Can you switch the order of the determinants when changing variables using the Jacobian?

Question

Let say we're changing the variables and we use the Jacobian to do this. Lets say we integrate in respect to $u$ and $v$, does it matter if we set up the integral like $\int\int\,\mathrm{d}u\mathrm{d}v$ or $\int\int\,\mathrm{d}v\mathrm{d}u$ after calculating the Jacobian?

Answer 1

What I think you are asking is (relevant to the question)

does the Jacobian change if we consider $$ \newcommand{\pwrt}[2]{\frac{\partial #1}{\partial #2}} J = \left|\det \pmatrix{ \pwrt xu & \pwrt xv\\ \pwrt yu & \pwrt yv}\right| $$ as opposed to $$ \newcommand{\pwrt}[2]{\frac{\partial #1}{\partial #2}} J = \left|\det \pmatrix{ \pwrt xv & \pwrt xu\\ \pwrt yv & \pwrt yu}\right| $$

The answer to this question is no. Switching rows/columns changes the sign but not the magnitude of the determinant of a matrix.