Question $9$

$4\mathrm{.}5$ pts

Let $C_1$ be the part of the circle from $(5,0)$ to $(0,5)$ in the first quadrant, $C_2$ be the line segment from $(0,5)$ to $(-5,0),$ and $C_3$ be the line segment from $(5,0)$ to $(-5,0).$

$$

If $F(x,y)$ is a conservative vector field on $R^2$ with

$,{\displaystyle \underset{C_1}{\overset{\phantom{}}{}}}F*dr=8$ and ${\displaystyle \underset{C_3}{\overset{\phantom{}}{}}}F*dr=2,$

then ${\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}F*dr=$

$($Be careful about the direction of each curve.$)$

Question $10$

$4\mathrm{.}5$ pts

If $f(x,y,z)$ is the potential function of the vector filed

$F(x,y,z)=($:$y{e}^{xy},x{e}^{xy},-2z$:$)$

so that $f(0,0,0)=4,$ find $f(1,0,1).$

$2$

$1$

$0$

$5$

$3$