Choosing useful coordinates for a problem in differential geometry

[SteveKeeling]

Does anyone know of an “Idiot’s Guide to Coordinate Transforms…”, or good rules of thumb to employ to determine the “proper” set of coordinates for a particular problem? I’m not really having any trouble with the mathematical machinery like finding the Jacobian, etc.; my problem is actually determining a useful set of coordinates. I would appreciate any effort to point me to a reference or in the proper direction. Thanks in advance,

 

[matt grime]

The coordinates to choose might be implied by the shape of the domain of the integration.

If you had say the diamond going through the points (1,0) (0,1) (-1,0) (0,-1) then the equations of the boundary are things like: x+/-y=+/-1
suggesting the best coordinates are u=x+y and v=x-y, as u and v range between -1 and 1.

Sometimes the integrand will imply what to pick: anything with a sqrt(x^2+y^2) might be tractable with polars.

Do lots of examples. It is the only way to learn to do, erm, examples.