$(1$ point$)$

Verify Stokes' theorem for the helicoid $(r,)=($:$rcos,rsin,$:$)$ where $(r,)$ lies in the rectangle $[0,1][0,\frac{}{2}],$ and $F$ is the vector field $F=($:$3z,9x,5y$:$).$ First, compute the surface integral:

${}_M(gradF)*dS={\displaystyle \underset{a}{\overset{b}{}}}{\displaystyle \underset{c}{\overset{d}{}}}f(r,)drd,$ where

$a=p,b=\frac{}{2},d=$

$,$ and

$f(r,)=5t$

$($use $"$t$"$ for theta$).$

Finally, the value of the surface integral is

Next compute the line integral on that part of the boundary from $(1,0,0)$ to $(0,1,\frac{}{2}).{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{a}{\overset{b}{}}}g()d,$ where

$a=,b=,$ and

$g()=($use $"$t$"$ for theta$).$