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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, June 6, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 6 \emph{Updated May 29}}
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\begin{document}

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\begin{questions}
    
\question Use the divergence theorem to calculate the flux of \ $\mathbf{F} = (x-y) \, \mathbf{i} + (y-z) \, \mathbf{j} + (z-x) \, \mathbf{k}$ \ out of the unit sphere \ $x^2 + y^2 + z^2 = 1$.

\medskip

\question Let $S$ be the boundary surface of a solid region $W$.  Show that
\begin{displaymath} \iint_S \mathbf{r} \boldsymbol{\cdot} \mathbf{n} \, dS = 3 \ \text{volume}(W) . \end{displaymath}
Explain this result geometrically.

\medskip

\question Let $W$ be the pyramid with top vertex $(0,0,1)$, and base vertices at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(1,1,0)$. \ Let $S$ be the closed boundary surface of $W$, oriented outward from $W$. \ Let \ $\mathbf{F}(x,y,z) = (x^2 y, \ 3 y^2 z, \ 9 z^2 x)$. \  Use Gauss' theorem to compute \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} \, d\mathbf{S}$.

\medskip

%% Rogawski 17.3.15
\question Let \ $\mathbf{F}(x,y,z) = (x + y, \ z, \ z - x)$ \ and let $\mathcal{S}$ be the boundary surface of the solid region between the paraboloid \ $z = 9 - x^2 - y^2$ \ and the $xy$-plane.  Evaluate \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} d\mathbf{S}$.

\medskip

\question Let $W$ be a symmetric elementary region in $\mathbb{R}^3$ with positively oriented closed boundary surface $\partial W$.  Show that
\begin{displaymath}
\iiint_W \frac{1}{r^2} \, dx \, dy \ dz = \iint_{\partial W} \frac{1}{r^2} \mathbf{r} \boldsymbol{\cdot} \mathbf{n} \ dS,
\end{displaymath}
where \ $\mathbf{r} = (x, y, z)$ \ and \ $r = \| \mathbf{r} \| = \sqrt{x^2 + y^2 + z^2}$.

\medskip

\question Let \ $\mathbf{r} = (x, y, z)$ \ and \ $r = \| \mathbf{r} \| = \sqrt{x^2 + y^2 + z^2}$.  Verify the following identities.
\begin{parts}
\part $\displaystyle \nabla\left(\frac{1}{r}\right) = -\frac{1}{r^3} \, \mathbf{r}$ \ for $r \not= 0$; and, in general, \ $\nabla (r^n) = n r^{n-2} \, \mathbf{r}$, \ and \ $\displaystyle \nabla \log(r) = \frac{1}{r^2} \, \mathbf{r}$.

\part $\displaystyle \nabla^2 \left(\frac{1}{r}\right) = 0$ \ for $r \not=0$; and, in general, \ $\nabla^2 r^n = n(n+1) r^{n-2}$.

\part $\displaystyle \boldsymbol{\nabla} \boldsymbol{\cdot} \left(\frac{1}{r^3} \, \mathbf{r}\right) = 0$; and, in general, \ $\boldsymbol{\nabla} \boldsymbol{\cdot} \left(r^n \, \mathbf{r}\right) = (n+3) r^n$

\part $\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{r} = \boldsymbol{0}$; \ and, in general, \ $\boldsymbol{\nabla} \boldsymbol{\times} \left(r^n \, \mathbf{r} \right) = \boldsymbol{0}$.
\end{parts}

\medskip

\question Let \ $\mathbf{F}(x,y,z) = y \mathbf{i} + [z \cos(yz) + x] \mathbf{j} + y \cos(yz) \mathbf{k}$.
\begin{parts}
\part Verify that $\mathbf{F}$ is irrotational.

\part Find a scalar potential for $\mathbf{F}$.
\end{parts}

\medskip

\question Let \ $\mathbf{F}(x, y, z) = [2 x y z + \sin(x)]\mathbf{i} + x^2 z \mathbf{j} + x^2 y \mathbf{k}$.

Evaluate $\displaystyle \int_{\mathbf{c}} \mathbf{F} \boldsymbol{\cdot} d\mathbf{s}$, \ where\ $\mathbf{c}(t) = (\cos^5(t), \; \sin^3(t), \; t^4)$ \ for \ $0 \leq t \leq \pi$.

\end{questions}

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