In multi-variable calculus we have the Change of Variables formula but it holds under certain assumptions: $f$ needs to be compactly supported continuous, then the map $T$ (the change of variable map) needs to be $C^1$ diffeomorphism of an open set. But suppose $T:[0,1]\times [0,2\pi]\to \mathbb R^2$ given by $T(r,\theta)=(r\cos\theta,r\sin\theta)$. In such a case $T$ is NOT defined on an open set.
How does one handle such situations where the conditions like $f$ being compactly supported, with support contained in range($T$). $T$ being defined on an open set, are not satisfied?
For concreteness, let $f(x,y)=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}$ then suppose I want to find $\int_{x^2+y^2\leq1} f(x,y)dxdy$. Please clarify how the change of variables can be used.
When the transformation formula is applied in a concrete situation one usually sets up an equality of the form $$\int_B f(x,y)\>{\rm d}(x,y)=\int_{\hat B}f\bigl(T(r,\theta)\bigr)J_T(r,\theta)\>{\rm d}(r,\theta)\ ,\tag{1}$$ whereby $T$ maps the compact parameter domain $\hat B$ "essentially bijectively" onto the given domain $B$. The functions appearing in the integrals $(1)$ are assumed to be so nice, that both integrals are well defined by themselves. The essential point is that they are equal. For the proof it is then sufficient to approximate $\hat B$ and $B$ by smaller domains $\hat B_\epsilon$ and $B_\epsilon$, whereby one then can make strict assumptions about bijectivity, nonvanishing of the Jacobian, etc., for the restriction $T\restriction\hat B_\epsilon$. Such arguments then allow to prove that the integrals $(1)$ are equal with an error $<\epsilon$; and as $\epsilon>0$ was arbitrary we are done. Note that a full proof of $(1)$ that really takes care of everything is indeed a very lengthy matter.