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: 64

Answer

Please see the figure attached.

Work Step by Step

We have $\rho = \csc \phi \cot \phi $. Write $\rho = \frac{1}{{\sin \phi }}\frac{{\cos \phi }}{{\sin \phi }}$ (1) ${\ \ \ }$ $\rho \sin \phi = \frac{{\cos \phi }}{{\sin \phi }}$ Recall 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. Dividing $z$ by $r$ gives $\frac{z}{r} = \frac{{\cos \phi }}{{\sin \phi }}$ Thus, equation (1) becomes $r = \frac{z}{r}$, ${\ \ \ }$ $z = {r^2}$ Since ${r^2} = {x^2} + {y^2}$, $z = {x^2} + {y^2}$ This is the equation of an elliptic paraboloid (see Eq. (3) of Section 13.6).
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