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This article attempts to conveniently list articles on some of the most useful coordinate charts in some of the most useful examples of Riemannian manifolds. The notion of a coordinate chart is fundamental to various notions of a ''manifold'' which are used in mathematics. In order of increasing ''level of structure:'' *topological manifold *smooth manifold *Riemannian manifold and semi-Riemannian manifold For our purposes, the key feature of the last two examples is that we have defined a metric tensor which we can use to integrate along a curve, such as a geodesic curve. The key difference between Riemannian metrics and semi-Riemannian metrics is that the former arise from bundling positive-definite quadratic forms, whereas the latter arise from bundling indefinite quadratic forms. A four-dimensional semi-Riemannian manifold is often called a Lorentzian manifold, because these provide the mathematical setting for metric theories of gravitation such as general relativity. For many topics in applied mathematics, mathematical physics, and engineering, it is important to be able to write the most important partial differential equations of mathematical physics *heat equation *Laplace equation *wave equation (as well as variants of this basic triad) in various coordinate systems which are ''adapted'' to any symmetries which may be present. While this may be how many students first encounter a non-Cartesian coordinate chart, such as the cylindrical chart on E3 (three-dimensional Euclidean space), it turns out that these charts are useful for many other purposes, such as writing down interesting vector fields, congruences of curves, or frame fields in a convenient way. Listing commonly encountered coordinate charts unavoidably involves some real and apparent overlap, for at least two reasons: *many charts exist in all (sufficiently large) dimensions, but perhaps only for certain families of manifolds such as spheres, *many charts most commonly encountered for specific manifolds, such as spheres, actually can be used (with an appropriate metric tensor) for more general manifolds, such as spherically symmetric manifolds. Therefore, seemingly any attempt to organize them into a list involves multiple overlaps, which we have accepted in this list in order to be able to offer a convenient if messy reference. We emphasize that ''this list is far from exhaustive''. ==Favorite surfaces== Here are some charts which (with appropriate metric tensors) can be used in the stated classes of Riemannian and semi-Riemannian surfaces: *isothermal chart *Radially symmetric surfaces: * *polar chart *Surfaces embedded in E3: * *Monge chart *Certain minimal surfaces: * *asymptotic chart (see also asymptotic line) Here are some charts on some of the most useful Riemannian surfaces (note that there is some overlap, since many charts of S2 have closely analogous charts on H2; in such cases, both are discussed in the same article): *Euclidean plane E2: * *Cartesian chart * *Maxwell chart *Sphere S2: * *polar chart (arc length radial chart) * *stereographic chart * *central projection chart * *axial projection chart * *Mercator chart *Hyperbolic plane H2: * *polar chart * *stereographic chart (Poincaré model) * *upper half space chart (another Poincaré model) * *central projection chart (Klein model) * *Mercator chart Favorite semi-Riemannian surface: *AdS2 (or S1,1) and dS2 (or H1,1): * *central projection * *equatorial trig ''Note:'' the difference between these two surfaces is in a sense merely a matter of convention, according to whether we consider either the cyclic or the non-cyclic coordinate to be timelike; in higher dimensions the distinction is less trivial. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of coordinate charts」の詳細全文を読む スポンサード リンク
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