I was wondering, when we apply the change of coordinates for an integral, if we need to show that the transformation is globally one-to-one. If it is simply a linear transformation, if the determinant Jacobian is not 0, then it is globally one-to-one. However, if the transformation is not linear, then the Jacobian not being 0 simply shows that it is locally one-to-one.
However, do I need to show that the function is globally one-to-one, within the bounds of the integral?
Also, is there any advice on how to do this? I am not even sure how to show that something like the spherical coordinates is one to one within some bounds.
Thank you!
No it is not enough. A standard example is
$f(x,y) = \begin{pmatrix} e^x\cos y \\ e^x\sin y \end{pmatrix}.\ f$ has non-vanishing Jacobian at all $\vec x\in \mathbb R^2$ but it is not injective.