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: 41

Answer

$$\phi (x,y) = e^{xy} $$

Work Step by Step

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