Problem Set 8

1. Green's theorem in the plane


2. Verifying Stokes' theorem

Let $S$ denote the part of the cone $x^2 + y^2 = z^2$, $z > 0$ which lies beneath the plane $x + 2z = 1$. Let $\mb{F}(x, y,z) = x\mb{j}$. Show that the projection of $\partial S$ vertically to the $xy$-plane is an ellipse. Parameterize $\partial S$ and determine $\int_{\partial S} \mb{F} \cdot \de{\mb{r}}$.

Show that $\de\mb{S} \cdot \mb{k} = \de{x}\de{y}$ on $S$ and verify Stokes' Theorem for $\mb{F}$ on $S$.


3. Verifying Stokes' Theorem for a torus

Let $0 < a < b$. Verify Stokes' Theorem when $\mb{F} = [y, z, x]$ and $\sigma$ is the top half of the torus generated by rotating the circle $(x - b)^2 + z^2 = a^2$ about the $z$-axis.


4. Corollary of Stokes' Theorem

The vecto field $\mb{F}(\mb{R})$ is defined by \[ \mb{F}(\mb{R}) = \int_C |\mb{r} - \mb{R}|^2 \, \de{\mb{r}} \]

where $\mb{r}$ lies on a simple closed curve $C$. Show that there are constant vector $\mb{A}$ and $\mb{B}$ such that $\mb{F}(\mb{R}) = \mb{R} \wedge \mb{A} + \mb{B}$. Deduce that \[ \nabla \wedge \mb{F} = -4\iint_S \de{\mb{S}} \]

where $S$ is any smooth surface spanning $C$.