\documentclass[12pt]{article}

\usepackage{precalc}
\usepackage{url}

\usepackage{fancyvrb}

\newcommand{\mhfe}{my helicopter is full of eels}
\newcommand{\ssss}[1]{\sqrt{\sqrt{\sqrt{\sqrt{#1}}}}}

\begin{document}

\VerbatimFootnotes

\pctitle{\LaTeX\ tutorial}

\section{Introduction}

\LaTeX\footnote{By the way, \LaTeX\ should be pronounced like
  ``lah-tek'', it should \emph{not} be pronounced ``lay-teks''!  If
  you are talking with someone who is familiar with \LaTeX\ and you
  say it ``lah-tek'', they will know that you are cool.  Conversely,
  if you say ``lay-teks'', they will dismiss you as a hopeless n00b.
  Just FYI.} is a powerful, free document typesetting system that is
especially suited to mathematics, and is the de facto standard in the
scientific publishing and academic communities.  It handles gracefully
formatting all manner of text as well as mathematical equations. For
example:
\begin{equation} 
    \sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}
\end{equation}

\begin{equation}
    \left[ \begin{array}{cc} 3 & 4 \\ 2 & 7 \end{array} \right] \cdot
    \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] =
    \left[ \begin{array}{c} 0 \\ -1 \end{array} \right]\
\end{equation}

\begin{equation}
    \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}}
\end{equation}


\section{Installing}

The first order of business is to install the \LaTeX\ system on your
computer.  Follow the instructions for the appropriate operating system.

\subsection{Installing on OS X}

Installing \LaTeX\ is easy on OS X.  You will just need to install the
MacTeX distribution, which includes \LaTeX\ itself as well as some
convenient related applications.  In particular, it includes TeXShop,
a program you can use to easily create and edit \LaTeX\ documents.

To obtain MacTeX, go to \url{http://www.tug.org/mactex/2008/}.
Download the file \texttt{MacTeX.mpkg.zip} and run the installer it
contains (probably just by double-clicking on it).  (As of this
writing, at 7:30pm on September 2, it appears that the newest version
of this file is in the process of being uploaded, so you may have to
wait a bit for it to become available.)  This will install a complete
\LaTeX\ system as well as some other supporting software.

\texttt{MacTeX.mpkg.zip} is rather large (1.15 GB) so if your internet
connection is too slow or you don't want to wait for it to download,
you can instead click on ``Smaller Packages'', and download and
install both the mactex-additions and BasicTeX packages.  This should
be fine for what you need to do, and you can always install more if
you ever find that there is something you need which wasn't included
in these smaller packages.

\subsection{Installing on Windows}

You will need two pieces of software: MiKTeX is a Windows
version of \LaTeX\ itself together with many useful extension
packages, and LEd is an editor you can use to create \LaTeX\
documents.  You don't \emph{need} a special editor to create \LaTeX\
documents; they are just plain text files so Notepad would do.
However, having an editor which knows about \LaTeX\ and has many of
the needed tools already built-in saves a lot of time and annoyance.

\subsubsection{MiKTeX}
\label{sec:miktex}

To download MiKTeX, go to \url{http://miktex.org/2.7/Setup.aspx},
scroll down, and click on the ``Basic MiKTeX 2.7'' installer.  Once
you have downloaded the installer, run it and follow the instructions
to install it.  If you have any questions, or problems, let me know (I
don't have a Windows computer to test it on, so I'm not sure exactly
what the installation process will be like).  After it is finished
installing, you should run the update wizard as recommended (it will
probably be accessible from the Start menu).

\subsubsection{LEd}
\label{sec:led}

To download LEd, go to
\url{http://www.latexeditor.org/download_main.html}, download the
installer, run it, and follow the instructions.  Again, if you have
any questions or problems, let me know.

Once LEd is installed, you should be able to use it to edit your
\LaTeX\ documents and preview the nicely typeset version.  LEd has a
number of additional features, such as menus from which you can select
special characters and symbols to insert into your document, a
built-in spell checker and thesaurus, and many other features which
you will probably never need!  Feel free to play around with it and
see what it can do.

