Consider the transformation that converts from polar to cartesian coordinates:
\begin{align} x &= r\cos{\theta} \\ y &= r\sin{\theta} \end{align}
To compute the change of coordinates, we would take the determinant of the matrix:
\begin{bmatrix} \partial x/\partial r & \partial x/\partial \theta \\ \partial y/\partial r & \partial y/\partial \theta \end{bmatrix}
Or, we could take the determinant of the matrix:
\begin{bmatrix} \partial x/\partial \theta & \partial x/\partial r \\ \partial y/\partial \theta & \partial y/\partial r \end{bmatrix}
One wold give me the factor $r$ and the other $-r$. Which one is "correct"? In general, how does one decide upon the order of variables (i.e. the order of columns and rows of the Jacboian matrix) to calculate the change of variables factor?