)|rank]](''U'') = ''n''−1
* The [[characteristic polynomial]]s of ''U'' is
:<math>p_U(\lambda) = (-1)^n\lambda^n.</math>
* ''U''''n'' = 0. This follows from the previous property by the [[Cayley–Hamilton theorem]].

* The [[permanent]] of ''U'' is ''0''.

The following properties show how ''U'' and ''L'' are related:
* ''L''T = ''U''; ''U''T = L

*The [[null space]]s of ''U'' and ''L'' are
:<math> N(U) = \operatorname{span}\{ (1,0,\ldots, 0)^T \}, </math>
:<math> N(L) = \operatorname{span}\{ (0,\ldots, 0, 1)^T \}.</math>
* The [[spectral theorem|spectrum]] of ''U'' and ''L'' is <math>\{0\}</math>. The [[algebraic multiplicity]] of ''0'' is ''n'', and its [[geometric multiplicity]] is ''1''. From the expressions for the null spaces, it follows that (up to a scaling) the only eigenvector for ''U'' is <math>(1,0,\ldots, 0)^T</math>, and the only eigenvector for ''L'' is <math>(0,\ldots, 0,1)^T</math>.

* For ''LU'' and ''UL'' we have
:<math>UL = I - \operatorname{diag}(0,\ldots, 0,1),</math>
:<math>LU = I - \operatorname{diag}(1,0,\ldots, 0).</math>
:These matrices are both idempotent, symmetric, and have the same rank as ''U'' and ''L''

* ''L''''n-a''''U''''n-a'' + ''L''''a''''U''''a'' = ''U''''n-a''''L''''n-a'' + ''U''''a''''L''''a'' = ''I'' (the [[identity matrix]]), for any integer ''a'' between 0 and ''n'' inclusive.

==Examples==

::<math>
S=\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix}; \quad A=\begin{pmatrix}
1 & 1 & 1 & 1 & 1 \\
1 & 2 & 2 & 2 & 1 \\
1 & 2 & 3 & 2 & 1 \\
1 & 2 & 2 & 2 & 1 \\
1 & 1 & 1 & 1 & 1
\end{pmatrix}.</math>

Then <math>
SA=\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 \\
1 & 2 & 2 & 2 & 1 \\
1 & 2 & 3 & 2 & 1 \\
1 & 2 & 2 & 2 & 1
\end{pmatrix}; \quad AS=\begin{pmatrix}
1 & 1 & 1 & 1 & 0 \\
2 & 2 & 2 & 1 & 0 \\
2 & 3 & 2 & 1 & 0 \\
2 & 2 & 2 & 1 & 0 \\
1 & 1 & 1 & 1 & 0
\end{pmatrix}.</math>

Clearly there are many possible permutations. For example, <math>S^{T}AS</math> is equal to the matrix ''A'' shifted up and left along the main diagonal.

:::::<math>
S^{T}AS=\begin{pmatrix}
2 & 2 & 2 & 1 & 0 \\
2 & 3 & 2 & 1 & 0 \\
2 & 2 & 2 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix}.</math>

==See also==
* [[Nilpotent matrix]]

==References==
[http://www.ee.imperial.ac.uk/hp/staff/dmb/matrix/special.html#Shift_Matrix Shift Matrix - entry in the Matrix Reference Manual]

{{unreferenced|date=June 2008}}

[[Category:Matrices]]
[[Category:Sparse matrices]]

Next-generation matrix

2013-09-05T10:26:20Z

Jitse Niesen: revert as before

{{Infobox enzyme
| Name = ACP-SH:acetate ligase
| EC_number = 6.2.1.35
| CAS_number =
| IUBMB_EC_number = 6/2/1/35
| GO_code =
| image =
| width =
| caption =
}}
'''ACP-SH:acetate ligase''' ({{EC number|6.2.1.35}}, ''HS-acyl-carrier protein:acetate ligase'', ''[acyl-carrier protein]:acetate ligase'', ''MadH'') is an [[enzyme]] with system name ''acetate:(acyl-carrier-protein) ligase (AMP-forming)''.<ref>{{cite journal | title = Malonate decarboxylase of Malonomonas rubra, a novel type of biotin-containing acetyl enzyme |author = Hilbi, H., Dehning, I., Schink, B. and Dimroth, P. |journal = Eur. J. Biochem. |year = 1992 |volume = 207 |pages = 117–123 |pmid = 1628643}}</ref><ref>{{cite journal | title = The acyl carrier protein of malonate decarboxylase of Malonomonas rubra contains 2′-(5"-phosphoribosyl)-3′-dephosphocoenzyme A as a prosthetic group |author = Berg, M., Hilbi, H. and Dimroth, P. |journal = Biochemistry |year = 1996 |volume = 35 |pages = 4689–4696 |pmid = 8664258}}</ref><ref>{{cite journal | title = Sequence of a gene cluster from Malonomonas rubra encoding components of the malonate decarboxylase Na+ pump and evidence for their function |author = Berg, M., Hilbi, H. and Dimroth, P. |journal = Eur. J. Biochem. |year = 1997 |volume = 245 |pages = 103–115 |pmid = 9128730}}</ref><ref>{{cite journal | title = Enzymic and genetic basis for bacterial growth on malonate |author = Dimroth, P. and Hilbi, H. |journal = Mol. Microbiol. |year = 1997 |volume = 25 |pages = 3–10 |pmid = 11902724}}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: ATP + [[acetat]]e + an [acyl-carrier protein] <math>\rightleftharpoons</math> AMP + [[diphosphate]] + an acetyl-[acyl-carrier protein]

This enzyme, from the anaerobic bacterium ''[[Malonomonas rubra]]'', is a component of the multienzyme complex [[EC 4.1.1.89]], biotin-dependent malonate decarboxylase.

== References ==
{{reflist}}

== External links ==
* {{MeshName|ACP-SH:acetate+ligase}}

[[Category:EC 6.2.1]]

Symplectic matrix

2013-09-04T08:56:27Z

Jitse Niesen: /* Symplectic transformations */ block diagonal form is now below, not above

{{More footnotes|date=November 2009}}

In mathematics, and in particular [[linear algebra]], a '''skew-symmetric''' (or '''antisymmetric''' or '''antimetric'''<ref>{{cite book |author=Richard A. Reyment, [[K. G. Jöreskog]], Leslie F. Marcus |title=Applied Factor Analysis in the Natural Sciences | publisher=Cambridge University Press | year= 1996 | isbn=0-521-57556-7 | page=68}}</ref>) '''matrix''' is a [[square matrix]] ''A'' whose [[transpose]] is also its negative; that is, it satisfies the condition {{nowrap|1=-''A'' = ''A''T.}} If the entry in the {{nowrap|1=''i'' th row}} and {{nowrap|1=''j'' th column}} is ''aij'', i.e. {{nowrap|1=''A'' = (''a''''ij'')}} then the skew symmetric condition is {{nowrap|1=''aij'' = −''aji''.}} For example, the following matrix is skew-symmetric:

:<math>\begin{bmatrix}
0 & 2 & -1 \\
-2 & 0 & -4 \\
1 & 4 & 0\end{bmatrix}.</math>

==Properties==

We assume that the underlying [[field (mathematics)|field]] is not of [[Characteristic (algebra)|characteristic]] 2: that is, that {{nowrap|1=1 + 1 ≠ 0}} where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Otherwise, a skew-symmetric matrix is just the same thing as a [[symmetric matrix]].

Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a [[vector space]]. Its [[dimension of a vector space|dimension]] is ''n''(''n''−1)/2.

Let Mat''n'' denote the space of {{nowrap|1=''n'' × ''n''}} matrices. A skew-symmetric matrix is determined by ''n''(''n'' − 1)/2 scalars (the number of entries above the [[main diagonal]]); a [[symmetric matrix]] is determined by ''n''(''n'' + 1)/2 scalars (the number of entries on or above the main diagonal). If Skew''n'' denotes the space of {{nowrap|1=''n'' × ''n''}} skew-symmetric matrices and Sym''n'' denotes the space of {{nowrap|1=''n'' × ''n''}} symmetric matrices and then since {{nowrap|1=Mat''n'' = Skew''n'' + Sym''n''}} and {{nowrap|1=Skew''n'' ∩ Sym''n'' = {0}}}, i.e.
:<math> \mbox{Mat}_n = \mbox{Skew}_n \oplus \mbox{Sym}_n , </math>
where ⊕ denotes the [[Direct sum of modules|direct sum]]. Let {{nowrap|1=A ∈ Mat''n''}} then
:<math> A = \frac{1}{2}(A - A^{\mathsf{T}}) + \frac{1}{2}(A + A^{\mathsf{T}}) . </math>
Notice that {{nowrap|1=½(''A'' − ''A''T) ∈ Skew''n''}} and {{nowrap|1=½(''A'' + ''A''T) ∈ Sym''n''.}} This is true for every [[square matrix]] ''A'' with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2.

Denote with <math>\langle \cdot,\cdot \rangle</math> the standard [[inner product]] on '''R'''''n''. The real ''n''-by-''n'' matrix ''A'' is skew-symmetric if and only if
:<math>\langle Ax,y \rangle = - \langle x, Ay\rangle \quad \forall x,y\in\Bbb{R}^n.</math>
This is also equivalent to <math>\langle x,Ax \rangle = 0</math> for all ''x'' (one implication being obvious, the other a plain consequence of <math>\langle x+y, A(x+y)\rangle =0</math> for all x and y).
Since this definition is independent of the choice of [[basis (linear algebra)|basis]], skew-symmetry is a property that depends only on the [[linear operator]] A and a choice of [[inner product]].

All [[main diagonal]] entries of a skew-symmetric matrix must be zero, so the [[trace of a matrix|trace]] is zero. If {{nowrap|1=''A'' = (''aij'')}} is skew-symmetric, {{nowrap|1=''aij'' = −''aji''}}; hence {{nowrap|1=''aii'' = 0.}}

3x3 skew symmetric matrices can be used to represent [[cross product]]s as matrix multiplications.

===Determinant===

Let ''A'' be a ''n''×''n'' skew-symmetric matrix. The [[determinant]] of ''A'' satisfies

:det(''A'') = det(''A''T) = det(−''A'') = (−1)''n''det(''A''). Hence {{nowrap|1=det(''A'') = 0}} when ''n'' is odd.
In particular, if ''n'' is odd, and since the underlying field is not of characteristic 2, the determinant vanishes. This result is called '''Jacobi's theorem''', after [[Carl Gustav Jacobi]] (Eves, 1980).

The even-dimensional case is more interesting. It turns out that the determinant of ''A'' for ''n'' even can be written as the square of a [[polynomial]] in the entries of ''A'', which was first proved by Cayley:<ref>{{cite journal
| last1 = Cayley
| first1 = Arthur
| authorlink1 = Arthur Cayley
| year = 1847
| title = Sur les determinants gauches
| trans_title = On skew determinants
| journal = Crelle's Journal
| volume = 38
| pages = 93–96
}}</ref>

:det(''A'') = Pf(''A'')2.

This polynomial is called the ''[[Pfaffian]]'' of ''A'' and is denoted Pf(''A''). Thus the determinant of a real skew-symmetric matrix is always non-negative. However this last fact can be proved in an elementary way as follows: the eigenvalues of a real skew-symmetric matrix are purely imaginary (see below) and to every eigenvalue there corresponds the conjugate eigenvalue with the same multiplicity; therefore, as the determinant is the product of the eigenvalues, each one repeated according to its multiplicity, it follows at once that the determinant, if it is not 0, is a positive real number.

The number of distinct terms ''s''(''n'') in the expansion of the determinant of a skew-symmetric matrix of order ''n'' has been considered already by Cayley, Sylvester, and Pfaff. Due to cancellations, this number is quite small as compared the number of terms of a generic matrix of order ''n'', which is ''n''!. The sequence ''s''(''n'') {{OEIS|A002370}} is
:1, 0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, …
and it is encoded in the [[exponential generating function]]
:<math>\sum_{n=0}^\infty \frac{s(n)}{n!}x^n=(1-x^2)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right).</math>
The latter yields to the asymptotics (for ''n'' even)
:<math>s(n)=\pi^ {-\frac{1}{2} } 2^ {\frac{3}{4}} \Gamma\left (3/4 \right) (n/e)^ {n-\frac{1}{4} } \left (1+O\big(\frac{1}{n}\big)\right).</math>

The number of positive and negative terms are approximatively a half of the total, although their difference takes larger and larger positive and negative values as ''n'' increases {{OEIS|A167029}}.

=== Spectral theory ===

Since a matrix is similar to its own transpose, they must have the same eigenvalues. It follows that the [[eigenvalue]]s of a skew-symmetric matrix always come in pairs ±λ (except in the odd-dimensional case where there is an additional unpaired 0 eigenvalue). From the spectral theorem, for a real skew-symmetric matrix the nonzero eigenvalues are all pure [[imaginary number|imaginary]] and thus are of the form ''i''λ1, −''i''λ1, ''i''λ2, −''i''λ2, … where each of the λ''k'' are real.

