$($Sections $16\mathrm{.}4,19\mathrm{.}2)$

Consider the surface

$S=\{(x,y,z)$:$z=\sqrt[\phantom{2}]{4-x^2-y^2}$;$z0\}$

oriented upward. Evaluate the flux integral

$2$

MAT$2615/$AS$4/0/2025$

where $F_{x,y,z}=(-y,x,1),$ by using Stokes's theorem.

$[$Hint: sketch $S$ and its boundary $C-$ then parametrise $C$ and apply Stokes$].$

$[10]$

$4.($Sections $16\mathrm{.}4,19\mathrm{.}2)$

Use Stokes's theorem to evaluate $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{F}_{*}d{r}_{?}$ where

$F_{x,y,z}=(2y(x-z),x^2+z^2,y^3)$

and $C$ is the positively oriented boundary of the part of the plane $2x+3y+4z=12$ in the positive octant $($i$.$e$.x0,y0$ and $z0).$

$[$Hint: sketch $C$ and the surface $S$ of which it is a boundary$-$ then apply Stokes and evaluate the surface integral$].$

$[10]$

$5.($Section $19\mathrm{.}3)$

Use Gauss' Theorem to evaluate the flux integral

${}_S{F}_{*}{n}_{d}S$

where $F_{x,y,z}=(x^3,y^3,z^3)$ and $S$ is the boundary of the volume bounded above by the sphere $x^2+y^2+z^2=4$ and below by the $xY-$plane.

$[$TOTAL: $67]$