\documentclass{proc-l}
\usepackage[margin=0.5in]{geometry}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{graphicx,changepage,xcolor, float, array}
\usepackage{fancyhdr}
\usepackage{amssymb}
\setlength{\parindent}{0in}
\pagestyle{fancy}
\fancyhf{}
\lhead{Sohail Farhangi, Pan Yan, Yilong Zhang}
\rhead{Recitation Handout for 3/25/2021}
\rfoot{Page \thepage}
%\lfoot{Our work for this problem}
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
\hskip -\arraycolsep
\let\@ifnextchar\new@ifnextchar
\array{#1}}
\makeatother
%\makeatletter
%\newcommand*{\rom}[1]{\expandafter\@slowromancap\romannumeral #1@}
%\makeatother
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\setlength{\headheight}{12pt}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\definecolor{auburn}{rgb}{0.43, 0.21, 0.1}
\definecolor{darkgreen}{rgb}{0.0, 0.4, 0.13}
\definecolor{darkpastelgreen}{rgb}{0.01, 0.75, 0.24}
\newcommand{\dotrule}[1]{%
\parbox[t]{#1}{\dotfill}}
\makeatletter
% new style {footmark}
\fancypagestyle{footmark}{
\ps@@fancy % use {fancy} style as a base of {footmark}
\fancyfoot[C]{\footmark}
}
% sets value of \footmark and sets the correct style for this page only
\newcommand\markfoot[1]{\gdef\footmark{#1}\thispagestyle{footmark}}
\makeatother
% Absolute value notation
\newcommand{\abs}[1]{\lvert#1\rvert}
% Blank box placeholder for figures (to avoid requiring any
% particular graphics capabilities for printing this document).
\newcommand{\blankbox}[2]{%
\parbox{\columnwidth}{\centering
% Set fboxsep to 0 so that the actual size of the box will match the
% given measurements more closely.
\setlength{\fboxsep}{0pt}%
\fbox{\raisebox{0pt}[#2]{\hspace{#1}}}%
}%
}
\begin{document}
\Huge
\begin{center}Math 2173 Spring 2021 Recitation Handout 10\end{center}
\huge
\vskip 15mm
Group Member 1: \underline{\hspace{140mm}}
\vskip 15mm
Group Member 2: \underline{\hspace{140mm}}
\vskip 15mm
Group Member 3: \underline{\hspace{140mm}}
\vskip 15mm
Group Member 4: \underline{\hspace{140mm}}
\vskip 15mm
Group Member 5: \underline{\hspace{140mm}}
\vskip 15mm
Group Member 6: \underline{\hspace{140mm}}
\vskip 10mm
\normalsize
Below is a checklist of instructions to follow when completing this assignment. Failure to follow these directions will result in penalty on your final score and/or in some problems not being graded. If multiple directions are not followed, then it is also possible that the assignment will not be accepted for any credit at all. Please contact your TA or make a post on the discussion boards for this course if you have any questions about this assignment or these directions.
\vskip 2mm
Sohail Farhangi: farhangi.3@osu.edu, \quad Pan Yan: yan.669@osu.edu, \quad Yilong Zhang: zhang.6100@osu.edu
\vskip 2mm
\begin{tabular}{|m{5mm}|m{185mm}|}
\hline
\multicolumn{2}{|c|}{Checklist of Instructions}\\
\hline
& Please clearly write the names of all group members working on this assignment in the spaces allotted above. \\
\hline
& This assignment must be completed by a group of 3, 4, 5, or 6 members. \\
\hline
& This assignment is to be uploaded to gradescope as a pdf file no later than \textcolor{red}{11:59 PM EST on Sunday, March 28}.\\
\hline
& The assignment will be uploaded by 1 group member, and that group member will be responsible for manually entering the names of all other collaborators into gradescope.\\
\hline
& This assignment must be completed using this template. You may either print this template to write on it and then scan it (pages ordered correctly) into a pdf file, or you may write directly on the template using programs such as notability.\\
\hline
& If you need more space than what is given to solve a given problem, then you will find blank pages provided at the end of this template. At the end of each problem section of this assignment you will find a space in which to indicate on what page your work is continued in case you used additional pages to complete your solution. You must provide the page number on which your work is continued in the alloted space, or write 'N/A' incase you did not use any additional pages.\\
\hline
& On the additional pages, you will also find space in which to indicate which problem the page is being used for, and if the page is used then that space must also be filled.\\
\hline
& To complete this handout, you may use your textbook, class notes, discussions with your TA and group members, and any resources that are available on Carmen. You should not receive any help from the MSLC or people outside of your group when solving these problems. You may discuss these problems on the Carmen discussion boards, but you should not provide your entire solution when answering a such question, you should only give a hint or a helpful idea.\\
\hline
\end{tabular}
\newpage
\huge
\iffalse
{\bf \textcolor{red}{Ungraded Optional} Problem 15.2.43:} Given the force field $\vec{F}=\langle x, y, z\rangle$, compute the work required to move an object on the tilted ellipse
$$\vec{r}(t)=\langle 4\cos(t), 4\sin(t), 4\cos(t) \rangle$$
for $0\le t \le 2\pi$.