Real skew-symmetric matrices are [[normal matrix|normal matrices]] (they commute with their [[adjoint matrix|adjoints]]) and are thus subject to the [[spectral theorem]], which states that any real skew-symmetric matrix can be diagonalized by a [[unitary matrix]]. Since the eigenvalues of a real skew-symmetric matrix are imaginary it is not possible to diagonalize one by a real matrix. However, it is possible to bring every skew-symmetric matrix to a [[block matrix|block diagonal]] form by an [[orthogonal matrix|orthogonal transformation]]. Specifically, every 2''n'' × 2''n'' real skew-symmetric matrix can be written in the form ''A'' = ''Q'' Σ ''Q''T where ''Q'' is orthogonal and
:<math>\Sigma = \begin{bmatrix}
\begin{matrix}0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\
0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \cdots & \begin{matrix}0 & \lambda_r\\ -\lambda_r & 0\end{matrix} \\
& & & & \begin{matrix}0 \\ & \ddots \\ & & 0 \end{matrix}
\end{bmatrix}</math>
for real λ''k''. The nonzero eigenvalues of this matrix are ±''i''λ''k''. In the odd-dimensional case Σ always has at least one row and column of zeros.

More generally, every complex skew-symmetric matrix can be written in the form ''A'' = ''U'' Σ ''U''T where ''U'' is unitary and Σ has the block-diagonal form given above with complex λ''k''. This is an example of the Youla decomposition of a complex square matrix.<ref>{{cite journal|doi=10.4153/CJM-1961-059-8|first=D. C. |last=Youla|title=A normal form for a matrix under the unitary congruence group|journal=Canad. J. Math. |volume=13|pages=694–704 |year=1961}}</ref>

==Alternating forms==

We begin with a special case of the definition. An '''alternating form''' φ on a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'', not of [[characteristic (algebra)|characteristic]] 2, is defined to be a [[bilinear form]]

: φ : ''V'' × ''V'' → ''K''

such that

: φ(''v'',''w'') = −φ(''w'',''v'').

This defines a form with desirable properties for vector spaces over fields of characteristic not equal to 2, but in a vector space over a field of characteristic 2, the definition fails, as every element is its own additive inverse. That is, symmetric and alternating forms are equivalent, which is clearly false in the case above. However, we may extend the definition to vector spaces over fields of characteristic 2 as follows:

In the case where the [[vector space]] ''V'' is over a field of arbitrary [[characteristic (algebra)|characteristic]] including characteristic 2, we may state that for all vectors ''v'' in ''V''

: φ(''v'',''v'') = 0.

This reduces to the above case when the field is not of characteristic 2 as seen below

: 0 = φ(''v'' + ''w'',''v'' + ''w'') = φ(''v'',''v'') + φ(''v'',''w'') + φ(''w'',''v'') + φ(''w'',''w'') = φ(''v'',''w'') + φ(''w'',''v'')

Whence,

: φ(''v'',''w'') = −φ(''w'',''v'').

Thus, we have a definition that now holds for vector spaces over fields of all characteristics.

Such a φ will be represented by a skew-symmetric matrix ''A'', ''φ(v, w) = vTAw'', once a [[basis (linear algebra)|basis]] of ''V'' is chosen; and conversely an ''n''×''n'' skew-symmetric matrix ''A'' on ''K''''n'' gives rise to an alternating form sending ''(v, w)'' to ''vTAw''.

==Infinitesimal rotations==
{{main|Euler's rotation theorem#Generators of rotations}}
{{main|Rotation matrix#Infinitesimal rotations}}
{{main|Infinitesimal strain theory#Infinitesimal rotation tensor}}
Skew-symmetric matrices over the field of real numbers form the [[tangent space]] to the real [[orthogonal group]] O(''n'') at the identity matrix; formally, the [[special orthogonal Lie algebra]]. In this sense, then, skew-symmetric matrices can be thought of as ''infinitesimal rotations''.

Another way of saying this is that the space of skew-symmetric matrices forms the [[Lie algebra]] o(''n'') of the [[Lie group]] O(''n'').
The Lie bracket on this space is given by the [[commutator]]:

:<math>[A,B] = AB - BA.\,</math>

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric:

:<math>[A,B]^{\mathsf{T}} = B^{\mathsf{T}}A^{\mathsf{T}}-A^{\mathsf{T}}B^{\mathsf{T}} = BA-AB = -[A,B] \, .</math>

The [[matrix exponential]] of a skew-symmetric matrix ''A'' is then an [[orthogonal matrix]] ''R'':

:<math>R=\exp(A)=\sum_{n=0}^\infty \frac{A^n}{n!}.</math>

The image of the [[exponential map]] of a Lie algebra always lies in the [[Connected space|connected component]] of the Lie group that contains the identity element. In the case of the Lie group O(''n''), this connected component is the [[special orthogonal group]] SO(''n''), consisting of all orthogonal matrices with determinant 1. So ''R'' = exp(''A'') will have determinant +1. Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that ''every'' orthogonal matrix with unit determinant can be written as the exponential of some skew-symmetric matrix. In the particular important case of dimension ''n''=2, the exponential representation for an orthogonal matrix reduces to the well-known [[complex number#Polar form|polar form]] of a complex number of unit modulus. Indeed, if n=2, a special orthogonal matrix has the form
:<math>\begin{bmatrix}
a & -b \\
b & \,a \end{bmatrix},</math>
with a2+b2=1. Therefore, putting ''a''=cos''θ'' and ''b''=sin''θ'', it can be written
:<math>\begin{bmatrix}
\cos\,\theta & -\sin\,\theta \\
\sin\,\theta & \,\cos\,\theta \end{bmatrix}=
\exp\left( \theta
\begin{bmatrix}
0 & -1 \\
1 &\,0 \end{bmatrix}
\right),
</math>
which corresponds exactly to the polar form cos''θ'' + ''i''sin''θ'' = e''iθ'' of a complex number of unit modulus.

The exponential representation of an orthogonal matrix of order ''n'' can also be obtained starting from the fact that in dimension ''n'' any special orthogonal matrix ''R'' can be written as R = Q S QT, where Q is orthogonal and S is a [[Block matrix#Block diagonal matrix|block diagonal matrix]] with <math>\scriptstyle\lfloor {n/2}\rfloor</math> blocks of order 2, plus one of order 1 if n is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ of the form above, S=exp(Σ), so that R = Q exp(Σ)QT = exp(Q Σ QT), exponential of the skew-symmetric matrix Q Σ QT. Conversely, the surjectivity of the exponential map, together with the above mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.

== Coordinate-free ==
More intrinsically (i.e., without using coordinates), skew-symmetric linear transformations on a vector space ''V'' with an [[inner product]] may be defined as the [[bivector]]s on the space, which are sums of simple bivectors ([[blade (geometry)|2-blades]]) <math>v \wedge w</math>. The correspondence is given by the map <math>v \wedge w \mapsto v^* \otimes w - w^* \otimes v,</math> where <math>v^*</math> is the covector dual to the vector <math>v</math>; in orthonormal coordinates these are exactly the elementary skew-symmetric matrices. This characterization is used in interpreting the [[curl (mathematics)|curl]] of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.

== Skew-symmetrizable matrix ==
An ''n''-by-''n'' matrix ''A'' is said to be '''skew-symmetrizable''' if there exist an invertible [[diagonal matrix]] ''D'' and skew-symmetric matrix ''S'' such that {{nowrap|1=''A'' = ''DS''.}} For '''real''' ''n''-by-''n'' matrices, sometimes the condition for ''D'' to have positive entries is added.<ref>{{cite arXiv |last1= Fomin |last2= Zelevinsky |first1= Sergey |first2= Andrei |eprint= math/010415 |title= Cluster algebras I: Foundations |year= 2001 |version= 1 | page=15}}</ref>

==See also==
*[[Symmetric matrix]]
*[[Skew-Hermitian matrix]]
*[[Symplectic matrix]]
*[[Symmetry in mathematics]]

==References==
{{reflist}}

==Further reading==
* {{cite book
|last=Eves
|first=Howard
|authorlink=Howard Eves
|title=Elementary Matrix Theory
|publisher=Dover Publications
|year=1980
|isbn=978-0-486-63946-8}}
* {{SpringerEOM
|urlname=S/s085720
|title=Skew-symmetric matrix
|last=Suprunenko|first=D. A.}}
*{{cite journal
|title=On the number of distinct terms in the expansion of symmetric and skew determinants.
|last=Aitken|first=A. C.
|year=1944
|journal=Edinburgh Math. Notes }}

== External links ==
* {{Cite web|url=http://mathworld.wolfram.com/AntisymmetricMatrix.html |title=Antisymmetric matrix|work= Wolfram Mathworld}}
* {{Cite web|url=http://www.tu-chemnitz.de/mathematik/hapack/|title=HAPACK - Software for (Skew-)Hamiltonian Eigenvalue Problems|

first1=Peter |last1=Benner|
first2=Daniel |last2=Kressner
}}
* {{Cite journal|doi=10.1145/355791.355799|title=Algorithm 530: An Algorithm for Computing the Eigensystem of Skew-Symmetric Matrices and a Class of Symmetric Matrices [F2]|year=1978|last1=Ward|first1=R. C.|last2=Gray|first2=L. J.|journal=ACM Transactions on Mathematical Software|volume=4|issue=3|pages=286}} [http://www.netlib.org/toms/530 Fortran] [http://jblevins.org/mirror/amiller/toms530.f90 Fortran90]

{{DEFAULTSORT:Skew-Symmetric Matrix}}
[[Category:Matrices]]

Talk:Spectrum (functional analysis)

2013-09-02T12:23:33Z

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Lebesgue constant (interpolation)

2013-06-10T11:54:47Z

Jitse Niesen: revert - the "References" section is for references cited or used when writing the Wikipedia article

[[Image:MorinSurfaceFromTheTop.PNG|thumb|220px|right|Morin surface seen from the top]]
[[Image:MorinSurfaceCrossView.PNG|thumb|220px|right|Morin surface seen from the side]]

The '''Morin surface''' is the [[half-way model]] of the [[Smale's paradox|sphere eversion]] discovered by [[Bernard Morin]]. It features fourfold rotational [[symmetry]].

If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through [[homotopy]] into a Morin surface, half of the outwardly visible Morin surface will be green, and half red:

[[Image:MorinSurfaceAsSphere'sInsideVersusOutside.PNG]] 
''Half of a Morin surface corresponds to the exterior (green) of the sphere to which it is homeomorphic, and the other symmetric half to the interior (red).''

Then, rotating the surface 90° around its axis of symmetry will exchange its colors, i.e. will exchange the inner-outer polarity of the orientable surface, so that retracing the steps of the homotopy at exactly the same position back to the original sphere after having so rotated the Morin surface will yield a sphere whose outer surface is red and whose inner surface is green: a sphere which has been turned inside out. The following is a summary of the eversion:

1. sphere: green outside, red inside... 
2. transforms into... 
3. Morin surface, 
3'. Morin surface rotated 90°... 
2'. inversely transforms into... 
1'. sphere: red outside, green inside.

==Structure of the Morin surface==
The Morin surface can be separated into four congruent quarter sections. These sections may be here called section East, section South, section West, and section North, or — respectively — section 0, section 1, section 2, and section 3. 
[[Image:MorinSurfaceSectionEast.PNG]] 
::''Section East of the Morin surface.''

The Morin surface has a quadruple point through which passes its axis of symmetry. This quadruple point is the starting point and the end point of six lines of double points. Each of the quarter sections is bounded by three of these lines of double points, so that each quarter section is homeomorphic to a triangle. Section East is now shown schematically: 
[[Image:MorinSurfaceQuarterSection.PNG|300px]] 
The diagram shows section East bounded by three loops: ABCDA, AEFGA, and AHIJA. The third loop, AHIJA, is a line of double points where section East intersects with itself. Loop ABCDA is only a line of double points when section East is joined to section West, and loop AEFGA is only a line of double points when section East is joined to section South. Point is the quadruple point which is actually the overlapping of four different points: A0, A1, A2, A3.

This is how section East is joined to the other sections: let each of its bounding loops be specified by an ordered quintuple of points, then
: <math> (A_1, B, C, D, A_3) = (A_1, D'', C'', B'', A_3) </math>
: <math> (A_2, E, F, G, A_3) = (A_2, H', I', J', A_3) </math>
: <math> (A_1, H, I, J, A_2) = (A_1, E''', F''', G''', A_2) </math>
where unprimed points belong to section 0 (East), primed points belong to section 1 (South), double-primed points belong to section 2 (West), and triple-primed points belong to section 3 (North).

