# Two Stokes' theorem problems

1) Use Stokes' Theorem to evaluate $$\displaystyle\iint_S\mathrm{curl}~\mathbf{F}\cdot d\mathbf{S}$$ $\mathbf{F}(x,y,z) = xyz~\mathbf{i}+ xy~\mathbf{j}+ x^2yz~\mathbf{k}$. $S$ consists of the top and four sides (but not the bottom) of the cube with vertices $(\pm9,\pm9,\pm9)$, oriented outward.

2) Use Stokes' Theorem to evaluate $$\displaystyle\iint_S\mathrm{curl}~\mathbf{F}\cdot d\mathbf{S}$$ $\mathbf{F}(x,y,z) = x^2z^2~\mathbf{i} + y^2z^2~\mathbf{j} + xyz~\mathbf{k}$. $S$ is the part of the paraboloid $z = x^2+y^2$ that lies inside the cylinder $x^2+y^2 = 25$, oriented upward.

I am pretty much lost.

• "I am pretty much lost" conveys no useful information. Do you understand what all the things are? Like, do you understand vector notation? Do you know what curl is and how to compute it? Do you know how to interpret ${\bf F}\cdot d{\bf S}$? Do you know what to do with the fact that the domain of integration is those panels of the cube? Please write down all of your thoughts, your specific difficulties, and any work or effort you've put into the problem. – arctic tern Dec 9 '15 at 22:26
• In particular, since each question says "use Stokes' theorem", what does Stokes' Theorem say? – user247327 Dec 9 '15 at 22:46

Stokes' theorem equates { the flux of $\operatorname{Curl}\mathbf{F}$ through any surface $S$ with an oriented boundary $C$ } with { a work integral on $C$ }. The theorem lets you replace a complicated surface $S$ in your integrals with a simpler surface $S'$ as long as the surface normals are pointing so as to give the same orientation for $C$.
In (i) you can replace $S$ by the top face of the cube ($0\le x\le 9,0\le y\le9,z=9$) with normal pointing downward. Picture the cube-without-top shrinking into the region where the top face would be; the surface normal ends up as $-\mathbf{k}$.
If (ii), the boundary $C$ of $S$ is the circle $x^2+y^2=25,z=25$. Imagine $S$ shrinking upward into $S'$, the disc with boundary $C$. The inward pointing normals on $S$ end up as an upward point normal $\mathbf{k}$ on $S'$.