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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Friday, December 8, 2023 (not graded)}
\newcommand{\myAssignmentTitle}{Homework Assignment 8}
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\begin{document}

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\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
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\begin{questions}
%8.4.19
\question Let \ $\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}$ \ and set \ $r = \|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2}$. \ Show that for a symmetric elementary region $W$ with outward-oriented $C^1$ boundary surface $\partial W$,
\begin{displaymath}
\iiint_W \frac{1}{r^2} \, dV = \iint_{\partial W} \frac{\mathbf{r}}{r^2} \boldsymbol{\cdot} \mathbf{n} \, dS.
\end{displaymath}
Note: This is a simple application of Gauss's divergence theorem if $(0,0,0) \not\in W$.  Be sure to carefully show the equation still holds when $(0,0,0) \in W$.

%8.3.6
\question Let $\mathbf{F}(x,y) = (x y, \; y^2)$.
\begin{parts}
\part Let $\mathbf{c}$ be the path \ $y = 2 x^2$ \ in $\mathbb{R}^2$ joining \ $(0,0)$ \ and \ $(1,2)$. \  Evaluate $\displaystyle \int_\mathbf{c} \mathbf{F} \boldsymbol{\cdot} d\mathbf{s}$.

\part Does the integral in part (a) depend on the choice of path joining $(0,0)$ to $(1,2)$?
\end{parts}

\bigskip

%8.3.13
\question Let \ $\displaystyle \mathbf{F}(x, y, z) = \left(e^x \sin(y)\right) \mathbf{i} + \left(e^x \cos(y)\right) \mathbf{j} + z^2 \mathbf{k}$.  Evaluate the integral \ $\displaystyle \int_\mathbf{c} \mathbf{F} \boldsymbol{\cdot} d\mathbf{s}$, \ where $\mathbf{c}$ is the path given by \ $\displaystyle \mathbf{c}(t) = \left(\sqrt{t}, \; t^3, \; \exp\left(\sqrt{t}\right)\right)$, \ with \ $0 \leq t \leq 1$.

\bigskip
%8.3.18
\question For each of the following vector fields $\mathbf{F}$, determine if it is a gradient field.  If it is, find a function $f$ such that $\nabla f = \mathbf{F}$.
\begin{parts}
\part $\displaystyle \mathbf{F}(x,y) = \left(2x + y^2 - y \sin(x), \; 2 x y + \cos(x)\right)$
\part $\displaystyle \mathbf{F}(x,y,z) = \left(6 x^2 z^2, \; 5 x^2 y^2, \; 4y^2 z^2 \right)$
\part $\displaystyle \mathbf{F}(x,y) = \left(y^3 + 1, \; 3 x y^2 + 1\right)$
\part $\displaystyle \mathbf{F}(x,y) = \left(2 x \cos(y), -x^2 \sin(y)\right)$
\end{parts} 

\bigskip

%Rogawski 16.3.19
\question Evaluate \ $\displaystyle \int_\mathbf{c} 2 x y z \, dx + x^2 z \, dy + x^2 6 \, dz$ \, over the path \ $\mathbf{c}(t) = \left(t^2, \; \sin\left( \frac{\pi}{4}t \right), \; \exp\left( t^2 - 2 t \right) \right)$

with \ $0 \leq t \leq 2$.

\bigskip

%Rogawski 16.3.19
\question Evaluate \ $\displaystyle \int_\mathcal{C} \sin(x) \, dx + z \cos(y) \, dy + \sin(y) \, dz$, \ where $\mathcal{C}$ is the ellipse defined by \ $4 x^2 + 9 y^2 = 36$

and oriented \emph{clockwise}.

\end{questions}

\end{document}
