change of variables in integral on
To go from the left-hand side to the right-hand side or vice versa, we can perform the formal substitutions:
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.
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 (http://planetmath.org/AreaFormula) on , and non-differentiable, but Lipschitz, functions . See  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. “http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07812-8/S0002-9904-1942-07812-8.pdfThe 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.
|Title||change of variables in integral on|
|Date of creation||2013-03-22 15:29:32|
|Last modified on||2013-03-22 15:29:32|
|Last modified by||stevecheng (10074)|
|Synonym||integral substitution formula|
|Synonym||integral substitution rule|