We first defined the Jacobian determinant as measuring the factor by which a transformation scales infinitesimal pieces of -dimensional volume. Now, with the change of variables formula and the mean value theorem in hand, we can pull out a macroscopic result.
for some , where .
Simply take in the change of variables formula
The mean value theorem now tells us that
for some between the maximum and minimum of on . But then since is connected, we know that there is some so that , as we asserted.