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.