\section{\LaTeX\ Basics}
\label{sec:latex-basics}

\subsection{The \LaTeX\ philosophy}
\label{sec:latex-philosophy}

The whole idea behind \LaTeX\ is that you should be able to specify
the \emph{content} of your document, without spending too much time
worrying about \emph{how it will look}; \LaTeX\ has a great set of
defaults for producing professional-looking documents, but also allows
you to tweak whatever you like.

This separation of content and layout is very different from the way
many other document processing systems work.  For example, when you
edit a Microsoft Word document, you see on your screen exactly how
your document will look when you print it out.  If you want some bold
text, you select the text and click on the ``bold'' icon, and the text
becomes bold.  If you want to make a new section with a title heading,
you have to type in the title, make the font bigger, and put the right
amount of space around it, and so on.  

With \LaTeX{}, on the other hand, editing your document and seeing how
it will look when printed are two entirely different things.  When you
edit your document, it is just a plain text file with some special
commands to tell \LaTeX\ how to lay out the document. If you want to
see how your document will look when printed, you must run \LaTeX\ on
your document in order to produce some sort of output file (such as a
PDF) which contains your nicely typeset document.  (LEd can perform
this step for you automatically.)  For example, if you
want to make some text bold, you surround it with the command
\verb|\textbf{...}|.  In order to create a new section, you simply
type something like \verb|\section{My section title}|.  When you later
run \LaTeX\ on your document, you will see bold text and a section
heading (with the title at a suitable size) in the output.

So, enough philosophy.  Let's get started creating your very first
\LaTeX\ document.  By the way, this tutorial itself is, of course, a
\LaTeX\ document!  As you are reading through the rest of this
tutorial, you should look at the source file \texttt{LaTeX-tutorial.tex}
to see the \LaTeX\ commands which produced the PDF you are reading.

\subsection{Document layout}
\label{sec:doc-layout}

A basic \LaTeX\ document looks something like this:

\begin{verbatim}
  \documentclass{article}

  % setup goes here

  \begin{document}

    % content goes here

  \end{document}
\end{verbatim}

The first line must always have a \verb|\documentclass| command, which
specifies what kind of document you are creating.  There are other
document types like \verb|book| and \verb|report|, but for the sorts
of documents you will be creating, you will never need to use anything
other than \verb|article|.  Notice the syntax of \LaTeX\ commands: a
backslash indicates a command, and any parameters to the command are
enclosed in curly braces, \verb|\like{this}|.\footnote{By the way, if
  you're wondering why everywhere in \texttt{LaTeX-tutorial.tex} I
  write \verb|\LaTeX\| with a backslash after it, it's because
  commands ignore any space that comes after them; to get a space you
  have to escape it with a backslash.  See the difference: \LaTeX\
  with a space, \LaTeX without a space.  You very rarely need to know
  this, however, since it only matters for commands in the middle of
  some text which take no parameters; most commands you'll be using
  either take some parameters, or are used in math mode, where the
  spacing is done for you automatically.}

The percent sign indicates a ``comment'': everything from a percent
sign until the end of the line will be ignored by \LaTeX{}, so you can
use this to write notes to yourself or others that will not be
included in the final output document.  In this case, the comment
\verb|% setup goes here| indicates the place where various setup
commands can be placed.  This section before the
\verb|\begin{document}| is called the \term{preamble}. You'll see some
  examples of commands that can go in the preamble later.

Finally, the content of the document must go between
\verb|\begin{document}| and \verb|\end{document}|.

Open up TeXShop or LEd and create a new file.  Copy the above document
skeleton into your blank file, and replace the ``content goes here''
comment with some content: for example, you could just type ``My very
first LaTeX document!''  Then figure out how to generate a typeset PDF
document as output.  There will probably be some sort of button in the
menu bar for doing this.  Then find the generated PDF file and open
it.  It should just be a blank page with the text that you entered as
your document content.

\subsection{Bits and pieces}
\label{sec:bits}

To make a new paragraph in a \LaTeX\ document, just separate the
paragraphs with a blank line.  Otherwise, \LaTeX\ generally ignores
any extra space you put in the middle of your text, turning multiple
spaces into a single space,
as
you
can
see
if              you                   take a look
          at the     \LaTeX\     code
  which generated
this                                    paragraph!

Because of the blank line above, this will be the first sentence of a
new paragraph.

