Exercise 9a

1.

Stewart, Section 14.8, Exercises 1, 2, 10, 19.

2.

Show by direct calculation that the conclusion of Stokes Theorem does not hold for $\mathbf{F}= \frac{1}{x^2 + y^2} (-y \mathbf{i}+ x \mathbf{j})$.

3.

Let $r = \sqrt{x^2 + y^2 + z^2}$, $\mathbf{r}= (x\mathbf{i}+ y \mathbf{j}+ z\mathbf{k})$

(a)

Find the gradient of $\frac{1}{r(z-r)}$. Find the curl of $-y \mathbf{i}+ x \mathbf{j}$.

(b)

Show that $\nabla \times \mathbf{F}= \mathbf{E}$, everywhere except on the non-negative part of the $z$-axis, where

$$\begin{split} \mathbf{F}& = \frac{Q}{4 \pi \epsilon_0} \frac... ...frac{Q}{4 \pi \epsilon_0} \frac{1}{r^3} \mathbf{r}, \end{split}$$

and $Q, \epsilon_0$ are strictly positive constants.

(c)

Let $C$ be the curve $x^2 + y^2 =1, z=0$ oriented in the anti-clockwise direction when viewed from above. Use Stokes theorem to show that if $S$ is any oriented surface with oriented boundary $C$, and $S$ does not intersect the non-negative part of the $z-axis$, then

$$\iint_S \mathbf{E}\cdot d\mathbf{S}= -\frac{Q}{2 \epsilon_0}$$

4.

(Optional) Suppose $\mathbf{G}$ is a vector field with $\nabla \cdot \mathbf{G}= 0$, everywhere in $3$-space. Define $\mathbf{H}$ in terms of $\mathbf{G}$ by

$$\begin{split} H_1 & = 0\\ H_2 & = \int_a^x G_3(u,y,z) du\\... ... - \int_a^x G_2(u,y,z) du + \int_b^y G_1(a,v,z) dv, \end{split}$$

where $a,b$ are arbitrary constants. Show by direct calculation that $\nabla \times \mathbf{H}= \mathbf{G}$ everywhere in $3$-space.