\includegraphics[scale=0.5]{Problem15243.PNG}
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our personal notes for this \textcolor{red}{Ungraded Optional} Problem are continued on page}
\clearpage
\fi
{\bf \textcolor{red}{Ungraded Optional} Problem:} Let $a$ be a positive number. Compute the circulation of the vector field
$$\vec{F}=\langle -y, x\rangle$$
on the circle $\mathcal{C}$ of radius $a$ centered at the origin with counterclock-wise orientation.
%\includegraphics[scale=0.5]{Problem15243.PNG}
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for \textcolor{red}{Ungraded Optional} is continued on page}
\newpage
\iffalse
{\bf \textcolor{red}{Ungraded Optional} Problem 15.2.46:} Given the force field $$\vec{F}=\frac{\langle x, y, z\rangle}{x^2+y^2+z^2}, $$
compute the work required to move an object on the line segment from $(1, 1, 1)$ to $(8,4,2)$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our personal notes for \textcolor{red}{Ungraded Optional} are continued on page}
\clearpage
\fi
\iffalse
{\bf \textcolor{red}{Ungraded Optional} Problem 15.2.44:} Given the force field $\vec{F}=\langle -y, x, z\rangle$, compute the work required to move an object on the helix
$$\vec{r}(t)=\langle 2\cos(t), 2\sin(t), \frac{t}{2\pi}\rangle$$
for $0\le t \le 2\pi$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our personal notes for this \textcolor{red}{Ungraded Optional} Problem are continued on page}
\clearpage
\fi
{\bf \textcolor{red}{Ungraded Optional} Problem:}
Let $\mathcal{C}$ be the unit circle with counterclockwise orientation. Compute the flux of $\vec{F}=\langle x,y \rangle$ across $\mathcal{C}$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our personal notes for this \textcolor{red}{Ungraded Optional} Problem are continued on page}
\clearpage
\iffalse
(a) Let $\mathcal{C}_1$ be given by $\vec{r}(t)=\langle 2 \cos (t), 2 \sin(t)\rangle$, $0\le t \le 2\pi$. Find the circulation of $\vec{F}=\langle x,y \rangle$ on $\mathcal{C}_1$.
\fi
\iffalse
{\bf Problem 15.2.28. (8 points):} Compute
\begin{equation}
\int_{\mathcal{C}} \frac{xy}{z} ds,
\end{equation}
where $\mathcal{C}$ is the line segment from $(1,4,1)$ to $(3,6,3)$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for Problem 15.2.28 is continued on page}
\newpage
\fi
\iffalse
{\bf Problem 15.2.38. (8 points):} Given $\vec{F}=\frac{\langle x, y\rangle}{x^2+y^2}$, and the curve $\mathcal{C}$ which is given by $\vec{r}(t)=\langle t, 4t\rangle$ from $t=1$ to $t=10$, compute
$$\int_{\mathcal{C}} \vec{F}\cdot \vec{T} ds.$$
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for Problem 15.2.38 is continued on page}
\newpage
\fi
{\bf Problem 15.2.47. (5 points):} Compute the circulation of
$$\vec{F}=\langle y-x, x\rangle$$
on the curve $\mathcal{C}$ which is given by $\vec{r}(t)=\langle 2\cos(t), 2\sin(t)\rangle$ for $0\le t\le 2\pi$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for Problem 15.2.47 is continued on page}
\newpage
{\bf (Modified) Problem 15.2.66 (5 points):}
Let $a$ be a positive number. Consider the vector field $\vec{F}=\langle y, x \rangle$ and the curve $\mathcal{C}$ given by $\vec{r}(t)=\langle a \cos(t), a\sin(t)\rangle$ for $0\le t \le 2\pi$. Compute the flux of $\vec{F}$ across $\mathcal{C}$. (Your answer should be in terms of $a$.)
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our personal notes for Problem 15.2.66 are continued on page}
\clearpage
\iffalse
{\bf Problem 15.2.47. (8 points):} Compute the circulation of
$$\vec{F}=\langle y-x, x\rangle$$
on the curve $\mathcal{C}$ which is given by $\vec{r}(t)=\langle 2\cos(t), 2\sin(t)\rangle$ for $0\le t\le 2\pi$.
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for Problem 15.2.47 is continued on page}
\newpage
\fi
\newpage
{\bf Problem 15.2.60 (4+4+4+1=13 points):} Consider the rotation field $\vec{F}=\langle -y, x\rangle$, and the three paths shown in the figure.
\begin{figure}[H]
\includegraphics{RotationFields.PNG}
\end{figure}
\begin{enumerate}
\item Compute the work required in the presence of the force field $\vec{F}$ to move an object on the curve $\mathcal{C}_1$.
\item Compute the work required in the presence of the force field $\vec{F}$ to move an object on the curve $\mathcal{C}_2$.
\item Compute the work required in the presence of the force field $\vec{F}$ to move an object on the curve $\mathcal{C}_3$.
\item Does it appear that the line integral $\int_{\mathcal{C}} \vec{F}\cdot \vec{T} ds$ is independent of the path, where $\mathcal{C}$ is any path from $(1,0)$ to $(0,1)$?
\end{enumerate}
\vskip 5mm
\hrule
\vfill
\hrule
%\markfoot{Our work for Problem 15.2.60 is continued on page}
\newpage
\vskip 5mm
\hrule
\vfill
\hrule
\markfoot{Our work for Problem 15.2.60 is continued on page}
\newpage
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\newpage
On this page my work for problem \underline{\hspace{50mm}} is continued.
\vskip 5mm
\hrule
\end{document}