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\begin{document}
\title{Formatting mathematics course papers in \LaTeX}
\author{{\small Math 2401 Differential Equations}\\ {\small Instructor: Barry McQuarrie} \\  {\small University of Minnesota, Morris}}

%\date{\today} % use this instead of the next command if you want the date the file was compiled
\date{Fall 2022}
\maketitle

\begin{abstract}
This document shows some of the technical details that arise when formatting mathematics papers in \LaTeX.
You will want to look at the source file ``\verb+Formatting Details LaTeX 2022.tex+'' to see the syntax that is used
to create the final typesetting.
{\bf Do not spend time trying to figure out
how to do more complicated things, if you wish to do something and don't know how to do it talk with me and we can
figure it out together.}
\end{abstract}

\section{Accessing \LaTeX}
\LaTeX{} is different from Word in that it requires processing by an installed
program. If you are considering pursuing math or physics graduate studies, you will likely want to learn \LaTeX{} at some point, even though it has a steeper learning curve than Word.

You can use a free Overleaf account to get started with \LaTeX{}, or you can download a distribution such as MikTex (windows) or MacTex (MacOS). You can learn
more about \LaTeX{} distributions at
\href{https://www.latex-project.org/get/}{https://www.latex-project.org/get/}.


If you wish to learn \LaTeX{} I can help--as mentioned, there is a steep learning curve but once you get the flow it goes quite well and produces beautiful results very quickly. All my lecture notes and tests are created in \LaTeX{} for this reason.

The best way to learn \LaTeX{} is to copy/paste/modify examples that do something similar to what you want to do, so I am providing the source files
in case anyone does want to play around with \LaTeX.


\section{Figures}

The image (high quality and with fonts of a reasonable size) can be saved as a
png file (or other image)
in the same directory as the source document before being imported into the document.
\begin{itemize}
\item Note that it is usually best to have figures saved as files with no spaces in the fine names.

\item The \LaTeX{} command that imports the image will also allow you to
center and resize the image (and rotate, but you generally do not need that).

%\item One of the powerful features of \LaTeX{} is it's ability to cross reference.
%
%\item If you add or delete figures, the cross referencing will update automatically.

\item If there is not enough space on the page for the figure where you put it, it will
automatically float to a point later in the document.
\end{itemize}

\subsection{Figure with one image}

\begin{figure}[h]
	\begin{center}
		\includegraphics[width = 0.3 \textwidth]{StyleGuideFig1.png}
	\end{center}
	\caption{\label{fig-ABC}Here is a figure with a caption. The caption should tell the reader what the image is showing us.}
\end{figure}



\subsection{Figure with multiple images}

You can include a tabular environment to create a  figure with multiple images,
as is shown in Figure~\ref{fig-37}.

\begin{figure}[h]
	\begin{center}
		\begin{tabular}{cc}
			\includegraphics[width = 0.4 \textwidth]{StyleGuideFig1.png} &amp;
			\includegraphics[width = 0.5 \textwidth]{StyleGuideFig2.png} \\
			(a) &amp; (b) \\
		\end{tabular}
	\end{center}
	\caption{\label{fig-37}(a) an implicit solution solution to an ODE over the direction field, and (b) the three explicit solutions. This will be useful
	when we look at existence and uniqueness of solutions for nonlinear ODEs.}
\end{figure}

\section{Tables}

Here we see an example of a Table.

\begin{table}[h]
	\begin{center}
		\begin{tabular}{l|l|l|l}
			$\gamma$            &amp; Roots of Characteristic Equation            &amp; Damping  &amp; Solution $u(t)$ \\ \hline
			$\gamma=2\sqrt{km}$ &amp; Real, multiplicity 2 ($r_1=r_2=-\gamma/2m$) &amp; Critical &amp; $(c_1 + c_2 t )e^{-\gamma t/2m}$ \\
			$\gamma&gt;2\sqrt{km}$ &amp; Real, distinct  ($r_1 \neq r_2$)            &amp; Over     &amp; $ c_1 e^{r_1 t} + c_2 e^{r_2 t}$ \\
			$\gamma&lt;2\sqrt{km}$ &amp; Complex Conjugates ($r_{1,2}=\lambda \pm i \mu)$&amp; Under&amp; $ c_1 e^{\lambda t} \cos \mu t + c_2 e^{\lambda t} \sin \mu t$\\
		\end{tabular}
		\caption{\label{tab-3G}The three types of damping for the damped free vibration.}
	\end{center}
\end{table}

Note that Figures and Tables may float to a different part of the document, unlike in Word where they
generally go exactly where you put them. As long as they do not appear too far away from
where they are referenced in the text that is fine, adjust locations as one of the final tasks in writing
a paper. You should
try and fix any awkward formatting by rephrasing, making images larger or smaller,
reordering parts of the text, or just inserting a page break using \verb+\newpage+.



