Solution: Finding a Jacobian In Exercises 3540, find the Jacobian for the indicated
Chapter 14, Problem 40(choose chapter or problem)
In Exercises 35 - 40, find the Jacobian
\(\frac{\partial(x, y, z)}{\partial(u, v, w)}\)
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
\(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\)
Cylindrical Coordinates
\(x=r \cos \theta, y=r \sin \theta, z=z\)
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 = r cos theta, y = r sin theta, z = z
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