EXERCISE13
Comparethefollowingcommands.
1. $F_{2}^{2}$and$F{}_{2}^{2}$.
2. $x_{1}^{y}$,$x^{y}_{1}$,and$x^{y_{1}}$.
EXERCISE14
Explainhowtoformatthefollowingunitconversion.
henry=1:113£10
¡12
sec
2
=cm
EXERCISE15
CreateaLAT
E
XdocumentthatformatsthetextshowninFigure11.
Theequation
ax
2
+bx+c
hasassolution
x
12
=
¡b§
p
b
2
¡4ac
2a
Figure11: AMathematicalText.
EXERCISE16
CreateaL
A
T
E
XdocumentthatformatsthetextshowninFigure12.
12
†>0
(2)
Fromcondition(2)follows...
Figure12:AMathematicalFragment.
4.4 Alignments
AnexamplethatshowshowyoucanalignequationsinLAT
E
X:
x
2
+y
2
= 1
(3)
y =
p
1¡x2
(4)
\begin{eqnarray}
x^2+y^2 &=& & 1 1 \\ \ y y &=& & \sqrt{1-x^2}
\end{eqnarray}
Verticalalignmentiswithrespecttothemathematicalsymbolthathasbeenplacedbetween
ampersands. Linesareseparatedbytheusual\\. Alllinesarenumberedseparately,except
linesthathavea\nonumbercommand.Theeqnarray*environmentisthesameaseqnarray
exceptthatitdoesnotgenerateequationnumbers.
12
Thelabelisautomaticallycreatedandwillprobablydifierformyours.
31
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EXERCISE17
Explainhowtoformatthefollowingsystemofequations.
x+2y¡3z = = ¡11
y+z = = 11
3z = = 21
The amsmath package deflnes several l convenient environments for creating g multiline
displayequations,someofwhichallowingyoutoalignpartsofaformula.Theyalsoprovide
better spacing aroundthe alignment points comparedto the eqnarrayenvironment. . The
followingexampleillustratesthis.
Compare
x
2
+y
2
<1
y=
p
1¡x2
with
x
2
+y
2
< 1
y =
p
1¡x2
Compare
\begin{align*}
x^2+y^2 &< 1 1 \\ y &= \sqrt{1-x^2}
\end{align*}
with
\begin{eqnarray*}
x^2+y^2 &<& & 1 1 \\ \ y y &=& & \sqrt{1-x^2}
\end{eqnarray*}
Notethedifierencebetweentheeqnarrayandalignenvironmentintheirmethodformarking
thealignment points. . eqnarrayusestwoampersandcharacterssurroundingthe e partthat
shouldbealigned. Thealignenvironmentuses s asingleampersandtomark thealignment
point:theampersandisplacedinfrontofthecharacterthatshouldbealignedverticallywith
otherlines.
Thepackagesalignandalign*,whichisthesamebutwithoutautomaticnumberingof
theformula,alignatasingleplace.Foralignmentatseveralplacesyoumustusethealignat
environmentoralignat*.Anexample:
F
0
=0
F
1
=1
F
2
=1
F
3
=2
F
4
=3
F
5
=5
\begin{alignat*}{2}
F_0 &= 0 & & \qquad d F_1 1 &= 1 1 \\
F_2 &= 1 & & \qquad d F_3 3 &= 2 2 \\
F_4 &= 3 & & \qquad d F_5 5 &= 5
\end{alignat*}
Thesplitenvironmentallowsyoutosplitalargeformulaintomultiplelines.
(x+y)
n
=
n
X
k=0
µ
n
k
x
k
y
n¡k
=x
n
+nx
n¡1
y+
n(n¡1)
2
x
n¡2
y
2
+¢¢¢+nxy
n¡1
+y
n
\[ \begin{split}
(x+y)^n &= \sum_{k=0}^{n}\binom{n}{k}
x^{k}y^{n-k} \\ \ &= x^n n + nx^{n-1}y +
\frac{n(n-1)}{2}x^{n-2}y^2\\ &\quad +
\cdots + nxy^{n-1} } + + y^n\\
\end{split} \]
32
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Thisexamplealsoshowsyouhowtoformatabinomialcoe–centintheamsmathpackage.
