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## Substitution in double integrals.

Consider a domain in the xy-plane. Suppose now that there exists an invertible function g=(g1,g2), defined over a domain in the uv-plane such that (x,y)=( g1(u,v), g2(u,v)). Moreover suppose that g is differentiable over (i.e. each gi is differentiable over ). Thus, we have: f(x,y)=f(g1(u,v), g2(u,v)).

Definition 7.23   Suppose that f, g1 and g2 have continuous first order partial derivatives. The determinant

is called the Jacobian (determinant) of g=(g1,g2), i.e. of the coordinate substitution .

Noah Dana-Picard
2001-05-30