Show that (Phi(theta, phi)=(a cos theta sin phi, b sin theta sin phi, c cos phi)) is

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Show that \(\Phi(\theta, \phi)=(a \cos \theta \sin \phi, b \sin \theta \sin \phi, c \cos \phi)\) is a parametrization of the ellipsoid
\[
\left(\frac{x}{a}ight)^{2}+\left(\frac{y}{b}ight)^{2}+\left(\frac{z}{c}ight)^{2}=1
\]
Then calculate the volume of the ellipsoid as the surface integral of \(\mathbf{F}=\frac{1}{3}\langle x, y, zangle\) (this surface integral is equal to the volume by the Divergence Theorem).

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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