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…

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