\section{Equations}\label{secequ}

The power of \LaTeX{} is in how beautifully it formats equations. You can
put an equation inline using dollar signs, \verb+$f(x) = 1/2$+ will yield
$f(x) = 1/2$. Note that for inline equations we do not stack the fractions.

For an unnumbered equation that it on it's own line (called a displayed equation), we can use \verb+\[f(x) = \frac{1}{2},\]+
which will produce
\[f(x) = \frac{1}{2},\]
where we included the comma since the equation is grammatically treated as part of the sentence. If we
wished to start a new sentence, we would have put a period. Also notice how we can stack fractions using
\verb+\frac{1}{2}+.

\subsection{Examples of unnumbered and numbered equations}

A system of two first order linear DEs
with aligned equal signs:
\begin{align}
	x^\prime_1 &amp;= p_{11}(t) x_1 + p_{12}(t) x_2 + g_1(t), \nonumber \\
	x^\prime_2 &amp;= p_{21}(t) x_1 + p_{22}(t) x_2 + g_2(t). \label{SysDE}
\end{align}
The system in \eqref{SysDE} written in matrix form:
\begin{align}
	\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}^\prime
	&amp;= \begin{bmatrix} p_{11}(t) &amp; p_{12}(t) \\ p_{21}(t) &amp; p_{22}(t) \end{bmatrix}
	\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
	+ \begin{bmatrix} g_1(t) \\ g_2(t) \end{bmatrix}. \label{SysDEMat}
\end{align}
The system in \eqref{SysDEMat} written in matrix notation:
\begin{align} {\bf x}^\prime &amp;= {\bf P}(t){\bf x} + {\bf g}(t). \label{SysDEMatNot} \end{align}
The bold face indicates a matrix or vector.


Here are some examples of more complicated equations, where I have included
punctuation as if these were finishing a sentence for you to see.

\[
	f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\: dx. 	
\]

\begin{align*}
f_{\text{avg}} &amp;= \frac{1}{b-a} \int_a^b f(x)\: dx. &amp;&amp;\text{(average value of function)}
\end{align*}


\begin{align}
	f_{\text{avg}} &amp;= \frac{1}{b-a} \int_a^b f(x)\: dx. &amp;&amp;\text{(average value of function)} \label{MyAVG}
\end{align}


I pretty much use the \verb+align+ structure for equations since it is so versatile.
The ampersand symbol \verb+&amp;+ is used to indicate what text should be aligned.

\begin{align}
	f_{\text{avg}} &amp;= \frac{1}{b-a} \int_a^b f(x)\: dx, \nonumber \\
	&amp;= \frac{1}{12-0} \int_0^{12} e^x \: dx, \nonumber\\
	&amp;= \frac{1}{12} e^x \Big|_0^{12}, \nonumber\\
	&amp;= \frac{1}{12} \left( e^{12} - 1 \right). \label{result}
\end{align}
Here's an unnumbered example with notes on each line
\begin{align*}
	\int \sin^2(x)\,dx &amp;=\int \frac{1-\cos(2x)}{2}\,dx, &amp;&amp;\text{(by trig identities)}\\
	&amp;=\int \frac{1}{2}\,dx-\frac{1}{2}\int\cos(2x)\,dx, \\
	&amp;=\frac{x}{2}-\frac{1}{4}\sin(2x)+C. &amp;&amp;\text{(for some constant $C$)}
\end{align*}


If you want to have a numbered piece-wise defined function, here's an example:
\begin{align}\label{Equation:piecewise}
	f(x)=
	\begin{cases}
		x^2 &amp;\text{if } x&gt;3,\\
		-\cos(x) &amp;\text{if } x\leq 3.
	\end{cases}
\end{align}

\subsection{Breaking an unnumbered equation across a line}

You just need to add some space using \verb+\quad+ to shift the second line over.

\begin{align}
f(x) &amp;\sim f(0) + f^\prime(0) x + \frac{f^{\prime\prime}(0)}{2!} x^2 \nonumber \\
     &amp;\quad \mbox{}+\frac{f^{\prime\prime\prime}(0)}{3!} x^3
\end{align}


