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Substitution in double integrals.

Consider a domain $\mathcal{D}$ in the xy-plane. Suppose now that there exists an invertible function g=(g1,g2), defined over a domain $\mathcal{R}$ in the uv-plane such that (x,y)=( g1(u,v), g2(u,v)). Moreover suppose that g is differentiable over $\mathcal{R}$ (i.e. each gi is differentiable over $\mathcal{R}$). 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

\begin{displaymath}J(u,v) =
\begin{vmatrix}\frac {\partial x}{\partial u} & \fr...
...l y}{\partial u} & \frac {\partial y}{\partial v}

is called the Jacobian (determinant) of g=(g1,g2), i.e. of the coordinate substitution $(x,y) \longrightarrow (u,v)$.

Theorem 7.24  

Noah Dana-Picard