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Suppose the equations $u\=u\left(x\,y\right)\,v\=v\left(x\,y\right)$ map a plane region $R$ in the $\mathrm{xy}$plane to a region $R\prime$ in the $\mathrm{uv}$plane. Suppose further that the mapping is invertible with equations $x\=x\left(u\,v\right)\,y\=y\left(u\,v\right)$ that map $R\prime$ back to $R$.
Let the Jacobian of the map from $R\prime$ to $R$ be given by
$\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\=verbar;\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}verbar;$



Then a double integral over $R$ is related to an equivalent double over $R\prime$ by the "formula"
$\int {\int}_{R}f\left(x\,y\right)\mathrm{dA}$ = $\int {\int}_{R\prime}f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)\left\frac{\partial \left(xcomma;y\right)}{\partial \left(ucomma;v\right)}\right\mathrm{dA}\prime$${}$



where $\mathrm{dA}\=\mathrm{dx}\mathrm{dy}$ or $\mathrm{dy}\mathrm{dx}$, and $\mathrm{dA}\prime \=\mathrm{du}\mathrm{dv}$ or $\mathrm{dv}\mathrm{du}$.
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The Jacobian of the map from $R$ to $R\prime$, that is $\frac{\partial \left(u\,v\right)}{\partial \left(x\,y\right)}$, is the reciprocal of the Jacobian $\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}$. This relationship can be exploited in cases where inverting the mapping equations is algebraically tedious. In some such cases, obtaining the reciprocal of $\frac{\partial \left(u\,v\right)}{\partial \left(x\,y\right)}$ and expressing it in terms of $u$ and $v$ might involve simpler manipulations than a direct calculation of $\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}$.
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