\documentclass{proc-l}
\usepackage[margin=0.5in]{geometry}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
\usepackage{graphicx,changepage,xcolor, float, array}
\usepackage{fancyhdr}
\setlength{\parindent}{0in}
\pagestyle{fancy}
\fancyhf{}
\lhead{Sohail Farhangi, Pan Yan, Yilong Zhang}
\rhead{Recitation Handout for 1/14/2021}
\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
\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}
\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 1\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 11:59 PM EST on Sunday, January 17.\\
\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
{\bf 13.6.77:} The electric field due to a point charge of strength $Q$ at the origin has a potential function $V(x,y,z) = kQ/r$, where $r^2 = x^2+y^2+z^2$ is the square of the distance between a variable point $P(x,y,z)$ at the charge, and $k > 0$ is a physical constant. The electric field is given by ${\bf E}(x,y,z) = -\nabla V(x,y,z)$.
\begin{itemize}
\item[{\bf a.}] Show that
\begin{equation}
{\bf E}(x,y,z) = kQ\langle \frac{x}{r^3}, \frac{y}{r^3}, \frac{z}{r^3}\rangle.
\end{equation}
\item[{\bf b.}] Show that $|{\bf E}| = kQ/r^2$. Explain why this relationship is called the inverse square law.
\end{itemize}
\vskip 5mm
\hrule
\newpage
\quad
\vfill
\hrule
\markfoot{Our work for problem 13.6.77 is continued on page}
\newpage
{\bf Problem 13.6.64:} Consider the function $F(x,y,z) = e^{xyz}$.
\begin{itemize}
\item[{\bf a.}] Write $F$ as a composite function $f\circ g$, where $f$ is a function of one variable and $g$ is a function of three variables.
\item[{\bf b.}] Calculate $\nabla F(x,y,z)$ as well as $\nabla g(x,y,z)$. Find a relationship between $\nabla F(x,y,z)$ and $\nabla g(x,y,z)$.
\end{itemize}
\vskip 5mm
\hrule
\newpage
\quad
\vfill
\hrule
\markfoot{Our work for problem 13.6.64 is continued on page}
\newpage
{\bf Problem 13.6.33 \textcolor{red}{ (NOT GRADED)}:} Consider the function $f(x,y) = 10-2x^2-3y^2$ and the point $P = (3,2)$.
\begin{itemize}
\item[{\bf a.}] Find the gradient field $\nabla f(x,y)$ of $f(x,y)$ and then evaluate it at $P$.
\item[{\bf b.}] Find the angles $\theta$ (with respect to the x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
\item[{\bf c.}] Write the directional derivative at $P$ as a function of $\theta$; call this function $g(\theta)$.
\item[{\bf d.}] Find the value of $\theta$ that maximizes $g(\theta)$ and find the maximum value.
\item[{\bf e.}] Verify that the value of $\theta$ that maximizes $g$ corresponds to the direction of the gradient vector at $P$. Verify that the maximum value of $g$ equals the magnitude of the gradient vector at $P$.
\end{itemize}
\vskip 5mm
\hrule
\newpage
\quad
\vfill
\hrule
\markfoot{Our work for problem 13.6.33 is continued on page}
\newpage
{\bf Problem 13.6.38:} Consider the function $f(x,y) = \ln(1+2x^2+3y^2)$ and the point $P = (\frac{3}{4},-\sqrt{3})$.
\begin{itemize}
\item[{\bf a.}] Find the gradient field $\nabla f(x,y)$ of $f(x,y)$ and then evaluate it at $P$.
\item[{\bf b.}] Find the angles $\theta$ (with respect to the x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
\item[{\bf c.}] Write the directional derivative at $P$ as a function of $\theta$; call this function $g(\theta)$.
\item[{\bf d.}] Find the value of $\theta$ that maximizes $g(\theta)$ and find the maximum value.
\item[{\bf e.}] Verify that the value of $\theta$ that maximizes $g$ corresponds to the direction of the gradient vector at $P$. Verify that the maximum value of $g$ equals the magnitude of the gradient vector at $P$.
\end{itemize}
\vskip 5mm
\hrule
\newpage
\quad
\vfill
\hrule
\markfoot{Our work for problem 13.6.38 is continued on page}
\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
\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}