Coordinate Transforms: Idiot's Guide & Rules of Thumb

[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.

 

[tacman]

Hello,

Thank you for your question. I understand your struggle with determining the proper set of coordinates for a particular problem. While I am not aware of an "Idiot's Guide" specifically for coordinate transforms, I can offer some general rules of thumb that may be helpful.

First, it is important to understand the purpose of coordinate transforms. They are used to change the coordinates of a point from one system to another. This can be useful when working with different coordinate systems, such as Cartesian, polar, or spherical coordinates.

One rule of thumb is to choose the coordinate system that best suits the problem at hand. For example, if you are working with a problem involving circular motion, polar coordinates may be more useful than Cartesian coordinates. If you are working with a problem involving objects moving in three-dimensional space, spherical coordinates may be more appropriate.

Another rule of thumb is to choose coordinates that simplify the problem. This could mean choosing coordinates that eliminate unnecessary variables or make the equations easier to work with. For example, if you are working with a problem involving a sphere, using spherical coordinates may simplify the equations and make the problem easier to solve.

It can also be helpful to visualize the problem and think about how the coordinates will affect the geometry. This can give you a better understanding of which coordinates will be most useful.

In addition, it may be helpful to consult textbooks or online resources for specific examples and applications of coordinate transforms. This can give you a better understanding of how to choose the proper coordinates for different types of problems.

I hope these rules of thumb are helpful in your search for the proper set of coordinates for your problem. Good luck!