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.
January 8, 2010
Posted by John Armstrong |
Analysis, Calculus |
10 Comments
