Simply put, does Fubini's Theorem hold for double integrals expressed as in polar terms? I.e., does the following hold given that all integration bounds are constants:
$$ \int_\alpha^\beta \int_a^b f(r, \theta) \, r \, \text{d}r \, \textrm{d}\theta \, \, \, \stackrel{?}{=} \, \, \, \int_a^b\int_\alpha^\beta f(r, \theta) \, r \, \textrm{d}\theta \, \text{d}r $$
Best Answer
The Funini's Theorem applies to any integral of the form $$\int\int fdudv$$ as far as the two dimensional case is concerned (same applies to any dimension). In other words, Fubini's theroem holds for any function of variables $u$ and $v$, independently of what those represent(if they are in polar or cartesian coordinates).
In reality, Fubini's theorem is about functions of multiple variables, and it treats those variables as abstract quantities. The same applies to any theorem that is regarded with multivariable calculus. So, you must see those variables as changing quantinties independent the one from the other and avoid, when possible, the geometric interpretation.