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# Inverse transform The equations $x=g(u, v), y=h(u, v)$ in Figure 15.57 transform the region $G$ in the $u v$ -plane into the region $R$ in the $x y$ -plane. Since the substitution transformation is one-to-one with continuous first partial derivatives, it has an inverse transformation and there are equations $u=\alpha(x, y), v=\beta(x, y)$ with continuous first partial derivatives transforming $R$ back into $G .$ Moreover, the Jacobian determinants of the transformations are related reciprocally by\begin{equation}\frac{\partial(x, y)}{\partial(u, v)}=\left(\frac{\partial(u, v)}{\partial(x, y)}\right)^{-1}\end{equation}Equation $(10)$ is proved in advanced calculus. Use it to find the area of the region $R$ in the first quadrant of the $x y$ -plane bounded by the lines $y=2 x, 2 y=x,$ and the curves $x y=2,2 x y=1$ for $u=x y$ and $v=y / x .$

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Baylor University

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