Solved: Calculus In Exercises 4954, find the Jacobians of

Chapter 3, Problem 52

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In Exercises 49–54, find the Jacobians of the functions. If x, y, and z are continuous functions of u, v, and w with continuous first partial derivatives, then the Jacobians J(u, v) and J(u, v, w) are

\(J(u, v)=\left|\begin{array}{ll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right| \text { and } J(u, v, w)=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\)

\(x=e^{u} \sin v, \quad y=e^{u} \cos v\)

Text Transcription:

J(u, v)=|frac partial x partial u & frac partial x partial v frac partial y partial u & frac partial y partial v| and  J(u, v, w)=|frac partial x partial u & frac partial x partial v & frac partial x partial w frac partial y partial u & frac partial y partial v & frac partial y partial w frac partial z partial u & frac partial z partial v & frac partial z partial w|

x=e^u sin v,  y=e^u cos v