The remaining three loops connect sections as follows:
:<math> (A_2, B', C', D', A_0) = (A_2, D''', C''', B''', A_0) </math>
:<math> (A_3, E', F', G', A_0) = (A_3, H'', I'', J'', A_0) </math>
:<math> (A_0, E'', F'', G'', A_1) = (A_0, H''', I''', J''', A_1). </math>

Section East has, considered just by itself, one loop of double points: AHIJA. If the surface is unwound and flattened the result will be the following: 
[[Image:MorinSurfaceQuarterSectionFlattened.PNG|300px]] 
which is homeomorphic to a triangle: 
[[Image:MorinSurfaceQuarterSectionTriangulated.PNG]]

Joining the four triangular sections at their seams will produce a [[tetrahedron]]: 
[[Image:MorinSurfaceQuarterSectionsJoined.PNG|500px]] 
which is homeomorphic to a sphere, which shows the Morin surface is a self-intersecting sphere.

==Morin surface gallery==
[[Image:QuartetOfMorinSurfaces(WithoutPassageBarriers).PNG|600px]] 
:''Four different views of the Morin surface: the first two are shown with "passage barriers" cut out, the last two are views from the "bottom".''

==See also==
*[[Smale's paradox]]

==External links==
*[http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/eversion.html "Turning a Sphere Inside Out"]
*[http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm A History of Sphere Eversions]

[[Category:Surfaces]]

LU decomposition

2012-08-21T13:04:09Z

Jitse Niesen: add requested references; add references to existence and uniqueness section

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Simpson's rule

2012-08-21T09:20:54Z

Jitse Niesen: /* Averaging the midpoint and the trapezoidal rules */ revert and explain big O notation

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2012-07-03T11:48:53Z

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2012-03-24T12:16:06Z

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{{Continuum mechanics|cTopic=[[Solid mechanics]]}}
In [[continuum mechanics]], the '''finite strain theory'''—also called '''large strain theory''', or '''large deformation theory'''—deals with [[Deformation (mechanics)|deformations]] in which both rotations and strains are arbitrarily large, i.e. invalidates the assumptions inherent in [[infinitesimal strain theory]]. In this case, the undeformed and deformed configurations of the [[Continuum mechanics|continuum]] are significantly different and a clear distinction has to be made between them. This is commonly the case with [[elastomer]]s, [[Plasticity (physics)|plastically-deforming]] materials and other [[fluid]]s and biological [[soft tissue]].

==Displacement==

[[File:Displacement of a continuum.svg|400px|right|thumb|Figure 1. Motion of a continuum body.]]
The displacement of a body has two components: a [[rigid body|rigid-body]] displacement and a deformation.
* A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.
* Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration <math>\kappa_0(\mathcal B)\,\!</math> to a current or deformed configuration <math>\kappa_t(\mathcal B)\,\!</math> (Figure 1).

A change in the configuration of a continuum body can be described by a [[displacement field (mechanics)|displacement field]]. A ''displacement field'' is a [[vector field]] of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. Relative displacement between particles occurs if and only if deformation has occurred. If displacement occurs without deformation, then it is deemed a rigid-body displacement.

===Material coordinates (Lagrangian description)===

The displacement of particles indexed by variable {{math|i}} may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration <math>P_i\,\!</math> and deformed configuration <math>p_i\,\!</math> is called the [[displacement (vector)|displacement vector]]. Using <math>\mathbf{X}\,\!</math> in place of <math>P_i\,\!</math> and <math>\mathbf{x}\,\!</math> in place of <math>p_i\,\!</math>, both of which are vectors from the origin of the coordinate system to each respective point, we have the [[Continuum mechanics#Lagrangian description|Lagrangian description]] of the displacement vector:

:<math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i\,\!</math>

Where <math>\mathbf e_i\,\!</math> is the [[unit vector]] that defines the [[Basis (linear algebra)|basis]] of the material (body-frame) coordinate system.

Expressed in terms of the material coordinates, the displacement field is:

:<math>\mathbf u(\mathbf X,t) = \mathbf b(\mathbf X,t)+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J\,\!</math>

The [[partial derivative]] of the displacement vector with respect to the material coordinates yields the ''material displacement gradient tensor'' <math>\nabla_{\mathbf X}\mathbf u\,\!</math>. Thus we have,
:<math>
\begin{align}
\mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \qquad &\text{or}& \qquad u_i = x_i-\delta_{iJ}X_J =x_i -X_i\\
\nabla_{\mathbf X}\mathbf u &= \nabla_{\mathbf X}\mathbf x - \mathbf I = \mathbf F - \mathbf I \qquad &\text{or}& \qquad \frac{\partial u_i}{\partial X_K} =\frac{\partial x_i}{\partial X_K}-\delta_{iK} = F_{iK} - \delta_{iK}
\end{align}
</math>
where <math>\mathbf F\,\!</math> is the [[Finite strain theory#Deformation gradient tensor|''deformation gradient tensor'']].

===Spatial coordinates (Eulerian description)===

In the [[Continuum mechanics#Eulerian description|Eulerian description]], the vector joining the positions of a particle <math>P\,\!</math> in the undeformed configuration and deformed configuration is called the [[displacement (vector)|displacement vector]]:

:<math>\mathbf U(\mathbf x,t)=U_i\mathbf E_i\,\!</math>

Where <math>\mathbf E_i\,\!</math> is the unit vector that defines the basis of the spatial (lab-frame) coordinate system.

Expressed in terms of spatial coordinates, the displacement field is:

:<math>\mathbf U(\mathbf x,t) = \mathbf b(\mathbf x,t)+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,\!</math>

The partial derivative of the displacement vector with respect to the spatial coordinates yields the ''spatial displacement gradient tensor'' <math>\nabla_{\mathbf x}\mathbf U\,\!</math>. Thus we have,
:<math>
\begin{align}
\mathbf U(\mathbf x,t) &= \mathbf x - \mathbf X(\mathbf x,t) \qquad &\text{or}& \qquad U_J = \delta_{Ji}x_i-X_J =x_J-X_J \\
\nabla_{\mathbf x}\mathbf U &= \mathbf I - \nabla_{\mathbf x}\mathbf X = \mathbf I -\mathbf F^{-1} \qquad &\text{or}& \qquad \frac{\partial U_J}{\partial x_k} = \delta_{Jk}-\frac{\partial X_J}{\partial x_k} = \delta_{Jk} - F^{-1}_{Jk} \,.
\end{align}
</math>

===Relationship between the material and spatial coordinate systems===

<math>\alpha_{Ji}\,\!</math> are the [[direction cosine]]s between the material and spatial coordinate systems with unit vectors <math>\mathbf E_J\,\!</math> and <math>\mathbf e_i\,\!</math>, respectively. Thus

:<math>\mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}\,\!</math>

The relationship between <math>u_i\,\!</math> and <math>U_J\,\!</math> is then given by

:<math>u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i\,\!</math>

Knowing that
:<math>\mathbf e_i = \alpha_{iJ}\mathbf E_J\,\!</math>
then
:<math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)\,\!</math>

===Combining the coordinate systems of deformed and undeformed configurations===

It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in <math>\mathbf b=0\,\!</math>, and the direction cosines become [[Kronecker delta]]s, i.e.

:<math>\mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}\,\!</math>

Thus in material (deformed) coordinates, the displacement may be expressed as:

:<math>\mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J\,\!</math>

And in spatial (undeformed) coordinates, the displacement may be expressed as:

:<math>\mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J \,\!</math>

==Deformation gradient tensor==
[[File:Deformation.png|400px|right|thumb|Figure 2. Deformation of a continuum body.]]
The deformation gradient tensor <math>\mathbf F(\mathbf X,t)=F_{jK}\mathbf e_j\otimes\mathbf I_K\,\!</math> is related to both the reference and current configuration, as seen by the unit vectors <math>\mathbf e_j\,\!</math> and <math>\mathbf I_K\,\!</math>, therefore it is a ''[[two-point tensor]]''.

Due to the assumption of continuity of <math>\chi(\mathbf X,t)\,\!</math>, <math>\mathbf F\,\!</math> has the inverse <math>\mathbf H= \mathbf F^{-1}\,\!</math>, where <math>\mathbf H\,\!</math> is the ''spatial deformation gradient tensor''. Then, by the [[implicit function theorem]],<ref name=Lubliner2008>{{cite book
| last = Lubliner
| first = Jacob
| title = Plasticity Theory (Revised Edition)
| publisher = Dover Publications
| year = 2008
| url = http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf
| isbn = 0-486-46290-0}}</ref> the [[Jacobian matrix and determinant|Jacobian]] determinant <math>J(\mathbf X,t)\,\!</math> must be [[nonsingular]], i.e. <math>J(\mathbf X,t)=\det \mathbf F(\mathbf X,t)\neq 0\,\!</math>

The ''material deformation gradient tensor'' <math>\mathbf F(\mathbf X,t)=F_{jK}\mathbf e_j\otimes\mathbf I_K\,\!</math> is a [[Tensor#Tensor rank|second-order tensor]] that represents the gradient of the mapping function or functional relation <math>\chi(\mathbf X,t)\,\!</math>, which describes the [[Continuum mechanics|motion of a continuum]]. The material deformation gradient tensor characterizes the local deformation at a material point with position vector <math>\mathbf X\,\!</math>, i.e. deformation at neighbouring points, by transforming ([[linear transformation]]) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function <math>\chi(\mathbf X,t)\,\!</math>, i.e. [[differentiable function]] of <math> \mathbf {X} \,\!</math> and time <math>t\,\!</math>, which implies that [[Fracture|crack]]s and voids do not open or close during the deformation. Thus we have,

:<math>
\begin{align}
d\mathbf{x} &=\frac {\partial \mathbf{x}} {\partial \mathbf {X}}\,d\mathbf{X}
\qquad &\text{or}& \qquad
dx_j =\frac{\partial x_j}{\partial X_K}\,dX_K \\
&= \nabla \chi(\mathbf X,t) \,d\mathbf{X} = \mathbf F(\mathbf X,t) \,d\mathbf{X}
\qquad &\text{or}& \qquad
dx_j =F_{jK}\,dX_K \,.
\end{align}
\,\!</math>

===Relative displacement vector===

Consider a [[Continuum mechanics|particle or material point]] <math>P\,\!</math> with position vector <math>\mathbf X=X_I\mathbf I_I\,\!</math> in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by <math>p\,\!</math> in the new configuration is given by the vector position <math>\mathbf{x}=x_i\mathbf e_i\,\!</math>. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.

Consider now a material point <math>Q \,\!</math> neighboring <math>P\,\!</math>, with position vector <math>\mathbf{X}+ \Delta \mathbf{X}=(X_I+\Delta X_I)\mathbf I_I\,\!</math>. In the deformed configuration this particle has a new position <math>q\,\!</math> given by the position vector <math>\mathbf{x}+ \Delta \mathbf{x}\,\!</math>. Assuming that the line segments <math>\Delta X\,\!</math> and <math>\Delta \mathbf x\,\!</math> joining the particles <math>P\,\!</math> and <math>Q\,\!</math> in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as <math>d\mathbf X\,\!</math> and <math>d\mathbf x\,\!</math>. Thus from Figure 2 we have

:<math>\begin{align}
\mathbf{x}+ d\mathbf{x}&= \mathbf{X}+d\mathbf{X}+\mathbf{u}(\mathbf{X}+d\mathbf{X}) \\
d\mathbf{x} &= \mathbf{X}-\mathbf{x}+d\mathbf{X}+\mathbf{u}(\mathbf{X}+d\mathbf{X}) \\
&= d\mathbf{X}+\mathbf{u}(\mathbf{X}+d\mathbf{X})-\mathbf{u}(\mathbf{X}) \\
&= d\mathbf{X}+d\mathbf{u} \\
\end{align}
\,\!</math>

where <math>\mathbf {du}\,\!</math> is the '''relative displacement vector''', which represents the relative displacement of <math>Q\,\!</math> with respect to <math>P\,\!</math> in the deformed configuration.

====Taylor approximation====

For an infinitesimal element <math>d\mathbf X\,\!</math>, and assuming continuity on the displacement field, it is possible to use a [[Taylor series expansion]] around point <math>P\,\!</math>, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle <math>Q\,\!</math> as
:<math>
\begin{align}
\mathbf{u}(\mathbf{X}+d\mathbf{X})&=\mathbf{u}(\mathbf{X})+d\mathbf{u} \quad & \text{or} & \quad u_i^* = u_i+du_i \\
&\approx \mathbf{u}(\mathbf{X})+\nabla_{\mathbf X}\mathbf u\cdot d\mathbf X \quad & \text{or} & \quad u_i^* \approx u_i+\frac{\partial u_i}{\partial X_J}dX_J \,.
\end{align}
\,\!</math>
Thus, the previous equation <math>d\mathbf x= d\mathbf{X}+d\mathbf{u}\,\!</math> can be written as
:<math>\begin{align}
d\mathbf x&=d\mathbf X+d\mathbf u \\
&=d\mathbf X+\nabla_{\mathbf X}\mathbf u\cdot d\mathbf X\\
&=\left(\mathbf I + \nabla_{\mathbf X}\mathbf u\right)d\mathbf X\\
&=\mathbf F d\mathbf X
\end{align}
\,\!</math>

=== Time-derivative of the deformation gradient ===
Calculations that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion into [[differential geometry]]<ref name=Yavari>A. Yavari, J.E. Marsden, and M. Ortiz, On spatial and material covariant balance laws in elasticity, Journal of Mathematical Physics, 47, 2006, 042903; pp. 1-53.</ref> but we avoid those issues in this article.

The time derivative of <math>\mathbf{F}</math> is
:<math>
\dot{\mathbf{F}} = \frac{\partial \mathbf{F}}{\partial t} = \frac{\partial}{\partial t}\left[\frac{\partial \mathbf{x}(\mathbf{X}, t)}{\partial \mathbf{X}}\right] = \frac{\partial}{\partial \mathbf{X}}\left[\frac{\partial \mathbf{x}(\mathbf{X}, t)}{\partial t}\right] = \frac{\partial}{\partial \mathbf{X}}\left[\mathbf{V}(\mathbf{X}, t)\right]
</math>
where <math>\mathbf{V}</math> is the velocity. The derivative on the right hand side represents a '''material velocity gradient'''. It is common to convert that into a spatial gradient, i.e.,
:<math>
\dot{\mathbf{F}} = \frac{\partial}{\partial \mathbf{X}}\left[\mathbf{V}(\mathbf{X}, t)\right] = \frac{\partial}{\partial \mathbf{x}}\left[\mathbf{v}(\mathbf{x},t)\right]\cdot\frac{\partial \mathbf{x}(\mathbf{X}, t)}{\partial \mathbf{X}}
= \boldsymbol{l}\cdot\mathbf{F}
</math>
where <math>\boldsymbol{l}</math> is the '''spatial velocity gradient'''. If the spatial velocity gradient is constant, the above equation can be solved exactly to give
:<math>
\mathbf{F} = e^{\boldsymbol{l}\, t}
</math>
assuming <math>\mathbf{F} = \mathbf{1}</math> at <math>t = 0</math>. There are several methods of computing the [[matrix exponential|exponential]] above.

Related quantities often used in continuum mechanics are the '''rate of deformation tensor''' and the '''spin tensor''' defined, respectively, as:
:<math>
\boldsymbol{d} = \tfrac{1}{2}\left(\boldsymbol{l} + \boldsymbol{l}^T\right) \,,~~
\boldsymbol{w} = \tfrac{1}{2}\left(\boldsymbol{l} - \boldsymbol{l}^T\right) \,.
</math>
The rate of deformation tensor gives the rate of stretching of line elements while the spin tensor indicates the rate of rotation or [[vorticity]] of the motion.

==Transformation of a surface and volume element==

To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as
:<math>
da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N}
\,\!</math>

where <math>da\,\!</math> is an area of a region in the deformed configuration, <math>dA\,\!</math> is the same area in the reference configuration, and <math>\mathbf{n}\,\!</math> is the outward normal to the area element in the current configuration while <math>\mathbf{N}\,\!</math> is the outward normal in the reference configuration, <math>\mathbf{F}\,\!</math> is the [[deformation gradient]], and <math>J = \det\mathbf{F}\,\!</math>.

The corresponding formula for the transformation of the volume element is
:<math>
dv = J~dV
\,\!</math>

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of Nanson's relation
|-
|To see how this formula is derived, we start with the oriented area elements
in the reference and current configurations:
:<math>
d\mathbf{A} = dA~\mathbf{N} ~;~~ d\mathbf{a} = da~\mathbf{n}
\,\!</math>
The reference and current volumes of an element are
:<math>
dV = d\mathbf{A}^{T}\cdot d\mathbf{L} ~;~~ dv = d\mathbf{a}^{T} \cdot d\mathbf{l}
\,\!</math>
where <math>d\mathbf{l} = \mathbf{F}\cdot d\mathbf{L}\,\!</math>.

Therefore,
:<math>
d\mathbf{a}^{T} \cdot d\mathbf{l}= dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L}
\,\!</math>
or,
:<math>
d\mathbf{a}^{T} \cdot \mathbf{F}\cdot d\mathbf{L} = dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L}
\,\!</math>
so,
:<math>
d\mathbf{a}^{T} \cdot \mathbf{F} = J~d\mathbf{A}^{T}
\,\!</math>
So we get
:<math>
d\mathbf{a} = J~\mathbf{F}^{-T} \cdot d\mathbf{A}
\,\!</math>
or,
:<math>
da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N}\qquad \qquad \square
\,\!</math>
|}

==Polar decomposition of the deformation gradient tensor==
[[File:Polar decomposition of F.png|400px|right|thumb|Figure 3. Representation of the polar decomposition of the deformation gradient]]

The deformation gradient <math>\mathbf{F}\,\!</math>, like any second-order tensor, can be decomposed, using the [[polar decomposition]] theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.

:<math>\mathbf{F} = \mathbf{R}\mathbf{U} = \mathbf{V} \mathbf{R}\,\!</math>

where the tensor <math>\mathbf{R}\,\!</math> is a [[proper orthogonal tensor]], i.e. <math>\mathbf R^{-1}= \mathbf R^T\,\!</math> and <math>\det \mathbf R=+1\,\!</math>, representing a rotation; the tensor <math>\mathbf{U}\,\!</math> is the ''right stretch tensor''; and <math>\mathbf{V}\,\!</math> the ''left stretch tensor''. The terms ''right'' and ''left'' means that they are to the right and left of the rotation tensor <math>\mathbf{R}\,\!</math>, respectively. <math>\mathbf{U}\,\!</math> and <math>\mathbf{V}\,\!</math> are both [[positive-definite matrix|positive definite]], i.e. <math>\mathbf x\cdot\mathbf U \cdot \mathbf x\ge 0 \,\!</math> and <math>\mathbf x\cdot\mathbf V \cdot \mathbf x\ge 0 \,\!</math>, and [[symmetric tensor]]s, i.e. <math>\mathbf U=\mathbf U^T\,\!</math> and <math>\mathbf V=\mathbf V^T\,\!</math>, of second order.

This decomposition implies that the deformation of a line element <math>d\mathbf X\,\!</math> in the undeformed configuration onto <math>d\mathbf x\,\!</math> in the deformed configuration, i.e. <math>d\mathbf x=\mathbf F \,d\mathbf X\,\!</math>, may be obtained either by first stretching the element by <math>\mathbf U\,\!</math>, i.e. <math>d\mathbf x'=\mathbf U \,d\mathbf X\,\!</math>, followed by a rotation <math>\mathbf R\,\!</math>, i.e. <math>d\mathbf x=\mathbf R \,d\mathbf x'\,\!</math>; or equivalently, by applying a rigid rotation <math>\mathbf R\,\!</math> first, i.e. <math>d\mathbf x'=\mathbf R \,d\mathbf X\,\!</math>, followed later by a stretching <math>\mathbf V\,\!</math>, i.e. <math>d\mathbf x=\mathbf V \,d\mathbf x'\,\!</math> (See Figure 3).

It can be shown that,

:<math>\mathbf V=\mathbf R\cdot\mathbf U \cdot \mathbf R^T\,\!</math>

so that <math>\mathbf U\,\!</math> and <math>\mathbf V\,\!</math> have the same [[eigenvalues]] or principal stretches, but different [[eigenvectors]] or ''principal directions'' <math>\mathbf{N}_i\,\!</math> and <math>\mathbf{n}_i\,\!</math>, respectively. The principal directions are related by

:<math>\mathbf{n}_i = \mathbf{R} \mathbf{N}_i. \,\!</math>

This polar decomposition is unique as <math>\mathbf F\,\!</math> is non-symmetric.

==Deformation tensors==
Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy-Green deformation tensors.

Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotation-independent measures of deformation in [[continuum mechanics]]. As a rotation followed by its inverse rotation leads to no change (<math>\mathbf{R}\mathbf{R}^T=\mathbf{R}^T\mathbf{R}=\mathbf{1}\,\!</math>) we can exclude the rotation by multiplying <math>\mathbf{F}\,\!</math> by its [[transpose]].

===The Right Cauchy-Green deformation tensor===
In 1839, [[George Green]] introduced a deformation tensor known as the ''right Cauchy-Green deformation tensor'' or ''Green's deformation tensor'', defined as:<ref name=note1>The [[IUPAC]] recommends that this tensor be called the Cauchy strain tensor.</ref><ref name=IUPAC>{{cite journal|author=A. Kaye, R. F. T. Stepto, W. J. Work,
J. V. Aleman (Spain), A. Ya. Malkin|year=1998|title=Definition of terms relating to the non-ultimate mechanical properties of polymers|journal= Pure & Appl. Chem|volume= 70|issue=3|pages=701–754|url=http://old.iupac.org/reports/1998/7003kaye/index.html}}</ref>

:<math>\mathbf C=\mathbf F^T\mathbf F=\mathbf U^2 \qquad \text{or} \qquad C_{IJ}=F_{kI}~F_{kJ} = \frac {\partial x_k} {\partial X_I} \frac {\partial x_k} {\partial X_J}.\,\!</math>

Physically, the Cauchy-Green tensor gives us the square of local change in distances due to deformation, i.e. <math>d\mathbf x^2=d\mathbf X\cdot\mathbf C d\mathbf X\,\!</math>

Invariants of <math>\mathbf{C}\,\!</math> are often used in the expressions for [[strain energy density function]]s. The most commonly used [[Invariants of tensors|invariants]] are
:<math>
\begin{align}
I_1^C & := \text{tr}(\mathbf{C}) = C_{II} = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \\
I_2^C & := \tfrac{1}{2}\left[(\text{tr}~\mathbf{C})^2 - \text{tr}(\mathbf{C}^2) \right]
= \tfrac{1}{2}\left[(C_{JJ})^2 - C_{IK}C_{KI}\right] = \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\
I_3^C & := \det(\mathbf{C}) = \lambda_1^2\lambda_2^2\lambda_3^2.
\end{align}
\,\!</math>
where <math>\lambda_i\,\!</math> are stretch ratios for the unit fibers that are initially oriented along the directions of three axis in the coordinate systems.

===The Finger deformation tensor===
The [[IUPAC]] recommends<ref name=IUPAC/> that the inverse of the right Cauchy-Green deformation tensor (called the Cauchy tensor in that document), i. e., <math>\mathbf C^{-1}</math>, be called the '''Finger tensor'''. However, that nomenclature is not universally accepted in applied mechanics.

:<math>\mathbf{f}=\mathbf C^{-1}=\mathbf F^{-1}\mathbf F^{-T} \qquad \text{or} \qquad f_{IJ}=\frac {\partial X_I} {\partial x_k} \frac {\partial X_J} {\partial x_k}\,\!</math>

===The Left Cauchy-Green or Finger deformation tensor===
Reversing the order of multiplication in the formula for the right Green-Cauchy deformation tensor leads to the ''left Cauchy-Green deformation tensor'' which is defined as:

:<math>\mathbf B=\mathbf F\mathbf F^T=\mathbf V^2 \qquad \text{or} \qquad B_{ij}=\frac {\partial x_i} {\partial X_K} \frac {\partial x_j} {\partial X_K}\,\!</math>

The left Cauchy-Green deformation tensor is often called the ''Finger deformation tensor'', named after [[Josef Finger]] (1894).<ref name=IUPAC/><ref>Eduardo N. Dvorkin, Marcela B. Goldschmit, 2006 [http://books.google.com/books?id=MVqa05_2QmAC&pg=PA25 Nonlinear Continua], p. 25, Springer ISBN 3-540-24985-0.</ref><ref name=note2>The [[IUPAC]] recommends that this tensor be called the Green strain tensor.</ref>

Invariants of <math>\mathbf{B}\,\!</math> are also used in the expressions for [[strain energy density function]]s. The conventional invariants are defined as
:<math>
\begin{align}
I_1 & := \text{tr}(\mathbf{B}) = B_{ii} = \lambda_1^2 + \lambda_2^2 + \lambda_3^2\\
I_2 & := \tfrac{1}{2}\left[(\text{tr}~\mathbf{B})^2 - \text{tr}(\mathbf{B}^2)\right]
= \tfrac{1}{2}\left(B_{ii}^2 - B_{jk}B_{kj}\right) = \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\
I_3 & := \det\mathbf{B} = J^2 = \lambda_1^2\lambda_2^2\lambda_3^2
\end{align}
\,\!</math>
where <math>J:=\det\mathbf{F}\,\!</math> is the determinant of the deformation gradient.

