## Calculus 8th Edition

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