If you need to put something in ``quotation marks,'' you need to use a
special syntax: use two backticks for the opening quote and two
apostrophes for the closing quote, like this: 
\verb|``quotation marks''|.  
If you use normal quotation mark characters by mistake, it
will look very bad: "like this".  Notice that the quotation marks in
front of the word `like' are facing the wrong way.

To make a new section in your document with the title Foo, you can type 
\verb|\section{Foo}|; to make a new subsection, type
\verb|\subsection{Foo}|.  The sections will be
automatically numbered for you (for example, see the section headings
in this document).  If you don't want the sections to be numbered, use
\verb|\section*| and \verb|\subsection*| instead of \verb|\section|
and \verb|\subsection|.  There is also a \verb|\subsubsection| command
but most of the time you shouldn't need it.

Any automatically-numbered item (sections, equations, figures\dots)
can be given a label; later in the document you can refer to the label
and the correct number will be automatically inserted.  This way, you
can refer to numbered things without knowing (or caring) what number
they will end up being given in the final document.  In order to give
something a label, put the command \verb|\label{blah}| right after
it---this will give it the label `blah' (you can use whatever name for
the label you like).  Then, later, you can use \verb|\ref{blah}| to
refer to the number that label `blah' has been given.  For example,
the current section is Section \ref{sec:bits}.  As you can see if you
look at the \LaTeX\ source for this paragraph, I didn't actually type
a number there---I just typed \verb|\ref{sec:bits}| to refer to the
label which I earlier gave to this section.

\subsection{Special environments}
\label{sec:special-environments}

\emph{Environments} are enclosed in \verb|\begin{foo}| \dots
  \verb|\end{foo}| pairs (where `foo' is the name of the environment),
and can specify some special way to typeset their contents.  For
example, you have already seen that the entire document contents must
be enclosed in a \verb|document| environment.  There are a few other
special environments you should be aware of.

The \verb|itemize| environment lets you make a bulleted list of items:

\begin{itemize}
  \item Like this.
  \item Each item inside the environment should be preceded by the
    special \verb|\item| command.
  \item For an example, look at the \LaTeX\ source that was used to
    produce this list.
\end{itemize}

The \verb|enumerate| environment lets you make a numbered list of
items:

\begin{enumerate}
  \item Like this.

  \item Again, each item should be preceded by the \verb|\item|
    command.

  \item The numbers are inserted automatically, so you can add,
    delete, or move items around without worrying about the numbering
    getting messed up!

    \begin{enumerate}
    \item You can even have nested \verb|enumerate| environments.
    \item Like this.
    \end{enumerate}

  \item One more item.

\end{enumerate}

\subsection{Mathematics}
\label{sec:mathematics}

To include a mathematical expression in the middle of some text,
enclose the mathematics in dollar signs.  For example, typing
\verb|$3+x=9$| produces $3+x=9$.  Notice how bad it looks if you don't
use dollar signs: 3+x=9.  

In mathematics mode (anything inside dollar signs) there are also a
huge number of commands you can use to produce special mathematical
symbols.

\begin{itemize}
  \item You can make superscripts using the \verb|^| (carat)
    character.  For example, \verb|$x^2$| produces $x^2$.  If the
    superscript consists of more than one character, be sure to
    enclose the superscript portion in curly braces so \LaTeX\ knows
    what should be included in the superscript.  For example, compare
    \verb|$x^i+2$|, which produces $x^i+2$, with \verb|$x^{i+2}$|,
    which produces $x^{i+2}$.


  \item You can make subscripts using \verb|_| (underscore).  For
    example, \verb|$x_3$| looks like this: $x_3$.

  \item You can make a square root using the
    \verb|\sqrt{...}| command.  For example, \verb|$\sqrt{x+2}$| looks
    like this: $\sqrt{x+2}$.

  \item You can make fractions using the \verb|\frac{...}{...}|
    command.  For example, \verb|$\frac{y+2}{5}$| looks like this:
    $\frac{y+2}{5}$.

