Hi I'm trying to integrate a function $e^{f(x,y)}$ over a x-y plane and I have some transforming variables : $$ u=x-y $$ $$ v=x+y $$ now the determinant of the jacobian matrix would be positive since $$ \begin{pmatrix} u \\ v \\ \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 1 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} $$ is the matrix. Therefore if I were to integrate over the u-v plane, I get positive values(as I expected). But the problem rises when I instead write $$ \begin{pmatrix} v \\ u \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} $$ now the determinant is negative. Since it's the exponential form function that I am integrating, it should give me a positive value but for this case it gives me a negative value. Now I know that somewhere 'flipping' has occurred, but how do I distinguish which one to use, $$ \begin{pmatrix} u \\ v \\ \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 1 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} $$ or $$ \begin{pmatrix} v \\ u \\ \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 1 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} $$ ??
Should I just always use the definite value of the determinant of the Jacobian matrix? Please help, thanks:)