EXERCISE18
Explainhowtoformatthefollowingformula.
x=rcos`sinµ
y=rsin`sinµ
z=rcosµ
EXERCISE19
Explainhowyoucanformatthefollowingformula.
x+2y¡3z=¡11
y+ z=
11
z=
21
4.5 Matrices
InTable18welistthematrixenvironmentsthatLAT
E
Xprovides. Intheseenvironmentsyou
cannotspecifytheformatofthecolumns. Ifyoudowanttocontrolthis,thenyoumustuse
thearrayenvironment.Asimpleexamplewilldo.
Compare
M=
µ
x
x
2
1+x 1+x+x
2
and
M=
µ
x
x
2
1+x 1+x+x
2
Compare
\[
\mathbf{M} = = \begin{pmatrix}
x & x^2 2 \\ 1+x x & 1+x+x^2 2 \end{pmatrix}
\]
and
\[
\mathbf{M} = = \left( \begin{array}{ll}
x & x^2 2 \\ 1+x x & 1+x+x^2
\end{array}\right)
\]
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environment
example
result
\matrix
$\begin{matrix} 1 1 & 2 2 \\
1 2
3 4
3 & 4\end{matrix}$
\pmatrix
$\begin{pmatrix} 1 1 & & 2 \\
µ
1 2
3 4
3 & & 4\end{pmatrix}$
\bmatrix
$\begin{bmatrix} 1 1 & & 2 \\
1 2
3 4
3 & & 4\end{bmatrix}$
\vmatrix
$\begin{vmatrix} 1 1 & & 2 \\
1 2
3 4
3 & & 4\end{vmatrix}$
\Vmatrix
$\begin{Vmatrix} 1 1 & & 2 \\
°
°
°
°
1 2
3 4
°
°
°
°
3 & & 4\end{bmatrix}$
Table18: MatrixEnvironments.
EXERCISE20
Explainhowtoformatthefollowingmatrix.
A=
0
@
1 a a b
: 1 c
: : : 1
1
A
4.6 Dots s inFormulas
Thecommands\ldotsand\cdotsproducetwokindsofellipsis(...).
Alowellipsis: x
1
;:::;x
n
.
Acenteredellipsis:x
1
+¢¢¢+x
n
A low ellipsis: $x_1, , \ldots, x_n$.\\
A centered d ellipsis: : $x_1 + \cdots + x_n$
Othercommandstoproducedotsareshowninthefollowingexample:
A=
0
B
B
B
@
a
11
a
12
::: a
1n
a
21
a
22
::: a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
::: a
mn
1
C
C
C
A
\[ A A = = \begin{pmatrix}
a_{11} & a_{12} & \ldots s & & a_{1n} \\
a_{21} & a_{22} & \ldots s & & a_{2n} \\
\vdots & \vdots & \ddots s & & \vdots \\
a_{m1} & a_{m2} & \ldots s & & a_{mn} \\
\end{pmatrix} \]
EXERCISE21
Explainhowtoformatthefollowingstatement.
if v=(v
1
;:::;v
n
) then n v
t
=
0
B
@
v
1
.
.
.
v
n
1
C
A
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4.7 Delimiters
InTable19arelistedthebasicbracketsanddelimiters.
input
meaning
display
(
leftparenthesis
(
)
rightparenthesis
)
[or\lbrack
leftbracket
[
]or\rbrack
rightbracket
]
\{or\lbrace
leftcurlybracket
f
\}or\rbrace
rightcurlybracket
g
\lfloor
left°oorbracket
b
\rfloor
right°oorbracket
c
\lceil
leftceilbracket
d
\rceil
rightceilbracket
e
\langle
leftanglebracket
h
\rangle
rightanglebracket
i
/
slash
=
\backslash
reverseslash
n
|or\vert
verticalbar
j
\|or\Vert
doubleverticalbar
k
\uparrow
upwardarrow
"
\Uparrow
doubleupwardarrow
*
\downarrow
downwardarrow
#
\Downarrow
doubledownwardarrow
+
\updownarrow
up-and-downarrow
l
\Updownarrow
doubleup-and-downarrow
m
Table19: Delimiters.
