I have observed that the Jacobian of the cylindrical and spherical coordinates is equal to the product of the magnitudes of the normal vectors. Is this true for all coordinate systems?
i.e.: $$\frac{\partial(x,y,z)}{\partial(u,v,w)} = \lvert \frac{\partial \mathbf r}{\partial u} \rvert \lvert \frac{\partial \mathbf r}{\partial v} \rvert\lvert \frac{\partial \mathbf r}{\partial w} \rvert$$