Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 47

Answer

The cylindrical coordinates: $\left( {r,\theta ,z} \right) = \left( {2\sqrt 2 ,0,2\sqrt 2 } \right)$.

Work Step by Step

We have in spherical coordinates $\left( {\rho ,\theta ,\phi } \right) = \left( {4,0,\frac{\pi }{4}} \right)$. The relations between cylindrical and spherical coordinates can be found using rectangular coordinates $x$, $y$, $z$ as in the following: $x = r\cos \theta = \rho \sin \phi \cos \theta $ $y = r\sin \theta = \rho \sin \phi \sin \theta $ $z = \rho \cos \phi $ So, $r = \rho \sin \phi $ and $z = \rho \cos \phi $. Whereas $\theta$ is the same in both cylindrical and spherical coordinates. In cylindrical coordinates: 1. the radial coordinate is $r = \rho \sin \phi = 4\sin \frac{\pi }{4} = 2\sqrt 2 $ 2. the angular coordinate $\theta=0$ 3. the $z$-coordinate satisfies $z = \rho \cos \phi = 4\cos \frac{\pi }{4} = 2\sqrt 2 $ Therefore, the cylindrical coordinates is $\left( {r,\theta ,z} \right) = \left( {2\sqrt 2 ,0,2\sqrt 2 } \right)$.
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