Change of Variables Theorem (Multivariable Calculus)

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Theorem

Let $\mathbf F: U \to V$ be a diffeomorphism between open subsets of $\R^2$.

Let $D^* \subset U$ and $D = \map {\mathbf F} {D^*} \subset V$ be bounded subsets.

Let $f: D \to \R$ be a bounded function.

Then

$\ds \iint_D \map f {x, y} \rd x \rd y = \iint_{D^*} \map f {\map {\mathbf F} {u, v} } \size {\map {\det D \mathbf F} {u, v} } \rd u \rd v$

where $\map {\det D \mathbf F} {u, v}$ is the Jacobian.

The equality means that the left-hand integral exists if and only if the right-hand integral does and that, if so, the two integrals are equal.


Proof


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Sources

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