For nearly incompressible materials, a slightly different set of invariants is used:
:<math>
(\bar{I}_1 := J^{-2/3} I_1 ~;~~ \bar{I}_2 := J^{-4/3} I_2 ~;~~ J=1) ~.
\,\!</math>

===The Cauchy deformation tensor===
Earlier in 1828,<ref>Jirásek,Milan; Bažant, Z. P. (2002) [http://books.google.com/books?id=8mz-xPdvH00C&pg=PA463 Inelastic analysis of structures], Wiley, p. 463 ISBN 0-471-98716-6</ref> [[Augustin Louis Cauchy]] introduced a deformation tensor defined as the inverse of the left Cauchy-Green deformation tensor, <math>\mathbf B^{-1}\,\!</math>. This tensor has also been called the '''Piola tensor'''<ref name=IUPAC /> and the '''Finger tensor'''<ref>J. N. Reddy, David K. Gartling (2000) [http://books.google.com/books?id=sv0VKLL5lWUC&pg=PA317 The finite element method in heat transfer and fluid dynamics], p. 317, CRC Press ISBN 1-4200-8598-0.</ref> in the rheology and fluid dynamics literature.

:<math>\mathbf{c}=\mathbf B^{-1}=\mathbf F^{-T}\mathbf F^{-1} \qquad \text{or} \qquad c_{ij}=\frac {\partial X_K} {\partial x_i} \frac {\partial X_K} {\partial x_j}\,\!</math>

===Spectral representation===
If there are three distinct ''principal stretches'' <math>\lambda_i \,\!</math>, the [[spectral decomposition]]s of <math>\mathbf{C}\,\!</math> and <math>\mathbf{B}\,\!</math> is given by

:<math> \mathbf{C} = \sum_{i=1}^3 \lambda_i^2 \mathbf{N}_i \otimes \mathbf{N}_i \qquad \text{and} \qquad \mathbf{B} = \sum_{i=1}^3 \lambda_i^2 \mathbf{n}_i \otimes \mathbf{n}_i\,\!</math>

Furthermore,

:<math> \mathbf U = \sum_{i=1}^3 \lambda_i \mathbf N_i \otimes \mathbf N_i ~;~~
\mathbf V = \sum_{i=1}^3 \lambda_i \mathbf n_i \otimes \mathbf n_i \,\!</math>
:<math> \mathbf R = \sum_{i=1}^3 \mathbf n_i \otimes \mathbf N_i ~;~~
\mathbf F = \sum_{i=1}^3 \lambda_i \mathbf n_i \otimes \mathbf N_i \,\!</math>

Observe that
:<math>
\mathbf{V} = \mathbf{R}~\mathbf{U}~\mathbf{R}^T =
\sum_{i=1}^3 \lambda_i~\mathbf{R}~(\mathbf{N}_i\otimes\mathbf{N}_i)~\mathbf{R}^T =
\sum_{i=1}^3 \lambda_i~(\mathbf{R}~\mathbf{N}_i)\otimes(\mathbf{R}~\mathbf{N}_i)
\,\!</math>
Therefore the uniqueness of the spectral decomposition also implies that <math>
\mathbf{n}_i = \mathbf{R}~\mathbf{N}_i \,\!</math>. The left stretch (<math>\mathbf{V}\,\!</math>) is also called the ''spatial stretch tensor'' while the right stretch (<math>\mathbf{U}\,\!</math>) is called the ''material stretch tensor''.

The effect of <math>\mathbf{F}\,\!</math> acting on <math>\mathbf{N}_i\,\!</math> is to stretch the vector by <math>\lambda_i\,\!</math> and to rotate it to the new orientation <math>\mathbf{n}_i\,\!</math>, i.e.,
:<math>
\mathbf{F}~\mathbf{N}_i = \lambda_i~(\mathbf{R}~\mathbf{N}_i) = \lambda_i~\mathbf{n}_i
\,\!</math>
In a similar vein,
:<math>
\mathbf{F}^{-T}~\mathbf{N}_i = \cfrac{1}{\lambda_i}~\mathbf{n}_i ~;~~
\mathbf{F}^T~\mathbf{n}_i = \lambda_i~\mathbf{N}_i ~;~~
\mathbf{F}^{-1}~\mathbf{n}_i = \cfrac{1}{\lambda_i}~\mathbf{N}_i ~.
\,\!</math>

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Examples
|-
|'''Uniaxial extension of an incompressible material'''

This is the case where a specimen is stretched in 1-direction with a [[stretch ratio]] of <math>\mathbf{\alpha=\alpha_1}\,\!</math>. If the volume remains constant, the contraction in the other two directions is such that <math>\mathbf{\alpha_1 \alpha_2 \alpha_3 =1}\,\!</math> or <math>\mathbf{\alpha_2=\alpha_3=\alpha^{-0.5}}\,\!</math>. Then:

:<math>\mathbf{F}=\begin{bmatrix} \alpha & 0 & 0 \\
0 & \alpha^{-0.5} & 0 \\
0 & 0 & \alpha^{-0.5} \end{bmatrix}\,\!</math>

:<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} \alpha^2 & 0 & 0 \\
0 & \alpha^{-1} & 0 \\
0 & 0 & \alpha^{-1} \end{bmatrix}\,\!</math>

'''Simple shear'''

<math>\mathbf{F}=\begin{bmatrix} 1 & \gamma & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}\,\!</math>

<math>\mathbf{B}=\begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\
\gamma & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}\,\!</math>

<math>\mathbf{C}=\begin{bmatrix} 1 & \gamma & 0 \\
\gamma & 1+\gamma^2 & 0 \\
0 & 0 & 1 \end{bmatrix}\,\!</math>

'''Rigid body rotation'''

<math>\mathbf{F}=\begin{bmatrix} \cos \theta & \sin \theta & 0 \\
- \sin \theta & \cos \theta & 0 \\
0 & 0 & 1 \end{bmatrix}\,\!</math>

<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} 1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix} = \mathbf{1}\,\!</math>
|}

===Derivatives of stretch===
[[Tensor derivative (continuum mechanics)|Derivatives]] of the stretch with respect to the right Cauchy-Green deformation tensor are used to derive the stress-strain relations of many solids, particularly [[hyperelastic material]]s. These derivatives are
:<math>
\cfrac{\partial\lambda_i}{\partial\mathbf{C}} =
\cfrac{1}{2\lambda_i}~\mathbf{N}_i\otimes\mathbf{N}_i =
\cfrac{1}{2\lambda_i}~\mathbf{R}^T~(\mathbf{n}_i\otimes\mathbf{n}_i)~\mathbf{R} ~;~~ i=1,2,3
\,\!</math>
and follow from the observations that
:<math>
\mathbf{C}:(\mathbf{N}_i\otimes\mathbf{N}_i) = \lambda_i^2 ~;~~~~\cfrac{\partial\mathbf{C}}{\partial\mathbf{C}} = \mathsf{I}^{(s)} ~;~~~~ \mathsf{I}^{(s)}:(\mathbf{N}_i\otimes\mathbf{N}_i)=\mathbf{N}_i\otimes\mathbf{N}_i.
\,\!</math>

===Physical interpretation of deformation tensors===
Let <math>\mathbf{X} = X^i~\boldsymbol{E}_i</math> be a Cartesian coordinate system defined on the undeformed body and let <math>\mathbf{x} = x^i~\boldsymbol{E}_i</math> be another system defined on the deformed body. Let a curve <math>\mathbf{X}(s)</math> in the undeformed body be parametrized using <math>s \in [0,1]</math>. Its image in the deformed body is <math>\mathbf{x}(\mathbf{X}(s))</math>.

The undeformed length of the curve is given by
:<math>
l_X = \int_0^1 \left| \cfrac{d \mathbf{X}}{d s}\cdot\cfrac{d \mathbf{X}}{d s} \right|~ds = \int_0^1 \left| \cfrac{d \mathbf{X}}{d s}\cdot\boldsymbol{I} \cdot\cfrac{d \mathbf{X}}{d s} \right|~ds
</math>
After deformation, the length becomes
:<math>
\begin{align}
l_x & = \int_0^1 \left| \cfrac{d \mathbf{x}}{d s}\cdot\cfrac{d \mathbf{x}}{d s} \right|~ds
= \int_0^1 \left| \left(\cfrac{d \mathbf{x}}{d \mathbf{X}}\cdot\cfrac{d \mathbf{X}}{d s}\right)\cdot
\left(\cfrac{d \mathbf{x}}{d \mathbf{X}}\cdot\cfrac{d \mathbf{X}}{d s}\right) \right|~ds \\
& = \int_0^1 \left| \cfrac{d \mathbf{X}}{d s}\cdot\left[
\left(\cfrac{d \mathbf{x}}{d \mathbf{X}}\right)^T\cdot \cfrac{d \mathbf{x}}{d \mathbf{X}}\right]
\cdot\cfrac{d \mathbf{X}}{d s} \right|~ds
\end{align}
</math>
Note that the right Cauchy-Green deformation tensor is defined as
:<math>
\boldsymbol{C} := \boldsymbol{F}^T\cdot\boldsymbol{F} = \left(\cfrac{d \mathbf{x}}{d \mathbf{X}}\right)^T\cdot \cfrac{d \mathbf{x}}{d \mathbf{X}}
</math>
Hence,
:<math>
l_x = \int_0^1 \left| \cfrac{d \mathbf{X}}{d s}\cdot\boldsymbol{C} \cdot\cfrac{d \mathbf{X}}{d s} \right|~ds
</math>
which indicates that changes in length are characterized by <math>\boldsymbol{C}</math>.

==Finite strain tensors==
The concept of ''strain'' is used to evaluate how much a given displacement differs locally from a rigid body displacement.<ref name=Lubliner2008/> One of such strains for large deformations is the ''Lagrangian finite strain tensor'', also called the ''Green-Lagrangian strain tensor'' or ''Green – St-Venant strain tensor'', defined as

:<math>\mathbf E=\frac{1}{2}(\mathbf C - \mathbf I)\qquad \text{or} \qquad E_{KL}=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right)\,\!</math>

or as a function of the displacement gradient tensor
:<math>\mathbf E =\frac{1}{2}\left[ (\nabla_{\mathbf X}\mathbf u)^T + \nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^T \cdot\nabla_{\mathbf X}\mathbf u\right]\,\!</math>
or
:<math>E_{KL}=\frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}+\frac{\partial U_M}{\partial X_K}\frac{\partial U_M}{\partial X_L}\right)\,\!</math>

The Green-Lagrangian strain tensor is a measure of how much <math>\mathbf C\,\!</math> differs from <math>\mathbf I\,\!</math>.

The ''Eulerian-Almansi finite strain tensor'', referenced to the deformed configuration, i.e. Eulerian description, is defined as

:<math>\mathbf e=\frac{1}{2}(\mathbf I - \mathbf c)
\qquad \text{or} \qquad
e_{rs}=\frac{1}{2}\left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,\!</math>

or as a function of the displacement gradients we have
:<math>e_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}-\frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j}\right)\,\!</math>

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of the Lagrangian and Eulerian finite strain tensors
|-
|A measure of deformation is the difference between the squares of the differential line element <math>d\mathbf X\,\!</math>, in the undeformed configuration, and <math>d\mathbf x\,\!</math>, in the deformed configuration (Figure 2). Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have,

:<math>d\mathbf{x}^2 - d\mathbf{X}^2=d\mathbf x\cdot d\mathbf x-d\mathbf X\cdot d\mathbf X
\qquad \text{or} \qquad
(dx)^2 - (dX)^2=dx_jdx_j-dX_M\,dX_M\,\!</math>

In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is

:<math> d\mathbf x = \frac {\partial \mathbf x} {\partial \mathbf X}\,d\mathbf X=\mathbf F \,d\mathbf{X} \qquad\text{or} \qquad dx_j=\frac{\partial x_j}{\partial X_M} \,dX_M\,\!</math>

Then we have,

:<math>\begin{align}
d\mathbf{x}^2&=d\mathbf x \cdot d\mathbf x \\
&= \mathbf F \cdot d\mathbf X \cdot \mathbf F \cdot d\mathbf X \\
&= d\mathbf X \cdot \mathbf F^T\mathbf F \cdot d\mathbf X \\
&= d\mathbf X\cdot\mathbf C\cdot d\mathbf X
\end{align}
\qquad \text{or} \qquad
\begin{align}
(dx)^2&=dx_j\,dx_j \\
&= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L \\
&= C_{KL}\,dX_K\,dX_L \\
\end{align}\,\!</math>

where <math>C_{KL}\,\!</math> are the components of the ''right Cauchy-Green deformation tensor'', <math>\mathbf C=\mathbf F^T \mathbf F\,\!</math>. Then, replacing this equation into the first equation we have,

:<math>\begin{align}
d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf X\cdot\mathbf C\cdot d\mathbf X-d\mathbf X\cdot d\mathbf X \\
&=d\mathbf X\cdot (\mathbf C - \mathbf I)\cdot d\mathbf X \\
&= d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \\
\end{align}\,\!</math>

or

:<math>\begin{align}
(dx)^2 - (dX)^2 &= \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}\,dX_K\,dX_L-dX_M\,dX_M \\
&= \left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right)\,dX_K\,dX_L \\
&=2E_{KL}\,dX_K\,dX_L
\end{align}\,\!</math>

where <math>E_{KL}\,\!</math>, are the components of a second-order tensor called the ''Green – St-Venant strain tensor'' or the ''Lagrangian finite strain tensor'',
:<math>\mathbf E=\frac{1}{2}(\mathbf C - \mathbf I)\qquad \text{or} \qquad E_{KL}=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right)\,\!</math>

