\documentclass[a4paper]{article} \usepackage{amsmath} \usepackage{hyperref} \title{\LaTeX\ Mathematics Examples} \author{Prof Tony Roberts} \begin{document} \maketitle \tableofcontents \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: \[ \vec\nabla\times\vec q =\vec i\left(\D yw-\D zv\right) +\vec j\left(\D zu-\D xw\right) +\vec k\left(\D xv-\D yu\right). \] Recall that typesetting multi-line mathematics is an art normally too hard for computer recipes. Nonetheless, if you need to be automatically flexible about multi-line mathematics, and you do not mind some rough typesetting, then perhaps invoke \verb|\parbox| to help as follows: % The \verb|breqn| package is not yet reliable enough for general use. \newcommand{\parmath}[2][0.8\linewidth]{\parbox[t]{#1}% {\raggedright\linespread{1.2}\selectfont\(#2\)}} \[ u_1=\parmath{ -2 \gamma \epsilon^{2} s_{2}+\mu \epsilon^{3} \big( \frac{3}{8} s_{2}+\frac{1}{8} s_{1} i\big)+\epsilon^{3} \big( -\frac{81}{32} s_{4} s_{2}^{2}-\frac{27}{16} s_{4} s_{2} s_{1} i+\frac{9}{32} s_{4} s_{1}^{2}+\frac{27}{32} s_{3} s_{2}^{2} i-\frac{9}{16} s_{3} s_{2} s_{1}-\frac{3}{32} s_{3} s_{1}^{2} i\big) +\int_a^b 1-2x+3x^2-4x^3\,dx } \] Also, sometimes use \verb|\parbox| to typeset multiline entries in tables. \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 \(a^2+b^2 =(p^2-q^2)^2+(2pq)^2 =p^4-2p^2q^2+q^4+4p^2q^2 =p^4+2p^2q^2+q^4 =(p^2+q^2)^2 =c^2\). \end{document}