Calculus 8th Edition

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

Chapter 16 - Vector Calculus - Review - Exercises - Page 1189: 14


$F$ is a conservative vector field and $$\int_CF.dr=2$$

Work Step by Step

As we are given that $F(x,y)=e^yi+(xe^y+e^z)j+ye^zk$ Since, $F=Pi+Qj+Rk$ will be conservative when $R_y=Q_z$,$P_y=Q_x$, and $P_z=R_x$ Thus,$R_y=Q_z=0$,$P_y=Q_x=0$, and $P_z=R_x=0$ This shows that the given vector field $F$ is conservative. By the fundamental theorem of line integrals, we have $\int_CF.dr=f(4,0,3)-f(0,2,0)=2$ Hence, the result is: $$\int_CF.dr=2$$
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