Question: For $\vec{F}(\vec{r})=\vec{r}$, find the flux on the surface of a sphere of radius $a$, oriented outward.

Solution: The easiest way to do this is to recall that $d\vec{A}=\vec{n}dA$. We have

\[\int_S\vec{F}\cdot d\vec{A}=\int_S\vec{F}\cdot\vec{n} dA=a\int_S dA\]

since the vectors $\vec{F}$ and $\vec{n}$ are parallel, with $\vec{n}$ being a unit vector (and hence the dot product is simply the magnitude of $\vec{F}$). Thus, we have

\[a\int_S dA=4\pi a^3.\]

Replacement Question: State something objectively coherent about Gauss’ Law.

Answer: I’ll accept a lot of answers here, but what I’m looking for is something along the lines of “electric flux on the surface of a closed object is proportional to the charge contained in the interior of the surface.” If you mumble something about Faraday cages, I’ll give it to you.

Question: State Green’s Theorem

Solution: If $C$ is a piecewise smooth, simple, closed curve that is the boundary of a region $R$ in the plane and oriented so that the region is on the left as one moves around the curve (equivalently, we move around the curve in a counter-clockwise fashion), and if $\vec{F}=F_1\vec{i}+F_2\vec{j}$ is a smooth vector field on an open region containing $R$ and $C$, then

\[\int_C\vec{F}\cdot d\vec{r}=\int_R\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right) dxdy.\]

Replacement Question: State the first ten or so words of the United States Declaration of Independence

Solution: When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another…