documentclass[20pt,a4paper]{extarticle}
usepackage[a4paper,margin=6mm]{geometry}
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+bright)left[1-frac{b}{a+b}right]=a,,
]
[
	sqrt{|xy|}leqleft|frac{x+y}{2}right|,
]
even when there is no matching delimiter
[
	int_a^bufrac{d^2v}{dx^2},dx
	=left.ufrac{dv}{dx}right|_a^b
	-int_a^bfrac{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},,quadtext{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 }xgeq 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*}
	(pwedge q)vee(pwedgeneg q) & = & pwedge(qveeneg q)
	quadtext{by distributive law}  \
	 & = & pwedge T quadtext{by excluded middle}  \
	 & = & p quadtext{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(itheta)=costheta +isintheta,,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_{qtoinfty}|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!}
	quadtext{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}{2pi}
	int_{-infty}^infty f(x)e^{-iomega x},dx,,
]
or
[
	dot{vec omega}=vec rtimesvec I,.
]





section{Command definition}

newcommand{Ai}{operatorname{Ai}} 
The Airy function, $Ai(x)$, may be incorrectly defined as this 
integral
[
	Ai(x)=intexp(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:
[
	vecnablatimesvec q
	=vec ileft(D yw-D zvright)
	+vec jleft(D zu-D xwright)
	+vec kleft(D xv-D yuright).
]

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][0.8linewidth]{parbox[t]{#1}%
    {raggedrightlinespread{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} ibig)+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} ibig) +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}