In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is
:<math> d\mathbf{X} = \frac {\partial \mathbf{X}} {\partial \mathbf {x}}d\mathbf{x}=\mathbf F^{-1} \, d\mathbf{x}=\mathbf{H} \,d\mathbf{x} \qquad \text{or} \qquad dX_M=\frac{\partial X_M}{\partial x_n}\, dx_n\,\!</math>

where <math>\frac{\partial X_M}{\partial x_n}\,\!</math> are the components of the ''spatial deformation gradient tensor'', <math>\mathbf{H}\,\!</math>. Thus we have

:<math>\begin{align}
d\mathbf{X}^2&=d\mathbf X \cdot d\mathbf X \\
&= \mathbf F^{-1} \cdot d\mathbf x \cdot \mathbf F^{-1} \cdot d\mathbf x \\
&= d\mathbf x \cdot \mathbf F^{-T}\mathbf F^{-1} \cdot d\mathbf x \\
&= d\mathbf x\cdot\mathbf c\cdot d\mathbf x
\end{align}
\qquad \text{or} \qquad
\begin{align}
(dX)^2&=dX_M\,dX_M \\
&= \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\
&= c_{rs}\,dx_r\,dx_s \\
\end{align}\,\!</math>

where the second order tensor <math>c_{rs}\,\!</math> is called ''Cauchy's deformation tensor'', <math>\mathbf c=\mathbf F^{-T}\mathbf F^{-1}\,\!</math>. Then we have,

:<math>\begin{align}
d\mathbf{x}^2 - d\mathbf{X}^2 &= d\mathbf x\cdot d\mathbf x-d\mathbf x\cdot\mathbf c\cdot d\mathbf x \\
&=d\mathbf x\cdot (\mathbf I - \mathbf c)\cdot d\mathbf x \\
&= d\mathbf x \cdot 2\mathbf e \cdot d\mathbf x \\
\end{align}\,\!</math>
or
:<math>\begin{align}
(dx)^2 - (dX)^2 &= dx_j\,dx_j-\frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s}\,dx_r\,dx_s \\
&= \left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,dx_r\,dx_s \\
&=2e_{rs}\,dx_r\,dx_s
\end{align}\,\!</math>

where <math>e_{rs}\,\!</math>, are the components of a second-order tensor called the ''Eulerian-Almansi finite strain tensor'',
:<math>\mathbf e=\frac{1}{2}(\mathbf I - \mathbf c)
\qquad \text{or} \qquad
e_{rs}=\frac{1}{2}\left(\delta_{rs} - \frac{\partial X_M}{\partial x_r}\frac{\partial X_M}{\partial x_s} \right)\,\!</math>

Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the ''displacement gradient tensor''. For the Lagrangian strain tensor, first we differentiate the displacement vector <math>\mathbf u(\mathbf X, t)\,\!</math> with respect to the material coordinates <math>X_M\,\!</math> to obtain the ''material displacement gradient tensor'', <math>\nabla_{\mathbf X}\mathbf u\,\!</math>

:<math>\begin{align}
\mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \\
\nabla_{\mathbf X}\mathbf u &= \mathbf F - \mathbf I \\
\mathbf F &= \nabla_{\mathbf X}\mathbf u + \mathbf I \\
\end{align}
\qquad \text{or} \qquad
\begin{align}
u_i& = x_i-\delta_{iJ}X_J \\
\delta_{iJ}U_J &= x_i-\delta_{iJ}X_J \\
x_i&=\delta_{iJ}\left(U_J+X_J\right) \\
\frac{\partial x_i}{\partial X_K}&=\delta_{iJ}\left(\frac{\partial U_J}{\partial X_K}+\delta_{JK}\right) \\
\end{align}
\,\!</math>

Replacing this equation into the expression for the Lagrangian finite strain tensor we have

:<math>\begin{align}
\mathbf E &= \frac{1}{2}\left(\mathbf F^T\mathbf F-\mathbf I\right) \\
&=\frac{1}{2}\left[ \left\{ (\nabla_{\mathbf X}\mathbf u)^T+\mathbf I\right\}\left( \nabla_{\mathbf X}\mathbf u+\mathbf I\right)-\mathbf I\right] \\
&=\frac{1}{2}\left[ (\nabla_{\mathbf X}\mathbf u)^T + \nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^T \cdot\nabla_{\mathbf X}\mathbf u\right] \\
\end{align}\,\!</math>
or
:<math>\begin{align}
E_{KL}&=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right) \\
&=\frac{1}{2}\left[\delta_{jM}\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\delta_{jN}\left(\frac{\partial U_N}{\partial X_L}+\delta_{NL}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left[\delta_{MN}\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\left(\frac{\partial U_N}{\partial X_L}+\delta_{NL}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left[\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\left(\frac{\partial U_M}{\partial X_L}+\delta_{ML}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}+\frac{\partial U_M}{\partial X_K}\frac{\partial U_M}{\partial X_L}\right)
\end{align}\,\!</math>

Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

:<math>e_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}-\frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j}\right)\,\!</math>
|}

=== Seth-Hill family of generalized strain tensors ===
[[B. R. Seth]] from the [[Indian Institute of Technology]], [[Kharagpur]] was the first to show that the Green and Almansi strain tensors are special cases of a more general [[strain measure]].<ref>{{Citation|last=Seth|first=B. R.|year=1961|title=Generalized strain measure with applications to physical problems|journal=MRC Technical Summary Report #248|publisher=Mathematics Research Center, United States Army, University of Wisconsin|pages=1–18.|url=http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0266913}}</ref><ref>{{Citation|last=Seth|first=B. R.|year=1962|title=Generalized strain measure with applications to physical problems|journal=IUTAM Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Mechanics, Haifa, 1962.}}</ref> The idea was further expanded upon by [[Rodney Hill]] in 1968.<ref>{{Citation|last=Hill|first=R.|year=1968|title=On constitutive inequalities for simple materials—I|journal=Journal of the Mechanics and Physics of Solids|volume=16|number=4|pages=229–242.}}</ref> The Seth-Hill family of strain measures can be expressed as

:<math>\mathbf E_{(m)}=\frac{1}{2m}(\mathbf U^{2m}- \mathbf I) = \frac{1}{2m}\left[\mathbf{C}^{m} - \mathbf{I}\right]\,\!</math>

For different values of <math>m\,\!</math> we have:

:<math>\begin{align}
\mathbf E_{(1)}&=\frac{1}{2}(\mathbf U^{2}- \mathbf I) = \frac{1}{2}(\mathbf{C}-\mathbf{I}) & \qquad \text{Green-Lagrangian strain tensor}\\
\mathbf E_{(1/2)}&=(\mathbf U- \mathbf I) = \mathbf{C}^{1/2}-\mathbf{I}& \qquad \text{Biot strain tensor}\\
\mathbf E_{(0)}&=\ln \mathbf U = \frac{1}{2}\,\ln\mathbf{C} & \qquad \text{Logarithmic strain, Natural strain, True strain, or Hencky strain} \\
\mathbf{E}_{(-1)} & = \frac{1}{2}\left[\mathbf{I}-\mathbf{U}^{-2}\right] & \qquad \text{Almansi strain}
\end{align}\,\!</math>

==Stretch ratio==
The '''stretch ratio''' is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration.

The stretch ratio for the differential element <math>d\mathbf X=dX\mathbf N\,\!</math> (Figure) in the direction of the unit vector <math>\mathbf N\,\!</math> at the material point <math>P\,\!</math>, in the undeformed configuration, is defined as

:<math>\Lambda_{(\mathbf N)}=\frac{dx}{dX}\,\!</math>

where <math>dx\,\!</math> is the deformed magnitude of the differential element <math>d\mathbf X\,\!</math>.

Similarly, the stretch ratio for the differential element <math>d\mathbf x=dx\mathbf n\,\!</math> (Figure), in the direction of the unit vector <math>\mathbf n\,\!</math> at the material point <math>p\,\!</math>, in the deformed configuration, is defined as

:<math>\frac{1}{\Lambda_{(\mathbf n)}}=\frac{dX}{dx}.\,\!</math>

The normal strain <math>e_{\mathbf N}\,\!</math> in any direction <math>\mathbf N\,\!</math> can be expressed as a function of the stretch ratio,

:<math>e_{(\mathbf N)}=\frac{dx-dX}{dX}=\Lambda_{(\mathbf N)}-1.\,\!</math>

This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity. Some materials, such as elastometers can sustain stretch ratios of 3 or 4 before they fail, whereas traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.001 (reference?)

==Physical interpretation of the finite strain tensor==

The diagonal components <math>E_{KL}\,\!</math> of the Lagrangian finite strain tensor are related to the normal strain, e.g.

:<math>E_{11}=e_{(\mathbf I_1)}+\frac{1}{2}e_{(\mathbf I_1)}^2\,\!</math>

where <math>e_{(\mathbf I_1)}\,\!</math> is the normal strain or engineering strain in the direction <math>\mathbf I_1\,\!</math>.

The off-diagonal components <math>E_{KL}\,\!</math> of the Lagrangian finite strain tensor are related to shear strain, e.g.

:<math>E_{12}=\frac{1}{2}\sqrt{2E_{11}+1}\sqrt{2E_{22}+1}\sin\phi_{12}\,\!</math>

where <math>\phi_{12}\,\!</math> is the change in the angle between two line elements that were originally perpendicular with directions <math>\mathbf I_1\,\!</math> and <math>\mathbf I_2\,\!</math>, respectively.

Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the [[Infinitesimal strain theory#Infinitesimal strain|infinitesimal strain tensor]]

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors
|-
|The stretch ratio for the differential element <math>d\mathbf X=dX\mathbf N\,\!</math> (Figure) in the direction of the unit vector <math>\mathbf N\,\!</math> at the material point <math>P\,\!</math>, in the undeformed configuration, is defined as

:<math>\Lambda_{(\mathbf N)}=\frac{dx}{dX}\,\!</math>

where <math>dx\,\!</math> is the deformed magnitude of the differential element <math>d\mathbf X\,\!</math>.

Similarly, the stretch ratio for the differential element <math>d\mathbf x=dx\mathbf n\,\!</math> (Figure), in the direction of the unit vector <math>\mathbf n\,\!</math> at the material point <math>p\,\!</math>, in the deformed configuration, is defined as

:<math>\frac{1}{\Lambda_{(\mathbf n)}}=\frac{dX}{dx}\,\!</math>

The square of the stretch ratio is defined as

:<math>\Lambda_{(\mathbf N)}^2=\left (\frac{dx}{dX}\right )^2\,\!</math>

Knowing that

:<math>(dx)^2=C_{KL}dX_KdX_L\,\!</math>

we have

:<math>\Lambda_{(\mathbf N)}^2=C_{KL}N_KN_L\,\!</math>

where <math>N_K\,\!</math> and <math>N_L\,\!</math> are unit vectors.

