|
|
| Line 1: |
Line 1: |
| : ''For other uses of "atlas", see [[Atlas (disambiguation)]].''
| |
|
| |
|
| In [[mathematics]], particularly [[topology]], one describes
| |
| a [[manifold]] using an '''atlas'''. An atlas consists of individual
| |
| ''charts'' that, roughly speaking, describe individual regions
| |
| of the manifold. If the manifold is the surface of the Earth,
| |
| then an atlas has its more common meaning. In general,
| |
| the notion of atlas underlies the formal definition of a [[manifold]].
| |
|
| |
|
| ==Charts==
| | If any of the aforementioned reasons resonates along with you, you may wish to consider finding a CCTV London company to handle your particularly ([http://justsamachar.com/business/airtel-in-africa-soon/?r=http://cctvdvrreviews.com justsamachar.com]) property's security needs. Some couples and [http://Eu4wiki.de/index.php?title=What_Ancient_Greeks_Knew_About_Cctv_16_Channel_Dvr_That_You_Still_Don_t families utilize] HD DVR as being a way to develop a fun bonding time. Jenis dvr cctv Below are some of the applications for CCTV o Monitoring traffic at traffic junctions o Keep watch on people at railway stations, airports etc o Production control in the factory.<br><br>Having a CCTV camera installed in a [http://www.rentanadviser.com/en/rss-reader/onlinerssreader.aspx?rssurl=http%3A%2F%2Fcctvdvrreviews.com Open source Dvr software for cctv] home, office or [http://Heaven.Ncity.net/xe/?document_srl=10408 business enables] you to observe the location from anywhere inside the world via computer honeywell digital video recorders or closed circuit television. If a vehicle is carrying female employees, these vehicle CCTV [http://www.assessor.shelby.tn.us/PropertySearchSales.aspx?id=D0255N%20H00010&ReturnUrl=http%3A%2F%2Fcctvdvrreviews.com cctv dvr hard drive failure] systems may be of next ([http://www.plattscsd.org/exit.asp?url=http://cctvdvrreviews.com http://www.plattscsd.org/]) great use by providing details of vehicle interiors at any honeywell video systems performance given point of time.<br><br>Finally, academic writing services save students some time. cctv 16 channel dvr com is that you just can basically be compensated for up to eight articles per month. Cctv camera dvr software free download [http://www.mokeling.com/link.php?url=http://cctvdvrreviews.com cctv dvr Tutorial] This fundamental portion of fashion dissertations needs to have these segments. we're trying to use our planning technique to get the most effective IELTS Writing score possible; it's not just a race to view who can write the most quickly. |
| | |
| The definition of an atlas depends on the notion of a ''chart''.
| |
| A '''chart''' for a [[topological space]] ''M'' is a [[homeomorphism]] <math>\varphi</math> from an [[open set|open subset]] ''U'' of ''M'' to an open subset of [[Euclidean space]]. The chart is traditionally recorded as the ordered pair <math> (U, \varphi)</math>.
| |
| | |
| ==Formal definition of atlas==
| |
| | |
| An '''atlas''' for a [[topological space]] ''M'' is a collection <math> \{(U_{\alpha}, \varphi_{\alpha})\}</math> of charts on ''M'' such that
| |
| <math> \bigcup U_{\alpha} = M</math>. If the codomain of each chart is the ''n''-dimensional [[Euclidean space]] and the atlas is connected, then ''M'' is said to be an ''n''-dimensional [[manifold]].
