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: 73

Answer

The spherical angles for Helsinki: $\left( {\theta ,\phi } \right) = \left( {25.0^\circ ,29.9^\circ } \right)$. The spherical angles for São Paulo: $\left( {\theta ,\phi } \right) = \left( {313.48^\circ ,113.52^\circ } \right)$.

Work Step by Step

1. Helsinki, Finland (60.1$^\circ$ N, 25.0$^\circ$ E) We have the latitude 60.1$^\circ$ N and the longitude 25.0$^\circ$ E. Since the latitude of Helsinki is north of the equator, $60.1^\circ = 90^\circ - \phi $, So, $\phi = 29.9^\circ $. Since the longitude of Helsinki lies to the east of Greenwich, $\theta = 25.0^\circ $. Thus, the spherical angles $\left( {\theta ,\phi } \right)$ for Helsinki: $\left( {\theta ,\phi } \right) = \left( {25.0^\circ ,29.9^\circ } \right)$. 2. São Paulo, Brazil (23.52$^\circ$ S, 46.52$^\circ$ W) We have the latitude 23.52$^\circ$ S and the longitude 46.52$^\circ$ W. São Paulo's latitude is south of the equator, so $23.52^\circ = \phi - 90^\circ $ and $\phi = 113.52^\circ $. Since 46.52$^\circ$ W refers to 46.52$^\circ$ in the negative $\theta$ direction, we have $\theta = 360^\circ - 46.52^\circ = 313.48^\circ $. Thus, the spherical angles $\left( {\theta ,\phi } \right)$ for São Paulo: $\left( {\theta ,\phi } \right) = \left( {313.48^\circ ,113.52^\circ } \right)$.
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