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In [[differential geometry]], the '''first fundamental form''' is the [[inner product]] on the [[tangent space]] of a [[surface]] in three-dimensional [[Euclidean space]] which is induced [[canonical form|canonically]] from the [[dot product]] of '''R'''<sup>''3''</sup>.  It permits the calculation of [[curvature]] and metric properties of a surface such as length and area in a manner consistent with the [[ambient space]]. The first fundamental form is denoted by the Roman numeral I,
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:<math>\!\mathrm{I}(x,y)= \langle x,y \rangle.</math>
 
Let ''X''(''u'',&nbsp;''v'') be a [[parametric surface]].  Then the inner product of two [[tangent vector]]s is
 
:<math>
\begin{align}
& {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\
& = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\
& = Eac + F(ad+bc) + Gbd,
\end{align}
</math>
 
where ''E'', ''F'', and ''G'' are the '''coefficients of the first fundamental form'''.
 
The first fundamental form may be represented as a [[symmetric matrix]].
 
:<math>\!\mathrm{I}(x,y) = x^T
\begin{pmatrix}
E & F \\
F & G
\end{pmatrix}y
</math>
 
==Further notation==
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.
:<math>\!\mathrm{I}(v)= \langle v,v \rangle = |v|^2</math>
 
The first fundamental form is often written in the modern notation of the [[metric tensor]]. The coefficients may then be written as <math>g_{ij}</math>:
:<math> \left(g_{ij}\right) = \begin{pmatrix}g_{11} & g_{12} \\g_{21} & g_{22}\end{pmatrix} =\begin{pmatrix}E & F \\F & G\end{pmatrix}</math>
 
The components of this tensor are calculated as the scalar product of tangent vectors ''X''<sub>1</sub> and ''X''<sub>2</sub>:
 
:<math>g_{ij} = X_i \cdot X_j</math>
 
for ''i'', ''j'' = 1, 2. See example below.
 
==Calculating lengths and areas==
 
The first fundamental form completely describes the metric properties of a surface. Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The [[line element]] ''ds'' may be expressed in terms of the coefficients of the first fundamental form as
:<math>ds^2 = Edu^2+2Fdudv+Gdv^2 \,</math>. <!--- "\," improves the display of this formula in Wikipedia. Do not delete --->
 
The classical area element given by <math> dA = |X_u \times X_v| \ du\, dv</math> can be expressed in terms of the first fundamental form with the assistance of [[Lagrange's identity]],
 
:<math>dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \langle X_u,X_v \rangle^2 } \ du\, dv = \sqrt{EG-F^2} \, du\, dv.</math>
 
===Example===
 
The unit [[sphere]] in '''R'''<sup>''3''</sup> may be parametrized as
 
:<math>X(u,v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix},\ (u,v) \in [0,2\pi) \times [0,\pi).</math>
 
Differentiating <math>X(u,v)</math> with respect to u and v yields
 
:<math>X_u = \begin{pmatrix} -\sin u \sin v \\ \cos u \sin v \\ 0 \end{pmatrix},\ X_v = \begin{pmatrix} \cos u \cos v \\ \sin u \cos v \\ -\sin v \end{pmatrix}.</math>
 
The coefficients of the first fundamental form may be found by taking the dot product of the [[partial derivatives]].
 
:<math>E = X_u \cdot X_u = \sin^2 v</math>
:<math>F = X_u \cdot X_v = 0</math>
:<math>G = X_v \cdot X_v = 1</math>
 
====Length of a curve on the sphere====
 
The [[equator]] of the sphere is a parametrized curve given by <math>(u(t),v(t))=(t,\frac{\pi}{2})</math> with t ranging from 0 to <math>2\pi</math>. The line element may be used to calculate the length of this curve.
 
:<math>\int_0^{2\pi} \sqrt{ E\left(\frac{du}{dt}\right)^2 + 2F\frac{du}{dt}\frac{dv}{dt} + G\left(\frac{dv}{dt}\right)^2 } \,dt = \int_0^{2\pi} \sin v \,dt = 2\pi \sin \frac{\pi}{2} = 2\pi</math>
 
====Area of a region on the sphere====
 
The area element may be used to calculate the area of the sphere.
 
