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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
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(1)
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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
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(2)
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where  is a differential mapping on
the interval ![[c,d]](/images/equations/ChangeofVariablesTheorem/Inline13.gif) and  is the interval
![[a,b]](/images/equations/ChangeofVariablesTheorem/Inline15.gif) 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.gif) |
(3)
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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.gif) |
(4)
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where  is the image of the original
region  ,
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(5)
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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.
Monthly 106, 497-501, 1999.
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