I am used to computing the Jacobian, when, say, changing from x,y coordinates to u,v coordinates, as computing the determinant of the derivative matrix of $x_u$,$x_v$, $y_u$, $y_v$, i.e., differentiating with respect to the new variables.
But I recently computed the Jacobian for a surface integral, and it is the magnitude of the cross-product of two partial derivative vectors, which gives the area of a parallelogram spanned by the two vectors -- makes perfect sense, intuitively.
But, why the difference in algorithm? Is the second way of computing the Jacobian ...strictly for surface integrals?
Thanks,