Verify Stokes' theorem for the helicoid $\backslash$Psi $($r$,\backslash$theta $)=($:rcos$\backslash$theta $,$rsin$\backslash$theta $,\backslash$theta :) where $($r$,\backslash$theta $)$ lies in the rectangle $[0,1]\backslash$times $[0,(\backslash$pi $)/(2)],$ and F is the vector field F$=($:$8$z$,8$x$,9$y:$).$

First, compute the surface integral:

$\_($M$)($grad$\backslash$times F$)*$dS$=\backslash$int$\_$a$^$b $\backslash$int$\_$c$^$d f$($r$,\backslash$theta $)$drd$\backslash$theta $,$ where

a$=$

$,$ b$=$

$,$ c$=$

$,$ d$=$

$,$ and

f$($r$,\backslash$theta $)=$

$($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 $(0,1,(\backslash$pi $)/(2)).$

$\backslash$int$\_$C F$*$dr$=\backslash$int$\_$a$^$b g$(\backslash$theta $)$d$\backslash$theta $,$ where

a$=$

$,$ b$=$

$,$ and

g$(\backslash$theta $)=$

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