  \item Some mathematical symbols can be typed directly from your
    keyboard, like $=$, $+$, $-$, $>$, and $<$.  However, there are a
    very large number of special mathematical symbols that do not
    correspond to a key on the keyboard but can be produced using a
    special \LaTeX\ command.  For example, $\infty$ (\verb|\infty|),
    $\to$ (\verb|\to|), $\geq$ (\verb|\geq|), $\leq$ (\verb|\leq|),
    $\neq$ (\verb|\neq|), $\cdot$ (\verb|\cdot|), and $\pi$
    (\verb|\pi|). There are many others, but you will learn them as
    we go along.

\end{itemize}

Sometimes you will want to typeset an equation, such as \[ x^3 + 15 =
33 \] by itself on a separate line, instead of in the middle of a
sentence like $x^3 + 15 = 33$.  To achieve this, you can just enclose
the equation in a \verb|\[ ... \]| pair, instead of dollar signs.
Sometimes you also want to give an equation a number so you can refer to it
later, like this:
\begin{equation}
  \label{eq:example}
  x^3 + 15 = 33
\end{equation}
We can solve equation \eqref{eq:example} to find that $x =
\sqrt[3]{18}$. To achieve this, enclose the equation in an
\verb|equation| environment.  Of course, you can give
equations \verb|\label|s just like sections.  To refer to equation
numbers with the parentheses included (as I did above), you can use
\verb|\eqref| instead of just \verb|\ref|.

\subsection{Making your own commands}
\label{sec:commands}

One final thing before you will know enough \LaTeX\ to get started.
There is a very helpful feature which lets you define your own
commands!  Let's say you find yourself typing ``my helicopter is full
of eels'' a lot, and it's getting kind of annoying to type it out
every time.  You can define a new command to generate this text, like
this:

\begin{verbatim}
  \newcommand{\mhfe}{my helicopter is full of eels}
\end{verbatim}

Any \verb|\newcommand|s should go in the preamble of your document,
that is, \emph{before} the \verb|\begin{document}|.

Once you have defined this command, you can just type \verb|\mhfe|
instead of the entire phrase, and the phrase will be included in the
output document: \mhfe.

You can even create commands which have missing details that you can
fill in later by providing parameters to the command.  For example,
suppose you find yourself often writing things like
$\sqrt{\sqrt{\sqrt{\sqrt{3}}}}$ and $\sqrt{\sqrt{\sqrt{\sqrt{5}}}}$.
It's annoying to have to type \verb|\sqrt{\sqrt{\sqrt{\sqrt{...}}}}|
every time, so you want to create a special command to save you some
typing.  The problem, of course, is that you might want to change the
number inside the square root each time.  The solution is to define a
command like this:

\begin{verbatim}
  \newcommand{\ssss}[1]{\sqrt{\sqrt{\sqrt{\sqrt{#1}}}}}
\end{verbatim}

The \verb|[1]| in square brackets says how many parameters the command
needs.  The \verb|#1| refers to whatever parameter is provided.  For example,
\verb|\ssss{\pi}| automatically turns into
\verb|\sqrt{\sqrt{\sqrt{\sqrt{\pi}}}}|, thus producing $\ssss{\pi}$. Of
course, you could also make a command with multiple parameters and use
\verb|#1|, \verb|#2|, and so on to access them. 

\subsection{\texttt{precalc.sty}}

I have created a special package file, \texttt{precalc.sty}, which
contains a number of commands and settings you can use while preparing
your solutions.  For now, simply place a copy of \texttt{precalc.sty}
in the same directory with any \texttt{.tex} file you create; then you
can use it by putting \verb|\usepackage{precalc}| in the preamble of
your \texttt{.tex} file.  The solution template file already contains
\verb|\usepackage{precalc}|, so you really don't have to worry about
it beyond making sure you have a copy of it in your working directory.

Even better, there is probably a way to ``install''
\texttt{precalc.sty} with MiKTeX so that you can use it automatically,
without having to put a copy in your working directory.  I don't know
how to do this with MiKTeX, but once you have MiKTeX installed we
could try to figure it out.

\section{Learning more}
\label{sec:learning}

If you want to learn more about \LaTeX\ and the underlying typesetting
system \TeX\ beyond what is covered in this tutorial, a great starting
place is to read \emph{The Not-So-Short Introduction to \LaTeX2e},
\url{http://tug.ctan.org/tex-archive/info/lshort/english/lshort.pdf}.
There is also much information which can be found at
\url{http://www.latex-project.org/} and \url{http://ctan.org}.


\end{document}