A theorem which effectively describes how lengths, areas, volumes, and generalized -dimensional
volumes (contents) are distorted by differentiablefunctions. 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-preservinglinear transformationiff ,
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.

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 and is the interval with and (Lax 1999). In two dimensions, the explicit statement
of the theorem is

(3)

and in three dimensions, it is

(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.

Jeffreys, H. and Jeffreys, B. S. "Change of Variable in an Integral." §1.1032 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 32-33, 1988.Kaplan, W. "Change of Variables in
Integrals." §4.6 in Advanced
Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 238-245, 1984.Lax,
P. D. "Change of Variables in Multiple Integrals." Amer. Math.
Monthly106, 497-501, 1999.