Under what conditions can we change the order of integration in any coordinate system?

Question

I thought of this question while mindlessly setting up another integral in spherical coordinates - I usually swap the orders of integration in spherical coordinates without thinking, and I'm not entirely sure why we can do so freely in spherical coordinates as opposed to other coordinate systems - I'm guessing it has something to do with the symmetry of the problem, can anyone help me define it more rigorously?

Answer 1

Once you have formulated your problem as a tripe integral in $r$, $\theta$, $\phi$ coordinates (or whatever they are called in the version of spherical coordinates you are used to), the integral is just a plain triple integral which you can think of being over a region in space with those coordinates as the usual rectangular coordinates. And so all the usual rules of that problem domain apply, including change of order in the integration. That is all there is to it!

Answer 2

The use of symmetry in integration is to create a change of variables to transform a domain of a set into another domain that's preferrably easier to integrate.

Integration is easy if we have rectangles. The change to spherical coordinates transforms a domain in $\mathbb{R}^3$ to a domain that's rectangular (with appropriate modifications in the expressions involved).

It is common then for the integral be easy to compute onwards. For a precise answer to your title question, the answer would be Fubini's theorem.