Presentation Notes.md

Introduction

Preliminaries

• Create an account on Sharelatex.com and log in to Projects page
• Navigate to the Github Repository: Repository Link
• Grab a copy of the example document handouts, start looking through and thinking about what you'd like to cover
• Sit close to the front!

Getting Started

• Introduce selves
• Mention that this is the first "intermediate" workshop and that feedback is appreciated

Live Coding

• Start with blank document

• Add some packages - stuff you'll almost always want/need

  • mathtools: Combines many amsmath packages
  • amssymb: You'll inevitably need a weird symbol
  • amsthm: Nice math environments
  • hyperref: Hyperlink all the things
  • float: Mitigates the living hell of image placement

• We'll pull in more specific stuff as needed

• Let's write a theorem: the quadratic equation

  • Use \textbf for bold, manually number it
  • Do the same for 'proof'
  • Math mode review: $$ \[ \]
    • Cue holy war!
    • $$ common in many tex environments (e.g. Mathjax, used on Math SE/WO), good for quick hacks but can introduce subtle bugs/spacing issues
    • \[ \] Does some extra syntax checking, ams package just redefines this to be its own (super nice) equation environment
  • Write combined roots with \pm
  • Then use comma to separate the two roots
  • Replace comma with \text{and}
  • Show spacing using \:
    • Cover other types of spacing:
      • \! a little less
      • \, a little more
      • \: medium
      • \; a lot
        • Just \ also
      • \quad so much space
      • \qquad wide open ranges
    • If you like operator spacing, define your own!
      • Binary (like $+$) or relational (like $<$)
      • \mathbin or \mathrel
      • ​

• Improving the theorem

  • Define theorem and proof environments: \newtheorem{theorem}{Theorem}
  • First parameter: custom name you give this environment
  • Second parameter: What text is displayed (in bold by default)
  • Optional parameters: When to break the numbering
    • section: By default, theorems are ordered 1,2,3... across all sections. This restarts numbering at every section break
    • theorem: Attaches the counter to theorem environments instead, restarts every time there's a new theorem
  • Call like a normal environment with begin/end
  • Supply optional parameters for Important Theorems (tm)
    • \begin{theorem}[Brouwer's Fixed Point Theorem]
  • Can also use theorem* to turn off numbering
  • Note: Proof is built-in! Gives us the QED box too!
    • Change the QED symbol: \renewcommand\qedsymbol{$\blacksquare$}

• Show sgn function theorem

• Show colspan theorem

  • Use theorem and proof environments this time
  • Show \operatorname{span} , then DeclareMathOperator
  • Use pmatrix
  • Show cdots, vdots, ddots
  • Show left/right modifiers

• Fancy letters! mathbb, mathcal, mathfrak, mathscr

• Over-stuff overline, overbrace, overset

• Colors: \usepackage[dvipsnames]{xcolor} {\color{MidnightBlue}2x}, {\color{BrickRed}2x}

• Tabular Environments

  • Alignment options:c, l, r

• Beamer

  • Frames
  • Itemize within frames and number to generate "transitions"
  • pause command if you just want to write text
  • Blocks:
    • block, alertblock, examples

• Tikz

  • Start with \draw or \fill , end with ;
    • These take params like color, thickness, etc
  • Lines: $(x_0, y_0) -- (x_1, y_1);$
  • Circles: $(x_0, y_0)$ circle (size pt);
  • Bezier: \draw (-2,2) .. controls (-1,0) and (1,0) .. (2,2);
    • Endpoints and control points
  • Ellipse: \fill[blue!50] (2.5,0) ellipse (1.5 and 0.5);
    • 2nd params are foci

• Macros \newcommand{\ff}[2]{\mathbb{F}_{#1^{#2}}}

  ​

Code References

Math Hackery

Column Space Theorem:

\DeclareMathOperator{\col}{col}
\DeclareMathOperator{\spn}{span}

...

\[
\operatorname{col}(A)=
\operatorname{col}\begin{pmatrix}
a_{11} & \cdots & a_{12}\\
\vdots & \ddots & \vdots\\
a_{m1} & \cdots & a_{mn}
\end{pmatrix}
=
\operatorname{span}\left\{
    \begin{pmatrix}
        a_{11}\\\vdots\\a_{m1}
    \end{pmatrix}
    , \, \ldots \, ,
    \begin{pmatrix}
        a_{1n}\\\vdots\\a_{mn}
    \end{pmatrix} 
\right\}\,.
\]

Align Environment:

\begin{align*}
(x+y)^3 &=(x+y)(x+y)^2\\
        &=(x+y)(x^2+2xy+y^2)\\
        &=x^3+2x^2y+xy^2+xy^2+2xy^2+y^3\\
        &=x^3+3x^2y+3xy^2+y^3\,.\tag*{\qedhere}
\end{align*}

Cases Environment

\[
\operatorname{sgn}(x):=
\begin{cases}
1   &\text{if $x>0$}\\
0   &\text{if $x=0$}\\
-1  &\text{if $x<0$\,.}
\end{cases}
\]

Overstuff

\[
a^m\cdot a^n
=
(\overbrace{a\cdots a}^{\text{$m$ factors}})\cdot (\overbrace{a\cdots a}^{\text{$n$ factors}})
\overset{(*)}{=}
\overbrace{a\cdots a\cdot a\cdots a}^{\text{$m+n$ factors}}
=
a^{m+n}
\,,
\]

Tabular Environments

\begin{center}
\begin{tabular}{ c | c | c }
 cell1 & cell2 & cell3 \\
 \hline
 cell4 & cell5 & cell6 \\  
 \hline
 cell7 & cell8 & cell9    
\end{tabular}
\end{center}

Tikz

\begin{tikzpicture}

% Lines
\draw[gray, thick] (-1,2) -- (2,-4);
\draw[gray, thick] (-1,-1) -- (2,2);

% Point
\filldraw[black] (0,0) circle (2pt) node[anchor=west] {Intersection point};
 
 % Bezier
 \draw (-2,2) .. controls (-1,0) and (1,0) .. (2,2);
 
 % Ellipse
 \fill[blue!50] (2.5,0) ellipse (1.5 and 0.5);
 
\end{tikzpicture}

Tables

Labels and Cross-Referencing

Source Code

Diagrams Using TikZ

Customization

Alternate Document Classes