Question $6(24$ marks$)$

Consider the force field vec$(F)=(x+2yz)hat()+(y+2xz)hat()+(z+2xy)hat(k).$

a$)$ Prove that vec$(F)$ is irrotational.

$(3$ marks$)$

b$)$ Find a potential function for it$.$

$(8$ marks$)$

c$)$ By using the potential function in b$),$ find the work done in moving a particle from the origin to $(1,2,3).$

$(3$ marks$)$

d$)$ By formulating a direct line integral, show that the same work done in c$)$ can be obtained by using a straight line path, $W={\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F),$vec$(dr).$

$(10$ marks$)$

Question $7(11$ marks$)$

Use Stoke's Theorem to evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*vec(dr)$ where $C$ is the boundary of the plane $2x+y+2z=2$ in the first octant given that vec$(F)=e^{-x}$hat$(i)+e^x$hat$(j)+e^z$hat$(k).$