Solved: In Exercises 3540, find the Jacobian for theindicated change of variables. Ifand

Chapter 14, Problem 39

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In Exercises 35-40, find the Jacobian \(\partial(x,\ y,\ z)/\partial(u,\ v,\ w)\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w), then the Jacobian of x, y, and z with respect to u, v, and w is

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

Spherical Coordinates

\(x=\rho \sin \phi \cos \theta,\ \ y=\rho \sin \phi \sin \theta,\ \ z=\rho \cos \phi\)

Text Transcription:

partial(x, y, z)/partial(u, v, w)

partial(x, y, z)/partial(u, v, w) = |partial x/ partial u  partial x/ partial v  partial x / partial w  _  partial y/partial u  partial y/ partial v  partial y/ partial w  _  partial z / partial u  partial z / partial v  partial z / partial w|

x=rho sin phi cos theta, y=rho sin phi sin theta, z=rho cos phi