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: 43


$$\phi (x,y,z) = xy z^2+K $$

Work Step by Step

Given $$\mathbf{F}=\left\langle y z^{2}, x z^{2}, 2 x y z\right\rangle$$ We need to find $\phi (x,y )$ such that $$ \frac{\partial \phi }{\partial x}=y z^2,\ \ \ \frac{\partial \phi }{\partial y}=x z^{2},\ \ \ \ \frac{\partial \phi }{\partial z}=2x y z$$ By integration we get \begin{align*} \phi (x,y,z)&=xy z^2+C_1\\ \phi (x,y,z)&= xy z^2 +C_2 \\ \phi (x,y,z)&=xy z^2 +C_3 \end{align*} Then we can choose $$\phi (x,y,z) = xy z^2+K $$
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