Order of variables when computing the Jacobian for the purposes of calculating the change of variables factor?

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?

• Why do we take the absolute value? Is it because ultimately, the determinant represents the "area conversion factor" of an infinitesimal piece of the space we are integrating we are integrating over, and areas (volumes in 3D, and hyper-volumes in $n$D) are always positive? May 22, 2016 at 0:33