List of coordinate charts

List of coordinate charts

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 manifoldFor 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.

Favorite Riemannian three-manifolds

Here are some charts which (with appropriate metric tensors) can be used in the stated classes of three-dimensional Riemannian manifolds:

*Diagonalizable manifolds:
**isothermal chart("Note:" not every three manifold admits an isothermal chart.)

*Axially symmetric manifolds:
**cylindrical chart
**parabolic chart
**hyperbolic chart
**prolate spheroidal chart (rational and trigonometric forms)
**oblate spheroidal chart (rational and trigonometric forms)
**toroidal chart

Here are some charts which can be used on some of the most useful Riemannian three-manifolds:

*Three-dimensional Euclidean space E3:
**cartesian
**polar spherical chart
**cylindrical chart
**elliptical cylindrical, hyperbolic cylindrical, parabolic cylindrical charts
**parabolic chart
**hyperbolic chart
**prolate spheroidal chart (rational and trigonometric forms)
**oblate spheroidal chart (rational and trigonometric forms)
**toroidal chart
**Cassini toroidal chart and Cassini bipolar chart

*Three-sphere S3
**polar chart
**stereographic chart
**Hopf chart

*Hyperbolic three-space H3
**polar chart
**upper half space chart (Poincaré model)
**Hopf chart

A few higher dimensional examples

*Sn
**Hopf chart

*Hn
**upper half space chart (Poincaré model)
**Hopf chart

Omitted examples

There are of course many important and interesting examples of Riemannian and semi-Riemannian manifolds which are not even mentioned here, including:
*Bianchi groups: there is a short list (up to local isometry) of three-dimensional real Lie groups, which when considered as Riemannian-three manifolds give homogeneous but (usually) non-isotropic geometries.
*other noteworthy real Lie groups,
*Lorentzian manifolds which (perhaps with some added structure such as a scalar field) serve as solutions to the field equations of various metric theories of gravitation, in particular general relativity. There is some overlap here; in particular:
*axisymmetric spacetimes such as Weyl vacuums possess various charts discussed here; the prolate spheroidal chart turns out to be particularly useful,
*de Sitter models in cosmology are, as manifolds, nothing other than H1,3 and as such possess numerous interesting and useful charts modeled after ones listed here.

In addition, one can certainly consider coordinate charts on complex manifolds, perhaps with metrics which arise from bundling Hermitian forms. Indeed, this natural generalization is just the tip of iceberg. However, these generalizations are best dealt with in more specialized lists.

ee also

*metric tensor
*List of mathematics lists, particularly:
**List of multivariable calculus topics
**List of Fourier analysis topics
**List of differential geometry topics


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… …   Wikipedia

  • List of selected stars for navigation — The selected stars for navigation are often used for sextant observations. Fifty eight selected navigational stars are given a special status in the field of celestial navigation. Of the approximately 6,000 stars visible to the naked eye under… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • List of United States Navy ratings — From left to right: a Special Warfare Operator 1st Class and a Boatswain s Mate 2nd Class. United States Navy ratings are general occupations that consist of specific skills and abilities. Each naval rating has its own specialty badge, which is… …   Wikipedia

  • Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia

  • Differentiable manifold — A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the Tropic of Cancer is a smooth curve, whereas in the first it has a sharp… …   Wikipedia

  • Manifold — For other uses, see Manifold (disambiguation). The sphere (surface of a ball) is a two dimensional manifold since it can be represented by a collection of two dimensional maps. In mathematics (specifically in differential geometry and topology),… …   Wikipedia

  • Spacetime — For other uses of this term, see Spacetime (disambiguation). Two dimensional analogy of spacetime distortion. Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. White lines do not represent the… …   Wikipedia

  • Topological manifold — In mathematics, a topological manifold is a Hausdorff topological space which looks locally like Euclidean space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”