\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)

\quad\text{by distributive law} \\

& = & 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!}

\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.

\)

\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}[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).

\]

\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\).