\section{Math}

\subsection{Equation}

This is an equation

\begin{equation}
    % A lonesome ( will have a fixed size
    % But a \left( will be scaled to contain whatever it has to
    \langle f, g \rangle_{\left[a, b\right]} 
    =
    \int_a^b f(x)\overline{g(x)} dx
    % \\  % The equation env ignores \\
    % a^2 + b^2 = c^2
    \label{eq:inner_product}  % You may label equations for referencing later
\end{equation}

\subsection{Gather}

This is the gather environment

\begin{gather*}
    E = mc^2
    \\
    E^2 = 
        \frac{                          % You
            mc^2                        % May
        }                               % Use
        {                               % As
            \sqrt{                      % Many
                1                       % Lines
                -                       % As
                \frac{                  % You
                    v^2                 % Want
                }
                {                       % Sometimes
                    c^2                 % It makes
                }                       % Things more
            }                           % Structured
        }                               % ...though not always...
\end{gather*}

\subsection{Align}

This is an \emph{anonymous} align environment

\begin{align*}
    24 &= 8 \cdot 3 \\  % Allows aligning glyphs by using &
    &= 4 \cdot 6 \\  % The glyph marked with & on each line
    &= 2 \cdot 2 \cdot 2 \cdot 3  % Will be aligned
\end{align*}

\subsection{Inlined math}

% Math can be inlined with \( ..my math.. \) or $..also my math..$
% $_$ is the old TeX way and is not advisable to use
% \(_\) is the more modern LaTeX way and is more robust
A \(\sum_n^k\) sum! And a $\frac{1}{n}$.

% The \[\] is a \displaystyle math environment
% There is no real reason to use it over the \begin syntax, but it's there. This env is displayed on its own line and is not a part of the paragraph in the same way $_$ and \(\) are.
You can also: \[\alpha\] do this. 

\subsection{Some symbols}

Greek

\begin{gather*}
    \psi,\ \Psi  % The "\ " thing forces the insert of a space, otherwise whitespace is ignored
    \\
    \alpha, \beta, \gamma, \delta  % Lower case with _small_ letters
    \\
    \Gamma, \Delta  % Upper case with _Capital_ letter
    \\
    \epsilon, \varepsilon  % Variants with _var_
\end{gather*}

Other things

% If you want a glyph, but don't know what it's called
% Either try something (most glyphs have sensible names)
% Or just search it up online, after a while you'll remember
\begin{gather*}
    \nabla, \partial
    \\
    \int, \iint, \oint
    \\
    \sum_{i = 1}^{n}, \prod_{1 \leq i \leq n}
    \\
    \frac{a + ib}{c + id}, i = 1, \dots, n
    \\
    \bar{x}, \vec{x}, x^\circ
\end{gather*}

Matrices (and vectors)

\begin{gather*}
    \begin{pmatrix}  % Parantheses
        1 & 2 & 3 \\  % Columns separated with &
        4 & 5 & 6 \\  % Rows separated with \\
        7 & 8 & 9
    \end{pmatrix}
    ,
    \begin{bmatrix}  % Brackets
        1 & 2 & 3 \\
        4 & 5 & 6 \\
        7 & 8 & 9
    \end{bmatrix}
    ,
    \begin{Bmatrix}  % _Curly_ brackets
        1 & 2 &  \\  % You can leave elements empty
        4 & \displaystyle\int_0^1x^2 dx & 6 \\  % You can make the elements complicated (and force roomy display with \displaystyle
        7 & 8 & 9
    \end{Bmatrix}
    ,
    \begin{pmatrix}  % Essentially a vector
        x_1 \\ x_2 \\ x_3  % Just because \\ causes linebreak, doesn't mean you need a new line in the source
    \end{pmatrix}
\end{gather*}

% Defining a new command, don't do it here, do it in the preamble or something similar
% newcommand takes a command name, a number of arguments and the "function body" of the command. The arguments to the new command being defined can be accessed by #<number>
\newcommand{\myvec}[1]{
\begin{pmatrix}
    #1
\end{pmatrix}
}

% Works like a charm
\begin{gather*}
    \myvec{x_1\\x_2\\x_3} \leftarrow \text{My vector command}
\end{gather*}