# Learn Latex – Mathematics for Machine Learning

\section{Delimiters}

See how the delimiters are of reasonable size in these examples
$\left(a+b\right)\left[1-\frac{b}{a+b}\right]=a\,,$
$\sqrt{|xy|}\leq\left|\frac{x+y}{2}\right|,$
even when there is no matching delimiter
$\int_a^bu\frac{d^2v}{dx^2}\,dx =\left.u\frac{dv}{dx}\right|_a^b -\int_a^b\frac{du}{dx}\frac{dv}{dx}\,dx.$

\section{Spacing}

Differentials often need a bit of help with their spacing as in
$\iint xy^2\,dx\,dy =\frac{1}{6}x^2y^3,$
whereas vector problems often lead to statements such as
$u=\frac{-y}{x^2+y^2}\,,\quad v=\frac{x}{x^2+y^2}\,,\quad\text{and}\quad w=0\,.$
Occasionally one gets horrible line breaks when using a list in mathematics such as listing the first twelve primes $$2,3,5,7,11,13,17,19,23,29,31,37$$\,.
In such cases, perhaps include \verb|\mathcode\,=”213B| inside the inline maths environment so that the list breaks: $$\mathcode\,=”213B 2,3,5,7,11,13,17,19,23,29,31,37$$\,.
Be discerning about when to do this as the spacing is different.

\section{Arrays}

Arrays of mathematics are typeset using one of the matrix environments as
in
$\begin{bmatrix} 1 & x & 0 \\ 0 & 1 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ y \\ 1 \end{bmatrix} =\begin{bmatrix} 1+xy \\ y-1 \end{bmatrix}.$
Case statements use cases:
$|x|=\begin{cases} x, & \text{if }x\geq 0\,, \\ -x, & \text{if }x< 0\,. \end{cases}$
Many arrays have lots of dots all over the place as in
$\begin{matrix} -2 & 1 & 0 & 0 & \cdots & 0 \\ 1 & -2 & 1 & 0 & \cdots & 0 \\ 0 & 1 & -2 & 1 & \cdots & 0 \\ 0 & 0 & 1 & -2 & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & \cdots & 1 & -2 \end{matrix}$

\section{Equation arrays}

In the flow of a fluid film we may report
\begin{eqnarray}
u_\alpha & = & \epsilon^2 \kappa_{xxx}
\left( y-\frac{1}{2}y^2 \right),
\label{equ} \\
v & = & \epsilon^3 \kappa_{xxx} y\,,
\label{eqv} \\
p & = & \epsilon \kappa_{xx}\,.
\label{eqp}
\end{eqnarray}
Alternatively, the curl of a vector field $(u,v,w)$ may be written
with only one equation number:
\begin{eqnarray}
\omega_1 & = &
\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\,,
\nonumber \\
\omega_2 & = &
\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\,,
\label{eqcurl} \\
\omega_3 & = &
\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\,.
\nonumber
\end{eqnarray}
Whereas a derivation may look like
\begin{eqnarray*}
(p\wedge q)\vee(p\wedge\neg q) & = & p\wedge(q\vee\neg q)
& = & p\wedge T \quad\text{by excluded middle} \\
& = & p \quad\text{by identity}
\end{eqnarray*}

\section{Functions}

Observe that trigonometric and other elementary functions are typeset
properly, even to the extent of providing a thin space if followed by
a single letter argument:
$\exp(i\theta)=\cos\theta +i\sin\theta\,,\quad \sinh(\log x)=\frac{1}{2}\left( x-\frac{1}{x} \right).$
With sub- and super-scripts placed properly on more complicated
functions,
$\lim_{q\to\infty}\|f(x)\|_q =\max_{x}|f(x)|,$
and large operators, such as integrals and
\begin{eqnarray*}
e^x & = & \sum_{n=0}^\infty \frac{x^n}{n!}
\overline{U_\alpha} & = & \bigcap_\alpha U_\alpha\,.
\end{eqnarray*}
In inline mathematics the scripts are correctly placed to the side in
order to conserve vertical space, as in
$$1/(1-x)=\sum_{n=0}^\infty x^n.$$

\section{Accents}

Mathematical accents are performed by a short command with one
argument, such as
$\tilde f(\omega)=\frac{1}{2\pi} \int_{-\infty}^\infty f(x)e^{-i\omega x}\,dx\,,$
or
$\dot{\vec \omega}=\vec r\times\vec I\,.$

\section{Command definition}

\newcommand{\Ai}{\operatorname{Ai}}
The Airy function, $\Ai(x)$, may be incorrectly defined as this
integral
$\Ai(x)=\int\exp(s^3+isx)\,ds\,.$

\newcommand{\D}{\frac{\partial #2}{\partial #1}}
\newcommand{\DD}{\frac{\partial^2 #2}{\partial #1^2}}
\renewcommand{\vec}{\boldsymbol{#1}}

This vector identity serves nicely to illustrate two of the new
commands:
$\vec\nabla\times\vec q =\vec i\left(\D yw-\D zv\right) +\vec j\left(\D zu-\D xw\right) +\vec k\left(\D xv-\D yu\right).$

\section{Theorems et al.}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\begin{definition}[right-angled triangles] \label{def:tri}
A \emph{right-angled triangle} is a triangle whose sides of length~$$a$$, $$b$$ and~$$c$$, in some permutation of order, satisfies $$a^2+b^2=c^2$$.
\end{definition}

\begin{lemma}
The triangle with sides of length~$$3$$, $$4$$ and~$$5$$ is right-angled.
\end{lemma}

This lemma follows from the Definition~\ref{def:tri} as $$3^2+4^2=9+16=25=5^2$$.

\begin{theorem}[Pythagorean triplets] \label{thm:py}
Triangles with sides of length $$a=p^2-q^2$$, $$b=2pq$$ and $$c=p^2+q^2$$ are right-angled triangles.
\end{theorem}

Prove this Theorem~\ref{thm:py} by the algebra

$$a^2+b^2 =(p^2-q^2)^2+(2pq)^2 =p^4-2p^2q^2+q^4+4p^2q^2 =p^4+2p^2q^2+q^4 =(p^2+q^2)^2 =c^2$$.

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