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The change of variables formula for multiple integrals. Let n be a positive integer.
 

Summary: The change of variables formula for multiple integrals.
Let n be a positive integer.
Theorem. Suppose U is an open subset of Rn
,
f : U Rn
and the following conditions hold:
(i) f is continuously differentiable;
(ii) f is univalent and
(iii) ker f(a) = {0} whenever a U.
Then
Ln
(f[A]) =
A
|det f(x)| dx
whenever A is a Lebesgue measurable subset of U.
Proof. We set
||x|| = max{|xi| : i = 1, . . . , n} for each x Rn
and note that || || is a norm on Rn
. We let ||| ||| be the corresponding norm on L(Rn
; Rn

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics