1. documentclass[20pt,a4paper]{extarticle}
2. usepackage[a4paper,margin=6mm]{geometry}
3. usepackage{amsmath}
4. usepackage{hyperref}
5.  
6. title{LaTeX Mathematics Examples}
7. author{Prof Tony Roberts}
8.  
9. begin{document}
10.  
11. maketitle
12.  
13. tableofcontents
14.  
15.  
16.  
17. section{Delimiters}
18.  
19. See how the delimiters are of reasonable size in these examples
20. [
21.         left(a+bright)left[1-frac{b}{a+b}right]=a,,
22. ]
23. [
24.         sqrt{|xy|}leqleft|frac{x+y}{2}right|,
25. ]
26. even when there is no matching delimiter
27. [
28.         int_a^bufrac{d^2v}{dx^2},dx
29.         =left.ufrac{dv}{dx}right|_a^b
30.         -int_a^bfrac{du}{dx}frac{dv}{dx},dx.
31. ]
32.  
33.  
34.  
35.  
36.  
37.  
38. section{Spacing}
39.  
40. Differentials often need a bit of help with their spacing as in
41. [
42.         iint xy^2,dx,dy
43.         =frac{1}{6}x^2y^3,
44. ]
45. whereas vector problems often lead to statements such as
46. [
47.         u=frac{-y}{x^2+y^2},,quad
48.         v=frac{x}{x^2+y^2},,quadtext{and}quad
49.         w=0,.
50. ]
51. 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),.
52. 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),.
53. Be discerning about when to do this as the spacing is different.
54.  
55.  
56.  
57.  
58.  
59.  
60. section{Arrays}
61.  
62. Arrays of mathematics are typeset using one of the matrix environments as
63. in
64. [
65.         begin{bmatrix}
66.                 1 & x & 0 \
67.                 0 & 1 & -1
68.         end{bmatrix}begin{bmatrix}
69.                 1  \
70.                 y  \
71.                 1
72.         end{bmatrix}
73.         =begin{bmatrix}
74.                 1+xy  \
75.                 y-1
76.         end{bmatrix}.
77. ]
78. Case statements use cases:
79. [
80.         |x|=begin{cases}
81.                 x, & text{if }xgeq 0,,  \
82.                 -x, & text{if }x< 0,.
83.         end{cases}
84. ]
85. Many arrays have lots of dots all over the place as in
86. [
87.         begin{matrix}
88.                 -2 & 1 & 0 & 0 & cdots & 0  \
89.                 1 & -2 & 1 & 0 & cdots & 0  \
90.                 0 & 1 & -2 & 1 & cdots & 0  \
91.                 0 & 0 & 1 & -2 & ddots & vdots \
92.                 vdots & vdots & vdots & ddots & ddots & 1  \
93.                 0 & 0 & 0 & cdots & 1 & -2
94.         end{matrix}
95. ]
96.  
97.  
98.  
99.  
100.  
101.  
102. section{Equation arrays}
103.  
104. In the flow of a fluid film we may report
105. begin{eqnarray}
106.         u_alpha & = & epsilon^2 kappa_{xxx}
107.         left( y-frac{1}{2}y^2 right),
108.         label{equ}  \
109.         v & = & epsilon^3 kappa_{xxx} y,,
110.         label{eqv}  \
111.         p & = & epsilon kappa_{xx},.
112.         label{eqp}
113. end{eqnarray}
114. Alternatively, the curl of a vector field $(u,v,w)$ may be written
115. with only one equation number:
116. begin{eqnarray}
117.         omega_1 & = &
118.         frac{partial w}{partial y}-frac{partial v}{partial z},,
119.         nonumber  \
120.         omega_2 & = &
121.         frac{partial u}{partial z}-frac{partial w}{partial x},,
122.         label{eqcurl}  \
123.         omega_3 & = &
124.         frac{partial v}{partial x}-frac{partial u}{partial y},.
125.         nonumber
126. end{eqnarray}
127. Whereas a derivation may look like
128. begin{eqnarray*}
129.         (pwedge q)vee(pwedgeneg q) & = & pwedge(qveeneg q)
130.         quadtext{by distributive law}  \
131.          & = & pwedge T quadtext{by excluded middle}  \
132.          & = & p quadtext{by identity}
133. end{eqnarray*}
134.  
