# Question 1

The electric field of a particle at the origin with charge $q$ is given by $\frac{q}{4\pi} \frac{\vec{r}}{|\vec{r}|^3}$ The electric field $\vec{E}$ of two charged particles at two different positions is obtained by adding the electric field of the first particle to the electric field of the second particle.**Write down**the electric field of a pair of particles where the first particle has charge $q$ and is at the origin and the second particle has charge $Q$ and is at position $\vec{a}$.

**Compute**the divergence of this electric field. Draw a picture showing the two particles and four closed surfaces $S_q, SQ_, S_{qQ}$ and $S$, where $S_q$ contains the charge $q$, $S_Q$ contains the charge $Q$, $S_{qQ}$ contains both charges and $S$ contains neither charge.

**Calculate**the flux integral of $\vec{E}$ for each of these four surfaces. Well-organized and explained responses will receive more credit.

# Question 2

Let $D \subset \mathbb{R}^3$ be a region in space. A function $f: D \to \mathbb{R}$ is said to be*harmonic*if it obeys $\nabla f = 0$ where $\nabla f = \vec{\nabla} \cdot \vec{\nabla} f$.

(i) Suppose $f$ is harmonic throughout $D$ and the boundary of $D$ is given by a closed surface $S = \partial D$.

**Show**that $\int_S dA \hat{N} \cdot \vec{\nabla} f = 0$

(ii)

**Show**that if $f$ is harmonic throughout $D$, then $\int_S dA \hat{N} \cdot (f \vec{\nabla} f) = \int_D dV | \vec{\nabla} f|^2$# Question 3

Suppose you are given a vector that depends smoothly on two parameters, i.e. $\vec{r} = \vec{r}(\alpha, \beta)$. Moreover you know that $\vec{r}(0, 0) = j + 2k$. Use calculus to**write down**an approximation for $\vec{r}(.01, .003)$.

**Find**approximately the area of the parallelogram with corners labeled by the vectors $\vec{r}(0, 0), \vec{r}(0, .003), \vec{r}(.01, 0)$ and $\vec{r}(.01, .003)$.

**Apply**both your formulae to the case $\vec{r} = \alpha^2 i + (1 + \alpha + \beta)j + (\alpha \beta + \beta + 2)k$

# Question 4

You probably already know that the volume of one hemisphere of a unit radius sphere is $2π/3$. Lets try to derive this result using the divergence theorem. Let the origin be at the center of a unit sphere and then**compute**the surface integral of the vector field $\vec{V} = \vec{r}$ over one hemisphere of the sphere (you may assume that the surface area of a unit sphere is $4π$).

**Explain**how to use the divergence theorem to rewrite this surface integral as a volume integral.

**Calculate**this volume integral and use this to

**find**the volume of a unit radius hemisphere.

# Question 5

Let $\vec{V}$ be any smooth vector field.**Show**that $\vec{\nabla} \cdot (\vec{\nabla} \times \vec{V}) = 0$

# Question 6

Consider a vector field $\vec{W}$ representing a*linear flow*where $\vec{W}$ points in the same direction and has the same length at every point in space.

**Find**a vector field $\vec{V}$ such that $W$ equals the curl of $\vec{V}$ (you may assume that your $x$-axis and $\vec{W}$ point in the same direction).

**Draw**pictures represetngin $\vec{V}$ and $\vec{W}$. Also

**draw**two closed curves $C_1$ and $C_2$ such that $\int_{C_1} d\vec{r} \cdot \vec{V} = 0$ and $\int_{C_2} d\vec{r} \cdot \vec{V} > 0$. Now

**draw**two surfaces $\Sigma$ and $\Sigma'$ such that $\int_{\Sigma} dA\hat{N} \cdot \vec{W} = \int_{\Sigma'} dA \hat{N} \cdot \vec{W} = \int_{C_2} d\vec{r} \cdot \vec{V}$ Finally

**draw**a surface $S$ for which $\int_S dA \hat{N} \cdot \vec{W} = 0$.

**Indicate**on every surface in your pictures the direction of the unit normal vector $\hat{N}$, and on each field closed curve the direction of integration.

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