\section{Cross Referencing}

If you were looking at the \LaTeX{} source, you will have noticed that I have
assigned a \verb+label+ to certain sections, figures, tables, and equations. You can refer to
these labels in the body of the text as needed. We use a \verb+~+ (an unbreakable space) to ensure the
numbers aren't split across a line.

\begin{itemize}
\item the syntax \verb+Section~\ref{secequ}+ gives Section~\ref{secequ}
\item the syntax \verb+Table~\ref{tab-3G}+ gives Table~\ref{tab-3G}
\item the syntax \verb+Figure~\ref{fig-37}+ gives Figure~\ref{fig-37}
\item the syntax \verb+Eq.~\eqref{MyAVG}+ gives Eq.~\eqref{MyAVG}
\item the syntax \verb+Eq.~\eqref{result}+ gives Eq.~\eqref{result}
\end{itemize}

Note you can label equations a variety of ways, just be consistent. Most people will just use the following
structure for equations ``\ldots as seen in \eqref{SysDEMatNot}, matrix notation is a compact way to
write an $n\times n$ system of equations.''.

Cross referencing in \LaTeX{} is super easy, and probably the most important advantage \LaTeX{} has over Word. But if you don't need to do a lot of cross referencing, Word work fine.



\section{Definitions and Theorems}

A sample definition, theorem, and proof:


\begin{defn}
If $f$ is a real-valued function defined on $[0,\infty)$, then the
{\it Laplace transform} of $f$ is defined as:
\[  \L[f(t)] = F(s) = \int_0^\infty e^{-st} f(t) \: dt. \]
\end{defn}

\begin{theorem}{\bf Existence of the Laplace transform.}
$\L[f(t)]$ exists if
\begin{itemize}
\item $f$ is piecewise continuous on $[0,\infty]$, and
\item $f$ is of exponential order as $t\rightarrow \infty$.
\end{itemize}

\end{theorem}



\begin{proof}

Since $f(t)$ is of exponential order, we have
$ \big|f(t)\big| \le K  e^{at}, \mbox{ for } t \ge T$.
Therefore,
\begin{align*}
F(s) = \L[f(t)] &amp;= \int_0^\infty e^{-st} f(t) \: dt, &amp;&amp;\text{(by definition)}\\
&amp;= \int_0^T e^{-st} f(t) \: dt + \int_T^\infty e^{-st} f(t) \: dt, &amp;&amp;\text{(property of integration)} \\
&amp;\le  \int_0^T e^{-st} f(t) \: dt + \int_T^\infty e^{-st} K  e^{at} \: dt, &amp;&amp;\text{($f$ is exponential order)} \\
&amp;\le  \int_0^T e^{-st} f(t) \: dt + K \int_T^\infty e^{-(s-a)t}\: dt,
\end{align*}
The last integral converges if $s-a&gt;0$, or $s&gt;a$.
Therefore, the Laplace transform exists.

\end{proof}


\section{Lists}

Here is an itemized list:
\begin{itemize}
 \item First item
 \item Second item
 \item Third item
\end{itemize}

Here is a numbered list:
\begin{enumerate}
 \item First
 \item Second
 \item Etc
\end{enumerate}

\begin{thebibliography}{9}
%%%%%%%%%%%%
% Barry prefers the bibliography to be listed in the order the references appear in the paper, but as long as you are consistent you can use other conventions.
% Bibliography items can be listed in alphabetical order by author's last name. If there are multiple authors, list them in alphabetical order. If there are multiple publications by the same author(s), list them in chronological order.
%
% You can decide on what labels make the most sense for you, since they only appear in the LaTeX code.
%%%%%%%%%%%%

\bibitem{mandelbrotset}
R. Devaney, The Mandelbrot Set Explorer, (2001). Retrieved August 15, 2007, from http://math.bu.edu/DYSYS/explorer/index.html

\bibitem{CGSR}
J. B. Carlin, A. Gelman, H. S. Stern, and D. B. Rubin, \emph{Bayesian Data Analysis}, Chapman and Hall, London, 1995.

\bibitem{techwriting}
A. Eisenberg, \emph{Guide to Technical Editing}, Oxford University, New York, 1992.

\bibitem{Met}
 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, â€œEquations of state calculations by fast computing machine,â€ \emph{J. Chem. Phys.} 21, pp. 1087-1091, 1953.
\end{thebibliography}


Your \LaTeX{} source file ends with a \verb+\end{document}+, and you can have things after that in the file, they
just won't appear in the pdf output


\end{document}


This will not appear in the pdf output!
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