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:

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.

Read more about List Of Coordinate ChartsFavorite Surfaces, Favorite Riemannian Three-manifolds, A Few Higher Dimensional Examples, Omitted Examples

Other articles related to "list of coordinate charts, list, charts, chart, coordinate charts":

List Of Coordinate Charts - Omitted Examples
... 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 ... such as Weyl vacuums possess various charts discussed here the prolate spheroidal chart turns out to be particularly useful, de Sitter models in ... In addition, one can certainly consider coordinate charts on complex manifolds, perhaps with metrics which arise from bundling Hermitian forms ...

Famous quotes containing the words list of, charts and/or list:

    Do your children view themselves as successes or failures? Are they being encouraged to be inquisitive or passive? Are they afraid to challenge authority and to question assumptions? Do they feel comfortable adapting to change? Are they easily discouraged if they cannot arrive at a solution to a problem? The answers to those questions will give you a better appraisal of their education than any list of courses, grades, or test scores.
    Lawrence Kutner (20th century)

    There’s one basic rule you should remember about development charts that will save you countless hours of worry.... The fact that a child passes through a particular developmental stage is always more important than the age of that child when he or she does it. In the long run, it really doesn’t matter whether you learn to walk at ten months or fifteen months—as long as you learn how to walk.
    Lawrence Kutner (20th century)

    Every morning I woke in dread, waiting for the day nurse to go on her rounds and announce from the list of names in her hand whether or not I was for shock treatment, the new and fashionable means of quieting people and of making them realize that orders are to be obeyed and floors are to be polished without anyone protesting and faces are to be made to be fixed into smiles and weeping is a crime.
    Janet Frame (b. 1924)