Learn Latex – Mathematics for Machine Learning

June 28, 2018

1. 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
\[
\int_a^bu\frac{d^2v}{dx^2}\,dx
=\left.u\frac{dv}{dx}\right|_a^b
-\int_a^b\frac{du}{dx}\frac{dv}{dx}\,dx.
\]

2. 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 \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.

3. 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
\[
\begin{matrix}
-2 & 1 & 0 & 0 & \cdots & 0 \\
1 & -2 & 1 & 0 & \cdots & 0 \\
0 & 1 & -2 & 1 & \cdots & 0 \\
0 & 0 & 1 & -2 & \ddots & \vdots \\
\vdots & \vdots & \vdots & \ddots & \ddots & 1 \\
0 & 0 & 0 & \cdots & 1 & -2
\end{matrix}
\]

4. 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)
\quad\text{by distributive law} \\
& = & p\wedge T \quad\text{by excluded middle} \\
& = & p \quad\text{by identity}
\end{eqnarray*}

5. 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!}
\quad\text{where }n!=\prod_{i=1}^n i\,, \\
\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
\(
1/(1-x)=\sum_{n=0}^\infty x^n.
\)

6. 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
\[
\dot{\vec \omega}=\vec r\times\vec I\,.
\]

7. 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}[2]{\frac{\partial #2}{\partial #1}}
\newcommand{\DD}[2]{\frac{\partial^2 #2}{\partial #1^2}}
\renewcommand{\vec}[1]{\boldsymbol{#1}}

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

8. 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 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~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~thm:py by the algebra

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

Share with: