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: 46

Answer

The spherical coordinates is $\left( {\rho ,\theta ,\phi } \right) = \left( {2\sqrt 3 ,\pi ,\frac{\pi }{3}} \right)$.

Work Step by Step

We have in cylindrical coordinates: $\left( {r,\theta ,z} \right) = \left( {3,\pi ,\sqrt 3 } \right)$. In spherical coordinates: 1. the radial coordinate is $\rho = \sqrt {{x^2} + {y^2} + {z^2}} = \sqrt {{r^2} + {z^2}} = \sqrt {{3^2} + {{\left( {\sqrt 3 } \right)}^2}} = 2\sqrt 3 $ 2. the angular coordinate $\theta=\pi$ 3. the angular coordinate $\phi$ satisfies $\cos \phi = \frac{z}{\rho } = \frac{{\sqrt 3 }}{{2\sqrt 3 }} = \frac{1}{2}$, ${\ \ }$ $\phi = \frac{\pi }{3}$ Therefore, the spherical coordinates is $\left( {\rho ,\theta ,\phi } \right) = \left( {2\sqrt 3 ,\pi ,\frac{\pi }{3}} \right)$.
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