The normal strain or engineering strain <math>e_{\mathbf N}\,\!</math> in any direction <math>\mathbf N\,\!</math> can be expressed as a function of the stretch ratio,

:<math>e_{(\mathbf N)}=\frac{dx-dX}{dX}=\Lambda_{(\mathbf N)}-1\,\!</math>

Thus, the normal strain in the direction <math>\mathbf I_1\,\!</math> at the material point <math>P\,\!</math> may be expressed in terms of the stretch ratio as

:<math>\begin{align}
e_{(\mathbf I_1)}=\frac{dx_1-dX_1}{dX_1}&=\Lambda_{(\mathbf I_1)}-1\\
&=\sqrt {C_{11}} -1=\sqrt{\delta_{11}+2E_{11}}-1\\
&=\sqrt{1+2E_{11}}-1\end{align}\,\!</math>

solving for <math>E_{11}\,\!</math> we have

<math> \begin{align}
2E_{11}&= \frac{(dx_1)^2 - (dX_1)^2}{(dX_1)^2} \\
E_{11}&= \left(\frac{dx_1-dX_1}{dX_1}\right)+ \frac {1}{2} \left(\frac{dx_1-dX_1}{dX_1}\right)^2 \\
&=e_{(\mathbf I_1)}+\frac{1}{2}e_{(\mathbf I_1)}^2\end{align}\,\!</math>

The ''shear strain'', or change in angle between two line elements <math>d\mathbf X_1\,\!</math> and <math>d\mathbf X_2\,\!</math> initially perpendicular, and oriented in the principal directions <math>\mathbf I_1\,\!</math> and <math>\mathbf I_2\,\!</math>, respectivelly, can also be expressed as a function of the stretch ratio. From the [[dot product]] between the deformed lines <math>d\mathbf x_1\,\!</math> and <math>d\mathbf x_2\,\!</math> we have

:<math>\begin{align}
d\mathbf x_1 \cdot d\mathbf x_2&=dx_1dx_2\cos\theta_{12} \\
\mathbf F \cdot d\mathbf X_1\cdot \mathbf F\cdot d\mathbf X_2&= \sqrt {d\mathbf X_1 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_1}\cdot \sqrt {d\mathbf X_2 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_2} \cos\theta_{12} \\
\frac{d\mathbf X_1\cdot \mathbf F^T\cdot\mathbf F\cdot d\mathbf X_2}{dX_1dX_2}&=\frac{\sqrt {d\mathbf X_1 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_1}\cdot \sqrt {d\mathbf X_2 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_2}}{dX_1dX_2}\cos\theta_{12}\\
\mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2&= \Lambda_{\mathbf I_1}\Lambda_{\mathbf I_2}\cos\theta_{12}\\
\end{align}\,\!</math>

where <math>\theta_{12}\,\!</math> is the angle between the lines <math>d\mathbf x_1\,\!</math> and <math>d\mathbf x_2\,\!</math> in the deformed configuration. Defining <math>\phi_{12}\,\!</math> as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have

:<math>\phi_{12}=\frac{\pi}{2}-\theta_{12}\,\!</math>
thus,

:<math>\cos\theta_{12}=\sin\phi_{12}\,\!</math>

then

:<math>\mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2= \Lambda_{\mathbf I_1}\Lambda_{\mathbf I_2}\sin\phi_{12}\,\!</math>

or

:<math>\begin{align}
C_{12}&=\sqrt{C_{11}}\sqrt{C_{22}}\sin\phi_{12}\\
2E_{12}+\delta_{12}&=\sqrt{2E_{11}+1}\sqrt{2E_{22}+1}\sin\phi_{12}\\
E_{12}&=\frac{1}{2}\sqrt{2E_{11}+1}\sqrt{2E_{22}+1}\sin\phi_{12}\end{align}\,\!</math>
|}

== Deformation tensors in curvilinear coordinates ==
A representation of deformation tensors in [[curvilinear coordinates]] is useful for many problems in continuum mechanics such as nonlinear shell theories and large plastic deformations. Let <math>\mathbf{x} = \mathbf{x}(\xi^1,\xi^2,\xi^3)</math> be a given deformation where the space is characterized by the coordinates <math>(\xi^1,\xi^2,\xi^3)</math>. The tangent vector to the coordinate curve <math>\xi^i</math> at <math>\mathbf{x}</math> is given by
:<math>
\mathbf{g}_i = \frac{\partial \mathbf{x}}{\partial \xi^i}
</math>
The three tangent vectors at <math>\mathbf{x}</math> form a basis. These vectors are related the reciprocal basis vectors by
:<math>
\mathbf{g}_i\cdot\mathbf{g}^j = \delta_i^j
</math>

Let us define a second-order tensor field <math>\boldsymbol{g}</math> (also called the [[metric tensor]]) with components
:<math>
g_{ij} := \frac{\partial \mathbf{x}}{\partial \xi^i}\cdot\frac{\partial \mathbf{x}}{\partial \xi^j} = \mathbf{g}_i\cdot\mathbf{g}_j
</math>
The [[curvilinear coordinates|Christoffel symbols of the first kind]] can be expressed as
:<math>
\Gamma_{ijk}
= \tfrac{1}{2}[(\mathbf{g}_i\cdot\mathbf{g}_k)_{,j} + (\mathbf{g}_j\cdot\mathbf{g}_k)_{,i} - (\mathbf{g}_i\cdot\mathbf{g}_j)_{,k}]
</math>
To see how the Christoffel symbols are related to the Right Cauchy-Green deformation tensor let us define two sets of bases
:<math>
\mathbf{G}_i := \frac{\partial \mathbf{X}}{\partial \xi^i} ~;~~ \mathbf{G}_i\cdot\mathbf{G}^j = \delta_i^j ~;~~ \mathbf{g}_i := \frac{\partial \mathbf{x}}{\partial \xi^i} ~;~~ \mathbf{g}_i\cdot\mathbf{g}^j = \delta_i^j
</math>

=== The deformation gradient in curvilinear coordinates ===
Using the definition of the [[curvilinear coordinates|gradient of a vector field]] in curvilinear coordinates, the deformation gradient can be written as
:<math>
\boldsymbol{F} = \boldsymbol{\nabla}_{\mathbf{X}}\mathbf{x} = \frac{\partial \mathbf{x}}{\partial \xi^i}\otimes\mathbf{G}^i = \mathbf{g}_i\otimes\mathbf{G}^i
</math>

=== The right Cauchy-Green tensor in curvilinear coordinates ===
The right Cauchy-Green deformation tensor is given by
:<math>
\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F} = (\mathbf{G}^i\otimes\mathbf{g}_i)\cdot(\mathbf{g}_i\otimes\mathbf{G}^i)
= (\mathbf{g}_i\cdot\mathbf{g}_j)(\mathbf{G}^i\otimes\mathbf{G}^j)
</math>
If we express <math>\boldsymbol{C}</math> in terms of components with respect to the basis {<math>\mathbf{G}^i</math>} we have
:<math>
\boldsymbol{C} = C_{ij}~\mathbf{G}^i\otimes\mathbf{G}^j
</math>
Therefore
:<math>
C_{ij} = \mathbf{g}_i\cdot\mathbf{g}_j = g_{ij}
</math>
and the Christoffel symbol of the first kind may be written in the following form.
:<math>
\Gamma_{ijk}
= \tfrac{1}{2}[C_{ik,j} + C_{jk,i} - C_{ij,k}]
= \tfrac{1}{2}[(\mathbf{G}_i\cdot\boldsymbol{C}\cdot\mathbf{G}_k)_{,j} + (\mathbf{G}_j\cdot\boldsymbol{C}\cdot\mathbf{G}_k)_{,i} - (\mathbf{G}_i\cdot\boldsymbol{C}\cdot\mathbf{G}_j)_{,k}]
</math>

===Some relations between deformation measures and Christoffel symbols===
Let us consider a one-to-one mapping from <math>\mathbf{X} = \{X^1,X^2,X^3\}</math> to <math>\mathbf{x} = \{x^1,x^2,x^3\}</math> and let us assume that there exist two positive definite, symmetric second-order tensor fields <math>\boldsymbol{G}</math> and <math>\boldsymbol{g}</math> that satisfy
:<math>
G_{ij} = \frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~g_{\alpha\beta}
</math>
Then,
:<math>
\frac{\partial G_{ij}}{\partial x^k} = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^k}~\frac{\partial X^\beta}{\partial x^j} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial^2 X^\beta}{\partial x^j \partial x^k}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial g_{\alpha\beta}}{\partial x^k}
</math>
Noting that
:<math>
\frac{\partial g_{\alpha\beta}}{\partial x^k} = \frac{\partial X^\gamma}{\partial x^k}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma}
</math>
and <math>g_{\alpha\beta} = g_{\beta\alpha}</math> we have
:<math>
\begin{align}
\frac{\partial G_{ij}}{\partial x^k} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^k}~\frac{\partial X^\beta}{\partial x^j} +
\frac{\partial^2 X^\alpha}{\partial x^j \partial x^k}~\frac{\partial X^\beta}{\partial x^i}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma} \\
\frac{\partial G_{ik}}{\partial x^j} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k} +
\frac{\partial^2 X^\alpha}{\partial x^j \partial x^k}~\frac{\partial X^\beta}{\partial x^i}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^k}~\frac{\partial X^\gamma}{\partial x^j}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma} \\
\frac{\partial G_{jk}}{\partial x^i} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k} +
\frac{\partial^2 X^\alpha}{\partial x^i \partial x^k}~\frac{\partial X^\beta}{\partial x^j}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^j}~\frac{\partial X^\beta}{\partial x^k}~\frac{\partial X^\gamma}{\partial x^i}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma}
\end{align}
</math>
Define
:<math>
\begin{align}
_{(x)}\Gamma_{ijk} & := \frac{1}{2}\left(\frac{\partial G_{ik}}{\partial x^j} + \frac{\partial G_{jk}}{\partial x^i} - \frac{\partial G_{ij}}{\partial x^k}\right) \\
_{(X)}\Gamma_{\alpha\beta\gamma} & := \frac{1}{2}\left(\frac{\partial g_{\alpha\gamma}}{\partial X^\beta} + \frac{\partial g_{\beta\gamma}}{\partial X^\alpha} - \frac{\partial g_{\alpha\beta}}{\partial X^\gamma}\right) \\
\end{align}
</math>
Hence
:<math>
_{(x)}\Gamma_{ijk} = \frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k} \,_{(X)}\Gamma_{\alpha\beta\gamma} + \frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k}~g_{\alpha\beta}
</math>
Define
:<math>
[G^{ij}] = [G_{ij}]^{-1} ~;~~ [g^{\alpha\beta}] = [g_{\alpha\beta}]^{-1}
</math>
Then
:<math>
G^{ij} = \frac{\partial x^i}{\partial X^\alpha}~\frac{\partial x^j}{\partial X^\beta}~g^{\alpha\beta}
</math>
Define the Christoffel symbols of the second kind as
:<math>
_{(x)}\Gamma^m_{ij} := G^{mk} \,_{(x)}\Gamma_{ijk} ~;~~
_{(X)}\Gamma^\nu_{\alpha\beta} := g^{\nu\gamma} \,_{(X)}\Gamma_{\alpha\beta\gamma}
</math>
Then
:<math>
\begin{align}
_{(x)}\Gamma^m_{ij} & = G^{mk}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k} \,_{(X)}\Gamma_{\alpha\beta\gamma} + G^{mk}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\frac{\partial x^k}{\partial X^\rho}~g^{\nu\rho}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~\frac{\partial x^k}{\partial X^\rho}~g^{\nu\rho}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\delta^\gamma_\rho~g^{\nu\rho}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~\delta^\beta_\rho~g^{\nu\rho}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~g^{\nu\gamma}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~g^{\nu\beta}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\frac{\partial x^m}{\partial X^\nu}~\delta^{\nu}_{\alpha}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}
\end{align}
</math>
Therefore
:<math>
_{(x)}\Gamma^m_{ij} = \frac{\partial x^m}{\partial X^\nu}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\frac{\partial x^m}{\partial X^\alpha}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}
</math>
The invertibility of the mapping implies that
:<math>
\begin{align}
\frac{\partial X^\mu}{\partial x^m}\,_{(x)}\Gamma^m_{ij} & = \frac{\partial X^\mu}{\partial x^m}~\frac{\partial x^m}{\partial X^\nu}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\frac{\partial X^\mu}{\partial x^m}~\frac{\partial x^m}{\partial X^\alpha}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j} \\
& = \delta^\mu_\nu~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\delta^\mu_\alpha~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j} \\
& = \frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\mu_{\alpha\beta} + \frac{\partial^2 X^\mu}{\partial x^i \partial x^j}
\end{align}
</math>
We can also formulate a similar result in terms of derivatives with respect to <math>x</math>. Therefore
:<math>
\begin{align}
\frac{\partial^2 X^\mu}{\partial x^i \partial x^j} & = \frac{\partial X^\mu}{\partial x^m}\,_{(x)}\Gamma^m_{ij} - \frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\mu_{\alpha\beta} \\
\frac{\partial^2 x^m}{\partial X^\alpha \partial X^\beta} & = \frac{\partial x^m}{\partial X^\mu}\,_{(X)}\Gamma^\mu_{\alpha\beta} - \frac{\partial x^i}{\partial X^\alpha}~\frac{\partial x^j}{\partial X^\beta} \,_{(x)}\Gamma^m_{ij}
\end{align}
</math>

== Compatibility conditions ==
{{Main|Compatibility (mechanics)}}
The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies. These allowable conditions leave the body without unphysical gaps or overlaps after a deformation. Most such conditions apply to simply-connected bodies. Additional conditions are required for the internal boundaries of multiply connected bodies.

=== Compatibility of the deformation gradient ===
The necessary and sufficient conditions for the existence of a compatible <math>\boldsymbol{F}</math> field over a simply connected body are
:<math>
\boldsymbol{\nabla}\times\boldsymbol{F} = \boldsymbol{0}
</math>

=== Compatibility of the right Cauchy-Green deformation tensor ===
The necessary and sufficient conditions for the existence of a compatible <math>\boldsymbol{C}</math> field over a simply connected body are
:<math>
R^\gamma_{\alpha\beta\rho} :=
\frac{\partial }{\partial X^\rho}[\,_{(X)}\Gamma^\gamma_{\alpha\beta}] -
\frac{\partial }{\partial X^\beta}[\,_{(X)}\Gamma^\gamma_{\alpha\rho}] +
\,_{(X)}\Gamma^\gamma_{\mu\rho}\,_{(X)}\Gamma^\mu_{\alpha\beta} -
\,_{(X)}\Gamma^\gamma_{\mu\beta}\,_{(X)}\Gamma^\mu_{\alpha\rho} = 0
</math>
We can show these are the mixed components of the [[Riemann-Christoffel curvature tensor]]. Therefore the necessary conditions for <math>\boldsymbol{C}</math>-compatibility are that the Riemann-Christoffel curvature of the deformation is zero.

