Problem $5$:

a$-$ A necessary and sufficient condition that $o\frac{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}}{b}ar(F)*\frac{d}{b}ar(r)=0$ for every closed curve C is that grad$\frac{}{b}ar(F)=0$ identically. Which theorem can be used to prove this statement.

b$-$ In Dynamics, If the work done by a force is independent of the path $($depends only on end points$),$ it will be called a conservative force. A conservative force satisfies the statement mentioned in case a of this problem. Can we say that the following force is conservative or not : $\frac{,}{b}ar(F)=\frac{2x{z}^3+6y}{b}ar()+\frac{6x-2yz}{b}ar()+\frac{3x^2{z}^2-y^2}{b}ar(k)$

Problem $6$:

Evaluate $\frac{{}_S}{b}ar(F)\frac{*}{b}ar(n)ds,$ where s is the unit sphere defined by

$S=\{(x,y,z)R^3$:$x^2+y^2+z^2=1\}$

And $\frac{?}{\text{b}\text{a}\text{r}}(F)$ is the vector field

$\frac{?}{\text{b}\text{a}\text{r}}(F)=\frac{x}{b}ar()+\frac{y^2}{b}ar()+8\frac{z^2}{b}ar(k)$