A theorem which effectively describes how lengths, areas, volumes, and generalized 
-dimensional
volumes (contents) are distorted by differentiable functions. In particular, the change of variables theorem
reduces the whole problem of figuring out the distortion of the content to understanding
the infinitesimal distortion, i.e., the distortion of the derivative
(a linear map), which is given by the linear map's
determinant. So 
is an area-preserving linear transformation iff 
,
and in more generality, if 
is any subset of 
, the content of its image is
given by 
times the content of the original. The change of variables
theorem takes this infinitesimal knowledge, and applies calculus
by breaking up the domain into small pieces and adds up
the change in area, bit by bit.
The change of variable formula persists to the generality of differential k-forms on manifolds, giving the formula
 |
(1)
|
under the conditions that 
and 
are compact connected oriented manifolds
with nonempty boundaries, 
is a smooth map which is an orientation-preserving
diffeomorphism of the boundaries.
In one dimension, the explicit statement of the theorem for 
a continuous function of 
is
 |
(2)
|
where 
is a differential mapping on the interval ![[c,d]](/images/equations/ChangeofVariablesTheorem/Inline13.svg)
and 
is the interval ![[a,b]](/images/equations/ChangeofVariablesTheorem/Inline15.svg)
with 
and 
(Lax 1999). In two dimensions, the explicit statement
of the theorem is
![int_Rf(x,y)dxdy=int_(R^*)f[x(u,v),y(u,v)]|(partial(x,y))/(partial(u,v))|dudv](/images/equations/ChangeofVariablesTheorem/NumberedEquation3.svg) |
(3)
|
and in three dimensions, it is
![int_Rf(x,y,z)dxdydz
=int_(R^*)f[x(u,v,w),y(u,v,w),z(u,v,w)]|(partial(x,y,z))/(partial(u,v,w))|dudvdw,](/images/equations/ChangeofVariablesTheorem/NumberedEquation4.svg) |
(4)
|
where 
is the image of the original region 
,
 |
(5)
|
is the Jacobian, and 
is a global orientation-preserving diffeomorphism
of 
and 
(which are open subsets of 
).
The change of variables theorem is a simple consequence of the curl theorem and a little de Rham cohomology.
The generalization to 
dimensions requires no additional assumptions other than the
regularity conditions on the boundary.