135.  
136.  
137.  
138.  
139.  
140. section{Functions}
141.  
142. Observe that trigonometric and other elementary functions are typeset
143. properly, even to the extent of providing a thin space if followed by
144. a single letter argument:
145. [
146.         exp(itheta)=costheta +isintheta,,quad
147.         sinh(log x)=frac{1}{2}left( x-frac{1}{x} right).
148. ]
149. With sub- and super-scripts placed properly on more complicated
150. functions,
151. [
152.         lim_{qtoinfty}|f(x)|_q
153.         =max_{x}|f(x)|,
154. ]
155. and large operators, such as integrals and
156. begin{eqnarray*}
157.         e^x & = & sum_{n=0}^infty frac{x^n}{n!}
158.         quadtext{where }n!=prod_{i=1}^n i,,  \
159.         overline{U_alpha} & = & bigcap_alpha U_alpha,.
160. end{eqnarray*}
161. In inline mathematics the scripts are correctly placed to the side in
162. order to conserve vertical space, as in
163. (
164.         1/(1-x)=sum_{n=0}^infty x^n.
165. )
166.  
167.  
168.  
169.  
170.  
171.  
172. section{Accents}
173.  
174. Mathematical accents are performed by a short command with one
175. argument, such as
176. [
177.         tilde f(omega)=frac{1}{2pi}
178.         int_{-infty}^infty f(x)e^{-iomega x},dx,,
179. ]
180. or
181. [
182.         dot{vec omega}=vec rtimesvec I,.
183. ]
184.  
185.  
186.  
187.  
188.  
189. section{Command definition}
190.  
191. newcommand{Ai}{operatorname{Ai}}
192. The Airy function, $Ai(x)$, may be incorrectly defined as this
193. integral
194. [
195.         Ai(x)=intexp(s^3+isx),ds,.
196. ]
197.  
198. newcommand{D}[2]{frac{partial #2}{partial #1}}
199. newcommand{DD}[2]{frac{partial^2 #2}{partial #1^2}}
200. renewcommand{vec}[1]{boldsymbol{#1}}
201.  
202. This vector identity serves nicely to illustrate two of the new
203. commands:
204. [
205.         vecnablatimesvec q
206.         =vec ileft(D yw-D zvright)
207.         +vec jleft(D zu-D xwright)
208.         +vec kleft(D xv-D yuright).
209. ]
210.  
211. 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:
212. % The verb|breqn| package is not yet reliable enough for general use.
213. newcommand{parmath}[2][0.8linewidth]{parbox[t]{#1}%
214.     {raggedrightlinespread{1.2}selectfont(#2)}}
215. [
216. 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 }
217. ]
218. Also, sometimes use verb|parbox| to typeset multiline entries in tables.
219.  
220.  
221. section{Theorems et al.}
222.  
223. newtheorem{theorem}{Theorem}
224. newtheorem{corollary}[theorem]{Corollary}
225. newtheorem{lemma}[theorem]{Lemma}
226. newtheorem{definition}[theorem]{Definition}
227.  
228. begin{definition}[right-angled triangles] label{def:tri}
229. 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).
230. end{definition}
231.  
232. begin{lemma}
233. The triangle with sides of length~(3), (4) and~(5) is right-angled.
234. end{lemma}
235.  
236. This lemma follows from the Definition~ref{def:tri} as (3^2+4^2=9+16=25=5^2).
237.  
238. begin{theorem}[Pythagorean triplets] label{thm:py}
239. Triangles with sides of length (a=p^2-q^2), (b=2pq) and (c=p^2+q^2) are right-angled triangles.
240. end{theorem}
241.  
242. Prove this Theorem~ref{thm:py} by the algebra (a^2+b^2 =(p^2-q^2)^2+(2pq)^2
243. =p^4-2p^2q^2+q^4+4p^2q^2
244. =p^4+2p^2q^2+q^4
245. =(p^2+q^2)^2 =c^2).
246.  
247.  
248. end{document}