MATH 21D

Final | Fall '15

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.

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

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

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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 $\vec{r} = \alpha^2 i + (1 + \alpha + \beta)j + (\alpha \beta + \beta + 2)k$

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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.

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Question 5

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

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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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