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The change of variables theorem states that given a diffeomorphism [itex]g:A \rightarrow B[/itex] between open sets, and a continuous function [itex]f:A \rightarrow R[/itex], then [itex]

\int _A f = \int _B f \circ g |Det Dg|[/itex] given that either one of the integrals exist.

I was wondering if anyone here could help explain to me the intuition behind the change of variables theorem. More specifically, I'm interested in the intuition (though formal arguments are welcome as well!) behind how the determinant function creeps into the equation and why the determinant of Dg? I'd be grateful for any insight into the theorem. Thanks!

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# Intuition for Change of Variables Theorem

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