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Question

# Find the Jacobian $\frac{\partial(x, y)}{\partial(u, v)} \text { or } \frac{\partial(x, y, z)}{\partial(u, v, w)}$.$x=u \cos v, y=u \sin v, z=w e^{u v}$

Solution

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\begin{aligned}\dfrac{\partial(x,y, z)}{\partial(u,v, w)}&=\begin{vmatrix}\dfrac{\partial x}{\partial u}& \dfrac{\partial x}{\partial v} & \dfrac{\partial x}{\partial w} \\ \\\dfrac{\partial y}{\partial u}& \dfrac{\partial y}{\partial v} & \dfrac{\partial y}{\partial w}\\ \\ \dfrac{\partial z}{\partial u} & \dfrac{\partial z}{\partial v} & \dfrac{\partial z}{\partial w}\end{vmatrix}\\&=\begin{vmatrix}\dfrac{\partial (u\cos{v})}{\partial u}& \dfrac{\partial (u\cos{v})}{\partial v} & \dfrac{\partial (u\cos{v})}{\partial w} \\ \\\dfrac{\partial (u\sin{v})}{\partial u}& \dfrac{\partial (u\sin{v})}{\partial v} & \dfrac{\partial (u\sin{v})}{\partial w}\\ \\ \dfrac{\partial (we^{uv})}{\partial u} & \dfrac{\partial (we^{uv})}{\partial v} & \dfrac{\partial (we^{uv})}{\partial w}\end{vmatrix}\\&=\begin{vmatrix}\cos{v}& -u\sin{v} & 0\\ \sin{v} & u\cos{v} & 0\\ wve^{uv} & wue^{uv} & e^{uv}\end{vmatrix}\\ &=\cos{v}\begin{vmatrix}u\cos{v} & 0\\ wue^{uv} &e^{uv}\end{vmatrix}-(-u\sin{v})\cdot \begin{vmatrix}\sin{v} & 0\\ wve^{uv} &e^{uv}\end{vmatrix}+0\\&=ue^{uv}\cos^2{v}+ue^{uv}\sin^2{v}\\&=ue^{uv}\end{aligned}

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