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. EDIT: this fact also provides an immediate explanation for the presence of the determinant in the denominator of a multivariate Gaussian (and by positive semi-definiteness of the covariance, that the square root makes sense).