I think the multivariate change-of-variable formula in integration (i.e. involving the determinant of the Jacobian) is still rather indispensable.  The treatment I'm most familiar with is in Folland, where, as far as I recall, it is only used to construct integration in polar coordinates (and I think there was only one exercise, concerning a further extension).  

One could perhaps say that the trick of computing the normalization to a Gaussian random variable, by way of passing through polar coordinates, uses determinants.