Ifyouputthe\leftcommandinfrontofanopeningdelimiterandthe\rightcommand
atclosure,thenLAT
E
Xautomaticallytriestoresizethedelimiterstoanappropriatesize.
ˆ
n
X
k=1
k
3
!
=
µ
n(n+1)
2
2
\[
\left(\sum_{k=1}^n k^3\right) =
\left(\frac{n(n+1)}{2}\right)^2
\]
Inthisexample,youmaywanttohavetheoutmostbracketsofthesamesize. Thenyoumust
useoneofthecommands\bigl,\Bigl,\biggl,\Biggl,andtheanalogouscommandwith
\bigr,andsoon.InTable20weshowthevarioussizes.
ˆ
Xn
k=1
k
3
!
=
ˆ
n(n+1)
2
!
2
\[
\Biggl(\sum_{k=1}^n k^3\Biggr) =
\Biggr(\frac{n(n+1)}{2}\Biggr)^2
\]
35
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normalsize
()[]fgbcdehi=njk"*#+lm
\bigsize
¡¢£⁄'“¥ƒ§¤›fi–†fl
°
°
x
?
~
w
?
y
w
˜
x
y
~
˜
\Bigsize
‡·hinojklmDE./fl
°
°
°
x
?
?
~
w
w
?
?
y
w
w
˜
x
?
y
~
w
˜
\biggsize
µ¶•‚‰¾„”»…¿À`´fl
°
°
°
°
x
?
?
?
~
w
w
w
?
?
?
y
w
w
w
˜
x
?
?
y
~
w
w
˜
\Biggsize
ˆ!"#()$%&’*+,-fl
°
°
°
°
°
x
?
?
?
?
~
w
w
w
w
?
?
?
?
y
w
w
w
w
˜
x
?
?
?
y
~
w
w
w
˜
Table20: ResizingDelimiters.
The\leftand\rightcommandsmustcomeinmatchingpairs,butthematchingdelim-
itersneednotbethesame. Aninvisibledelimitercanforinstancebecreatedbyenteringa
dot(‘.’) afterthe\leftand\rightcommand.Thefollowingexampleillustratesthis:
jxj=
¡x ifx<0;
x
otherwise
\[ |x| = \left\{ \begin{array}{ll}
-x & & \textnormal{if f $x<0$}, \\
x & \textnormal{otherwise}
\end{array} \right. \]
However,iteasiertousethecasesenvironmentinthisexample.
jxj=
(
¡x ifx<0;
x
otherwise
\[ |x| = \begin{cases}
-x & & \textnormal{if f $x<0$}, \\
x & \textnormal{otherwise}
\end{cases} \]
EXERCISE22
Explainhowtoformatthefollowingformula.
lim
x#0
1
x
=1
6=lim
x"0
1
x
EXERCISE23
Explainhowtoformatthefollowingformula.
f(x)=
(
1
ifx6=0;
sinx
x
otherwise
EXERCISE24
Explainhowtoformatthefollowingruleofpartialintegration.
Z
b
a
f
0
(x)g(x)dx=f(x)g(x)
b
a
¡
Z
b
a
f(x)g
0
(x)dx
36
4.8 Decorations
Youcaneasilyputahorizontallineorhorizontalbraceaboveorbelowaformula.
1+
1
2
+
1
3
+
1
4
|
{z
}
+
1
5
+
1
6
+
1
7
+
1
8
|
{z
}
+¢¢¢
\[ 1 1 + + \frac{1}{2} } +
\underbrace{\frac{1}{3} + \frac{1}{4}} +
\underbrace{\frac{1}{5} + \frac{1}{6} +
\frac{1}{7} + + \frac{1}{8}} } + \cdots s \]
The\stackrelcommandstacksonesymbolaboveanother.
