Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
List of coordinate charts
Summary
This article lists 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
The key feature of the last two examples is a defined metric tensor which can be used 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), 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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.