Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 17 - Line and Surface Integrals - 17.1 Vector Fields - Exercises - Page 919: 45


$$\phi (x,y,z) = \sin (x y z)+K $$

Work Step by Step

Given $$\mathbf{F}=\langle y z \cos (x y z), x z \cos (x y z), x y \cos (x y z)\rangle$$ We need to find $\phi(x,y,z)$ such that $$\frac{\partial \phi }{\partial x} = y z \cos (x y z),\ \ \ \ \frac{\partial \phi }{\partial y} = x z \cos (x y z),\ \ \ \ \frac{\partial \phi }{\partial z} =xy \cos (x y z) $$ Then \begin{align*} \phi(x,y,z)&= \sin (x y z)+C_1\\ \phi(x,y,z)&= \sin (x y z)+C_1\\ \phi(x,y,z)&= \sin (x y z)+C_1 \end{align*} Hence, we can choose $$\phi(x,y,z)= \sin (x y z)+K$$
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