Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 7

Answer

Domain: all real numbers satisfy $x^{2}+y^{2}+z^{2}\ne0$ Range: $(0,\infty)$ Level surfaces: $f(x,y,z)=c$ (positive constant)

Work Step by Step

Function: $f(x,y,z)=\frac{1}{x^{2}+y^{2}+z^{2}}$ Domain: x, y, and z can be any real numbers except zeros at the same time, that is $x^{2}+y^{2}+z^{2}\ne0$ Range: $f(x,y,z)\gt0$ Level surfaces: $f(x,y,z)=c$ (positive constant) $x^{2}+y^{2}+z^{2}=\frac{1}{c}$ represents a series of spheres. For example, when $c=1$, $x^{2}+y^{2}+z^{2}=1$ is a sphere shown in the figure.
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