## The Geometric Interpretation of the Jacobian Determinant

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.

If is injective and continuously differentiable on an open region , and is a compact, connected, Jordan measurable subset of , then we have

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.