Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.5 Curl and Divergence - 16.5 Exercises - Page 1149: 17


$f(x,y,z)=xe^{yz}+K$; Conservative

Work Step by Step

The vector field $F$ is conservative when $curl F=0$ When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$ Now, $curl F=[(xe^{yz}-xyze^{yz})-(xe^{yz}-xyze^{yz})]i+[(ye^{yz}-ye^{yz})]j+[(ze^{yz}-ze^{yz})-k=0$ Thus, we have the vector field $F$ is conservative. Consider $f(x,y,z)=x(e^{yz})+g(y,z)$ and $f_y=xz(e^{yz})+g_y$ or, $g'(y)=0$ Thus, we have $g_y=h(z)$ Also, $f_y=xz(e^{yz}) \implies g(y,z)=y \sin z+h(z)$ Now, we have $f(x,y,z)=xe^{yz}+h(z) \implies f_z=xy(e^{yz})+h'(z)$ Thus, we get $h'(z)=0$ Hence, we have $f(x,y,z)=xe^{yz}+K$
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