I am currently learning about polar coordinate transformation, especially for integrating over certain regions. Let's say we have to calculate
$\int_{n}{xy \; dx dy}$
Then I think the correct transformation would be (note the functional determinant $r$)
$\int_{m}{r cos \phi \cdot r sin \phi \cdot r \; d\phi dr}$
First question: Is that correct?
My second question is: I have some examples a friend of mine once wrote for an exam. The task was for the following region
$ B = \{ (x,y) \in R : 0 \leq x^2 + y^2 \leq 1, x \leq y \} $
to calculate the integral
$\int_{B}{x^2+y^2 \; dx dy}$
And he began with
$\int_{B}{x^2+y^2 \; dx dy} = \int_{0}^{\frac{5\pi}{4}} \int_{0}^{1} r^2 sin^2 \phi + r^2 cos^2 \phi \; d\phi dr$
Second question: Where is the functional determinant here? Is it missing or am I wrong assuming there should belong one?
I'd be happy if anyone could help me with some insight on this. Thanks!