Chapter 8: Chapter 8

a) Verify Green's Theorem for the vector field \(F^{-}(x, y)=(-y, x)\) where $\mathrm{C}^{-}(\mathrm{t})=(\mathrm{r} \cos \mathrm{t}, \mathrm{r} \sin \mathrm{t})-\pi \leq \mathrm{t} \leq \pi$ and \(\mathrm{R}=(\mathrm{P}:|\mathrm{P}| \leq \mathrm{r})\). b) Evaluate $\oint_{\mathrm{C}}\left(\mathrm{x}^{2}+2 \mathrm{y}^{2}\right) \mathrm{dx}\( where \)\mathrm{C}$ is the square with vertices \((1,1),(1,-1),(-1,-1),(-1,1)\)

Use Green's Theorem to find: a) \(\mathrm{y}^{2} \mathrm{~d} \mathrm{x}-\mathrm{xdy}\) clockwise around the triangle whose vertices are at \((0,0),(0,1),(1,0)\). b) The integral of the vector field $\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y})=(\mathrm{y}+3 \mathrm{x}, 2 \mathrm{y}-\mathrm{x})$ counterclockwise around the ellipse \(4 \mathrm{x}^{2}+\mathrm{y}^{2}=4\)

Verify Green's Theorem for \(\int_{C}\left(x^{2}-y^{2}\right) d x+2 x y d y\), where \(\mathrm{C}\) is the clockwise boundary of the square formed by the lines \(\mathrm{x}=0, \mathrm{x}=2, \mathrm{y}=0\), and \(\mathrm{y}=2\)

Use Green's Theorem to deduce the integral formula: $\iint_{R}\left[\left(\partial^{2} u / \partial x^{2}\right)+\left(\partial^{2} u / \partial y^{2}\right)\right] d x d y=\int_{C}(\partial u / \partial n) d x$ where s is the arc length along \(\mathrm{C}\) and \(\mathrm{n}\) is the outer normal to \(\mathrm{C}\).

Show that if the flow \(\mathrm{F}^{\rightarrow}\) is defined in a rectangle \(\mathrm{R}\), and has zero divergence at every point of \(R\), then the rate of flow across every closed path in \(\mathrm{R}\) is zero.