:<math>\int_0^{\pi} \int_0^{2\pi} \sqrt{ EG-F^2 } \ du\, dv = \int_0^{\pi} \int_0^{2\pi} \sin v \, du\, dv = 2\pi \left[-\cos v\right]_0^{\pi} = 4\pi</math>
 
==Gaussian curvature==
 
The [[Gaussian curvature]] of a surface is given by  
 
:<math> K = \frac{\det \mathit{II}}{\det I} = \frac{ LN-M^2}{EG-F^2 }, </math>
 
where ''L'', ''M'', and ''N'' are the coefficients of the [[second fundamental form]].
 
[[Theorema egregium]] of [[Carl Friedrich Gauss|Gauss]] states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that ''K'' is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the [[Gaussian curvature#Alternative_formulas|Brioschi formula]].
 
==See also==
*[[Metric tensor]]
*[[Second fundamental form]]
 
==External links==
*[http://mathworld.wolfram.com/FirstFundamentalForm.html First Fundamental Form &mdash; from Wolfram MathWorld]
*[http://planetmath.org/encyclopedia/FirstFundamentalForm.html PlanetMath: first fundamental form]
 
{{curvature}}
 
[[Category:Differential geometry of surfaces]]
[[Category:Differential geometry]]
[[Category:Surfaces]]

Latest revision as of 23:21, 6 January 2015

Obtaining gifts for mothers will leave many clueless- but there a few things all mothers generally speaking will like to receive. An individual are are a market obtain mom a gift, are generally some basics that will probably be an ideal choice.

It extremely important to give peace possibility to and promote it in a silent spot. You cannot proceed with shouting peace at work or in regards to the streets. Discover however say it can be a quite non offensive way, without sounding political. By wearing the symbol around your neck, ears or finger, you just cannot only promote the concept of world peace but also promote peace within quite neighborhood.



Observe her diamond jewelry tastes to put together a while. Does she like contemporary ring styles or maybe traditional calls? Engagement rings took on many designs latest years years, however many women prefer the traditional solitaire diamond diamond engagement ring that never gone away from style. The cut for the diamond gemstone is essential. There are princess cuts and round cuts.

Make sure you enable the glaze dry thoroughly a person begin proceed to another one step- even overnight. There are a few ways to require a bail to your pendant. Final like to drill a dent through extremely best top within the pendant, front to back, and attach a clip bail. This will cover some of your image, but it really does provide you with a very nice effect. Actually like to screw a very small eye-screw into the top of the pendant and can then be include a jump ring carry out their necklace through. Lastly, you will get glue-on bails for factor effect. Finish with a sequence or necklace of choosing and eureka! Your very own scrabble tile instant!

Women wear necklace to their neck. Women are wearing Pendants and necklaces since years. However, since the inception, necklaces have changed a property. The pattern, kinds have replaced. Now Pendants are produced of many materials like gold, silver and yellow metal. In today's economy, the prices of gold are soaring and thus, many women are buying silver jewelry. One will obtain the best designs in a number of styles within the market. Moreover, the regarding pendant made of silver is actually comparatively new. A silver pendant is a smaller piece of ornament, which an individual hangs at the neck from a chain. A solitaire can be very popular the actual planet market and ladies love put on it.

CURB YOUR ENTHUSIASM - Well, around attempt to bluff. Don't show too much interest. Should you decide to find something you like, act nonchalant about it - as it's the 100th similar piece you've seen that day of the week. Ask casually and throw a few negative comments on the jewelry's solution. Why do this? Vendors recognise interested customers and within a place exactly like a flea market or a normal store high is no fixed price; haggling actually natural as breathing. Another important thing is to haggle; these vendors sell their products for the highest possible price simply they could still significantly lower the cost.

The designs that purchased have this quality of appealing to everyone. Without regard for the looks, these items can be worn by all. People can also add a glamour quotient to them by having other accessories on them and changing their apparel. Designs and make of these ornaments are such they will look good with all dresses.

Pear - You'll find the pear shape continues to be popular, specifically in pendants. It reflects light nicely is definitely good sizes. Stay on the sting of finishing touches with diamond rings.

If you have any kind of questions pertaining to where and ways to utilize Scalar Energy Pendant Fake, you could contact us at our webpage.

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