| |
| | |
| ==Transition maps==
| |
| | |
| {{ Annotated image | caption=Two charts on a manifold
| |
| | image=Two coordinate charts on a manifold.svg
| |
| | image-width = 250
| |
| | annotations =
| |
| {{Annotation|45|70|<math>M</math>}}
| |
| {{Annotation|67|54|<math>U_\alpha</math>}}
| |
| {{Annotation|187|66|<math>U_\beta</math>}}
| |
| {{Annotation|42|100|<math>\varphi_\alpha</math>}}
| |
| {{Annotation|183|117|<math>\varphi_\beta</math>}}
| |
| {{Annotation|87|109|<math>\tau_{\alpha,\beta}</math>}}
| |
| {{Annotation|90|145|<math>\tau_{\beta,\alpha}</math>}}
| |
| {{Annotation|55|183|<math>\mathbf R^n</math>}}
| |
| {{Annotation|145|183|<math>\mathbf R^n</math>}}
| |
| }}
| |
| A transition map provides a way of comparing two charts of an atlas.
| |
| To make this comparison, we consider the composition of one chart
| |
| with the inverse of the other. This composition is not well-defined
| |
| unless we restrict both charts to the intersection of their domains
| |
| of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)
| |
| | |
| To be more precise, suppose that <math>(U_{\alpha}, \varphi_{\alpha})</math> and <math>(U_{\beta}, \varphi_{\beta})</math> are two charts for a manifold ''M'' such that <math>U_{\alpha} \cap U_{\beta}</math> is non-empty.
| |
| The '''transition map''' <math> \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})</math> is the map defined by
| |
| | |
| : <math>\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.</math>
| |
| | |
| Note that since <math>\varphi_{\alpha}</math> and <math>\varphi_{\beta}</math> are both homeomorphisms, the transition map <math> \tau_{\alpha, \beta}</math> is also a homeomorphism.
| |
| | |
| ==More structure== | |
| | |
| One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of [[differentiation (mathematics)|differentiation]] of functions on a manifold, then it is necessary to construct an atlas whose transition functions are [[differentiable]]. Such a manifold is called [[Differentiable manifold|differentiable]]. Given a differentiable manifold, one can unambiguously define the notion of [[tangent vectors]] and then [[directional derivative]]s.
| |
| | |
| If each transition function | |
| is a [[smooth map]], then the atlas is called a
| |
| [[smooth structure|smooth atlas]], and the manifold itself is called [[Differentiable manifold#Definition|smooth]].
| |
| Alternatively, one could require that the transition maps
| |
| have only ''k'' continuous derivatives in which case the atlas is
| |
| said to be <math> C^k </math>.
| |
| | |
| Very generally, if each transition function
| |
| belongs to a [[pseudo-group]] <math> {\mathcal G} </math>
| |
| of [[homeomorphism]]s of [[Euclidean space]], | |
| then the atlas is called a <math> {\mathcal G}</math>-atlas.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| {{refbegin}}
| |
| *{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}
| |
| *{{cite book | first = Mark R. | last = Sepanski | year = 2007 | title = Compact Lie Groups | publisher = Springer-Verlag | isbn = 978-0-387-30263-8}}
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| *[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd
| |
| | |
| [[Category:Differential topology]]
| |
If any of the aforementioned reasons resonates along with you, you may wish to consider finding a CCTV London company to handle your particularly (justsamachar.com) property's security needs. Some couples and families utilize HD DVR as being a way to develop a fun bonding time. Jenis dvr cctv Below are some of the applications for CCTV o Monitoring traffic at traffic junctions o Keep watch on people at railway stations, airports etc o Production control in the factory.
Having a CCTV camera installed in a Open source Dvr software for cctv home, office or business enables you to observe the location from anywhere inside the world via computer honeywell digital video recorders or closed circuit television. If a vehicle is carrying female employees, these vehicle CCTV cctv dvr hard drive failure systems may be of next (http://www.plattscsd.org/) great use by providing details of vehicle interiors at any honeywell video systems performance given point of time.
Finally, academic writing services save students some time. cctv 16 channel dvr com is that you just can basically be compensated for up to eight articles per month. Cctv camera dvr software free download cctv dvr Tutorial This fundamental portion of fashion dissertations needs to have these segments. we're trying to use our planning technique to get the most effective IELTS Writing score possible; it's not just a race to view who can write the most quickly.