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change of variables in integral on
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(Theorem)
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This theorem is a generalization of the substitution rule for integrals from one-variable calculus.
To go from the left-hand side to the right-hand side or vice versa, we can perform the formal substitutions:
The volume scaling factor
is sometimes called the Jacobian or Jacobian determinant.
Theorem 1 is typically applied when integrating over
using polar coordinates, or when integrating over
using cylindrical or spherical coordinates.
Intuitively speaking, the image of a small cube centered at , under a differentiable map is approximately the parallelogram resulting from the linear mapping
applied on that cube. If the volume of the original cube is , then the volume of the image parallelogram is
. The integral formula in Theorem 1 follows for an arbitrary set by approximating it by many numbers of small cubes, and taking limits.
Proofs of Theorem 1 can be obtained by making this procedure rigorous; see [7], [1], or [3].
A slightly stronger version of the theorem that does not require to be a diffeomorphism (i.e. that is a bijection and has non-singular derivative) is:
Theorem 2 Let
be continuously differentiable on an open subset of
. Then for any measurable function
, and any measurable set
,
where
counts the number of pre-images in of .
Observe that Theorem 2 (as well as its proof) includes a special case of Sard's Theorem.
The idea of Theorem 2 is that we may ignore those pieces of the set that transform to zero volumes, and if the map is not one-to-one, then some pieces of the image may be counted multiple times in the left-hand integral.
These formulas can also be generalized for Hausdorff measures on
, and non-differentiable, but Lipschitz, functions . See [4] or other geometric measure theory books for details.
- 1
- T. M. Flett. ``On Transformations in
and a Theorem of Sard''. American Mathematical Monthly, Vol. 71, No. 6 (Jun-Jul 1964), p. 623-629.
- 2
- Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
- 3
- Miguel De Guzman. ``Change-of-Variables Formula Without Continuity''. American Mathematical Monthly, Vol. 87, No. 9 (Nov 1980), p. 736-739.
- 4
- Frank Morgan. Geometric Measure Theory: A Beginner's Guide, second ed. Academic Press, 1995.
- 5
- James R. Munkres. Analysis on Manifolds. Westview Press, 1991.
- 6
- Arthur Sard. ``The Measure of the Critical Values of Differentiable Maps''. Bulletins of the American Mathematical Society, Vol. 48 (1942), No. 12, p. 883-890.
- 7
- J. Schwartz. ``The Formula for Change in Variables in a Multiple Integral''. American Mathematical Monthly, Vol. 61, No. 2 (Feb 1954), p. 81-95.
- 8
- Michael Spivak. Calculus on Manifolds. Perseus Books, 1998.
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Cross-references: theory, measure, functions, Lipschitz, multiple, one-to-one, map, Transform, Sard's theorem, derivative, non-singular, bijection, proofs, limits, numbers, linear mapping, parallelogram, differentiable map, cube, image, spherical coordinates, polar coordinates, determinant, Jacobian, factor, scaling, volume, side, Calculus, integrals, measurable set, measurable function, open subsets, diffeomorphism
There is 1 reference to this entry.
This is version 5 of change of variables in integral on , born on 2005-08-28, modified 2007-08-03.
Object id is 7349, canonical name is ChangeOfVariablesInIntegralOnMathbbRn.
Accessed 5813 times total.
Classification:
| AMS MSC: | 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables) | | | 26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume) | | | 28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions) |
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Pending Errata and Addenda
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