=== Compatibility of the left Cauchy-Green deformation tensor ===
No general sufficiency conditions are known for the left Cauchy-Green deformation tensor in three-dimensions. Compatibility conditions for two-dimensional <math>\boldsymbol{B}</math> fields have been found by Janet Blume.<ref name=Blume>{{cite journal|author=Blume, J. A.|year= 1989|title=Compatibility conditions for a left Cauchy-Green strain field|doi=10.1007/BF00045780|journal= J. Elasticity|volume=21|pages=271–308}}</ref><ref name=acharya>{{cite journal|author=Acharya, A.|url=http://imechanica.org/files/B-compatibility.pdf|year=1999|title=On Compatibility Conditions for the Left Cauchy–Green Deformation Field in Three Dimensions|journal=Journal of Elasticity|volume =56|issue=2 |pages=95–105|doi=10.1023/A:1007653400249}}</ref>

==See also==
* [[Infinitesimal strain]]
* [[Compatibility (mechanics)]]
* [[Curvilinear coordinates]]
* [[Piola-Kirchhoff stress tensor]], the stress tensor for finite deformations.
* [[Stress measures]]

==References==
{{reflist}}

==Further reading==
*{{cite book
| last = Dill | first = Ellis Harold
| title =Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity
| publisher = CRC Press
| year = 2006
| location = Germany
| url = http://books.google.com/?id=Nn4kztfbR3AC
| isbn = 0-8493-9779-0}}

*{{cite book
| last = Dimitrienko | first = Yuriy
| title = Nonlinear Continuum Mechanics and Large Inelastic Deformations
| publisher = Springer
| year = 2011
| location = Germany
| url = http://books.google.com/books?as_isbn=9789400700338
| isbn = 978-94-007-0033-8}}

*{{cite book
| last = Hutter | first = Kolumban
| coauthors = Klaus Jöhnk
| title = Continuum Methods of Physical Modeling
| publisher = Springer
| year = 2004
| location = Germany
| url = http://books.google.com/?id=B-dxx724YD4C
| isbn = 3-540-20619-1}}

*{{cite book
| last = Lubarda
| first = Vlado A.
| title = Elastoplasticity Theory
| publisher = CRC Press
| year = 2001
| url = http://books.google.com/?id=1P0LybL4oAgC
| isbn = 0-8493-1138-1 }}

*{{cite book
| last = Macosko
| first = C. W.
| authorlink =
| coauthors =
| title = Rheology: principles, measurement and applications
| publisher = VCH Publishers
| year = 1994
| isbn = 1-56081-579-5}}

*{{cite book
| last = Mase
| first = George E.
| title = Continuum Mechanics
| publisher = McGraw-Hill Professional
| year = 1970
| url = http://books.google.com/?id=bAdg6yxC0xUC
| isbn = 0-07-040663-4 }}

*{{cite book
| last = Mase
| first = G. Thomas
| coauthors = George E. Mase
| title = Continuum Mechanics for Engineers
| publisher = CRC Press
| year = 1999
|edition= Second
| url = http://books.google.com/?id=uI1ll0A8B_UC
| isbn = 0-8493-1855-6 }}

*{{cite book
| last = Nemat-Nasser
| first = Sia
| title = Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials
| publisher = Cambridge University Press
| year = 2006
| location = Cambridge
| url = http://books.google.com/?id=5nO78Rt0BtMC
| isbn = 0-521-83979-3}}

*{{Cite book
| last = Rees | first = David
| title = Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications
| publisher = Butterworth-Heinemann
| year = 2006
| url = http://books.google.com/?id=4KWbmn_1hcYC
| isbn = 0-7506-8025-3}}

== External links ==
*[http://www.imechanica.org/node/3786 Prof. Amit Acharya's notes on compatibility on iMechanica]

[[Category:Tensors]]
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Non-Newtonian fluids]]
[[Category:Solid mechanics]]

[[it:Deformazione]]

Collocation method

2012-03-23T15:24:39Z

Jitse Niesen: mention Gauss–Legendre method

Hi there. Let me start by introducing the author, her name is Sophia Boon but she by no means really liked that title. My working day occupation is a travel agent. Her family lives in Ohio. My spouse doesn't like it the way I do but what I really like doing is caving but I don't have the time lately. Here is my page ... best psychic ([http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ Click Link])

Rank-into-rank

2012-03-16T10:14:46Z

Jitse Niesen: remove {{technical}} (revert) - this topic is so specialistic and technical that it cannot be made accessible to general readers, and no specific concern was raised

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Nonstandard finite difference scheme

2012-03-01T14:13:46Z

Jitse Niesen: /* Example of a Nonstandard Finite Difference Model */ typo in equation

In [[astronomy]], a '''bolometric correction''' is a correction that must be made to the absolute [[Magnitude (astronomy)|magnitude]] of an object in order to convert an object's [[visible magnitude]] to its [[Bolometric_magnitude#Bolometric_magnitude|bolometric magnitude]]. Mathematically, such a calculation can be expressed:
:<math> BC = M_b - M_v\!\,</math>

The following is subset of a table from Kaler<ref name="kalerpaper"/> (p. 263) listing the bolometric correction for a range of stars. For the full table, see the referenced work.

{| class="wikitable" border="1"
|-
! Class
! Main Sequence
! Giants
! Supergiants
|-
| O3
| -4.3
| -4.2
| -4.0
|-
| G0
| -0.10
| -0.13
| -0.1
|-
| G5
| -0.14
| -0.34
| -0.20
|-
| K0
| -0.24
| -0.42
| -0.38
|-
| K5
| -0.66
| -1.19
| -1.00
|-
| M0
| -1.21
| -1.28
| -1.3
|}

The bolometric correction is large both for early type (hot) stars and for late type (cool) stars. The former because a substantial part of the produced radiation is in the ultraviolet, the latter because a large part is in the infrared. For a star like our Sun, the correction is only marginal because the Sun radiates most of its energy in the visual wavelength range.

The bolometric correction scale is set by the absolute magnitude of the Sun and an adopted [[bolometric magnitude]] for the [[Sun]]. The choice of adopted solar absolute magnitude, bolometric correction, and absolute bolometric magnitude are not arbitrary, although some classic references have tabulated mutually incompatible values for these quantities
.<ref name=tableprovisoBC/> The bolometric scale historically had varied somewhat in the literature, with the Sun's bolometric correction in V-band varying from -0.19 to -0.07 magnitude. Since the Sun is also a variable star, and there are minor differences in adopted solar luminosity values (albeit at a subtle level), in 1999 two [[IAU]] commissions
(Commissions 25: Stellar Photometry and Polarimetry, and Commission 36: Theory
of Stellar Atmospheres) agreed to separate the definition of bolometric correction and magnitude from the variable Sun. The 1999 [[IAU]] statements define that absolute bolometric magnitude zero correlates to a bolometric luminosity of 3.055e28 [[Watt]]s. This particular luminosity was selected as the zero-point for the absolute bolometric magnitude scale so that the Sun's luminosity (3.842e26 Watts) would correspond to absolute bolometric magnitude 4.75
(the value that was most commonly used by most astronomers). As the Sun has an apparent V magnitude of -26.75, and absolute V magnitude of 4.82, then the IAU bolometric magnitude scale implies that the bolometric correction for the Sun (with effective temperature of 5778 K) is -0.07 magnitude.<ref name=mamajeksun /> The new IAU definition means that theoretical evolutionary models for stars can define brightnesses in terms of bolometric and absolute magnitudes on a scale that is tied to a physical quantity (the luminosity zero-point of 3.055e28 [[Watt]]s) rather than to the Sun (which is intrinsically variable, and there are systematic uncertainties in the value of the solar flux constant as measured at 1 AU).

==See also==
*[[absolute magnitude]]

==External links==
*http://www.peripatus.gen.nz/Astronomy/SteMag.html - contains table of bolometric corrections
*http://articles.adsabs.harvard.edu//full/1996ApJ...469..355F/0000360.000.html - contains detailed tables<ref name=tablesof_bolometric_correction/> of bolometric corrections (note that these second set of tables are consistent with a bolometric magnitude of 4.73<ref name=tableprovisoBC/> for the Sun and also be aware that there are misprint<ref name=tableprovisoBC/> errors for a few of the figures in the tables)

==References==
{{reflist|30em|refs=

<ref name=kalerpaper>{{cite journal | title = Stars and their spectra: An Introduction to the Spectral Sequence | author = Kaler, James B. | year = 1989 | pages = 300}}</ref>

<ref name=tablesof_bolometric_correction>{{citation | display-authors=1 | last1=Flower | first1=Phillip J.| title=Transformations from Theoretical Hertzsprung-Russell Diagrams to Color-Magnitude Diagrams: Effective Temperatures, B-V Colors, and Bolometric Corrections | url=http://articles.adsabs.harvard.edu//full/1996ApJ...469..355F/0000360.000.html | journal=[[The Astrophysical Journal]] | volume=469 | page=355 |date=September 1996 | doi=10.1086/177785| bibcode=1996ApJ...469..355F }}</ref>

<ref name=tableprovisoBC>{{cite journal | author=Torres, Guillermo | title=On the Use of Empirical Bolometric Corrections for Stars | url=http://iopscience.iop.org/1538-3881/140/5/1158/fulltext/ | journal=[[The Astronomical Journal]] |date=November 2010 | volume=140 | issue=5 | pages=1158–1162 | layurl=http://iopscience.iop.org/1538-3881/140/5/1158/fulltext/aj363350t3_ascii.txt | doi=10.1088/0004-6256/140/5/1158 | bibcode=2010AJ....140.1158T |arxiv = 1008.3913 }}</ref>

<ref name=mamajeksun>{{cite web |url=http://www.pas.rochester.edu/~emamajek/sun.txt |title=Basic Astronomical Data for the Sun (BADS)|author=Eric Mamajek |date=April 12, 2012|work= |publisher= |accessdate=6 May 2012}}</ref>

}}

[[Category:Observational astronomy]]

Hypot

2012-02-28T16:49:00Z

Jitse Niesen: == See also == * Alpha max plus beta min algorithm (which is apparently an algorithm for implementing hypot)

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今回のデニムショートパンツを身に着けている。
喪ウッド[ひづめ]日付：2013年8月25日15時46分15秒は気にしないのQQのバイヤーを使用したいが、
「愛のすべての種類、 [http://www.rheintalverlag.ch/newsletter/gaga.php ��ߥ��]

Direct multiple shooting method

2012-02-27T19:41:59Z

Jitse Niesen: remove section which has been empty for years

== Uggs Suomesta jotka ovat läsnä niiden ympäristössä. ==

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== Ugg Online Store Finland 8 pistettä tammikuussa ==

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== Free Run 3 10. Ratkaise antaa ja saat tyytyväisyyttä näin ==

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Talk:Random Fibonacci sequence

2010-04-22T09:47:58Z

Jitse Niesen: moved Talk:Viswanath's constant to Talk:Random Fibonacci sequence: main topic of the article (its scope expanded over time)

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Central subgroup

2009-05-05T10:57:45Z

Jitse Niesen: add reference

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

== Iterated functions ==
Given an [[endomorphism]] ''f'' on a [[set (mathematics)|set]] ''X''
:<math>f: X \to X</math>
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
:<math>\ f^n(x) = x</math>
where <math>f^n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive integer ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n''.

If there exists distinct ''n'' and ''m'' such that
:<math>f^n(x) = f^m(x)</math>
then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>(f^n)^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

:<math>|(f^n)^\prime|\ne 1,</math>

and that it is ''attractive'' if

:<math>|(f^n)^\prime|< 1</math>

and it is ''repelling'' if

:<math>|(f^n)^\prime|> 1.</math>

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

=== Examples ===
* A period-one point is called a [[fixed point (mathematics)|fixed point]].

== Dynamical system ==
Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],
:<math>\Phi: \mathbb{R} \times X \to X</math>
a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' ≥ 0 so that
:<math>\Phi(t, x) = x\,</math>
The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

=== Properties ===
* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)\,</math> for all ''s'' in '''R'''
* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

==See also==
* [[Limit cycle]]
* [[Limit set]]
* [[Stable manifold|Stable set]]
* [[Sharkovsky's theorem]]
* [[Stationary point]]
* [[Periodic points of complex quadratic mappings]]

{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}

[[Category:Limit sets]]

Template:Citation needed

2007-03-15T06:04:51Z

Jitse Niesen: move category before superscript text, per request in Template_talk:Fact#Incorrect_behavior

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User talk:Jitse Niesen/Archive1

2005-08-04T18:34:40Z

Jitse Niesen: archive

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User talk:82.225.223.90

2005-05-03T17:29:11Z

Jitse Niesen: welcome, thank you, and hints

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