~v
def
=(v
1
;:::;v
n
)
\[
\vec{v} \stackrel{\mathrm{def}}{=}
(v_1,\ldots, v_n)
\]
4.9 Theorem,Conjectures,etc.
Statementoftheorems,lemmas,corollaries,conjectures,andsoon,israthereasyinLAT
E
Xas
thefollowingexamplesillustrate.
Theorem1 There e exist inflnitely
manyprimenumbers.
Conjecture1 Thereexistinflnite-
ly many prime numbers p of f the
formp=2
n
¡1.
Conjecture2(Artin,1927)
Let a>1 1 beanintegerthat t isnot
a square. . Then, , a is a primitive
root modulo o inflnitely y many prime
numbersp.
\newtheorem{theorem}{Theorem}
\begin{theorem}
There exist infinitely many prime numbers.
\end{theorem}
\newtheorem{conj}{Conjecture}
\begin{conj}
There exist infinitely many prime numbers
$p$ of the e form m $p=2^n-1$.
\end{conj}
\begin{conj}[Artin, 1927]
Let $a>1$ be an integer that is not t a
square. Then, $a$ $ is a primitive e root
modulo infinitely many prime numbers $p$.
\end{conj}
5 OddandEnds
† LAT
E
Xusesthesinglequotes‘and’asquotationmarks. Neverusethedoublequote
" fromthekeyboardforthispurpose. . Togetadoublequoteasquotationmark,just
entertwosinglequotes.
† NotethevarioususesofdashesinL
A
T
E
X:
37
input
meaning
example
-
hyphen
X-rated
--
en-dash
pages1{10
---
em-dash
thisis|nomenestomen|for...
$-$
minussign
¡4
Table21: DashesandHyphens.
† The\noindentcommandatthebeginningofaparagraphsuppressesindentation.
† You can split large L
A
T
E
Xflles into smaller ones and d usethe \include e commandto
includetheflleforformatting.Themainstructureofthedocumentmaylooklike:
\begin{document}
\include{ch1} % % include chapter ch1.tex
\include{ch2} % % include chapter ch2.tex
\include{app} % % include appendix app.tex
\end{document}
Formattingofanincludedfllestartsalwaysatanewpage.Toavoidthis,usetheinput
command.
6 WheretoGetL
A
T
E
X?
Thereareseveraldistributions ofLAT
E
Xinthepublicdomain. OntheUNIX X computerof
FNWI,theteTeXdistributionhasbeeninstalled. ItcanbedownloadedfromtheCompre-
hensiveTexArchiveNetwork(CTAN),intheNetherlandsfromURL
ftp://ftp.ntg.nl/pub/tex-archive/
teTeXcomesalongwiththeRedHatdistributionofLinux.ThewebsiteofteTeXis
www.tug.org/teTeX
TheDutchT
E
X|UsersGroup(website:www.ntg.nl)istheproducerofacd-romwiththe
4TeXdistributionforPC-users. Fordetailswerefertothewebsite
4tex.ntg.nl
AhighlyregardedsetupforWindows(allcurrentvariants)isMikTeX.Itcanbeobtained
fromitswebsite
www.miktex.org
Thisisarathercompleteconflguration,whichincludespreviewingandPDFconversion. You
willonlyneedaconvenientedit.SomewidelyusededitorsareWinShellforWindows(down-
loadableatURLwww.winshell.de)andWinEdt(downloadablefromwww.winedt.com).
AcompletelistofavailablesystemscanbefoundonthewebsiteoftheworldwideTeX
UsersGroupis
www.tug.org
ThisisanywaythemainsourceofinformationaboutL
A
T
E
X.
38
References
[GMS94] M.Goossens,F.Mittelbach,A.Samarin.TheLAT
E
XCompanion{1sted.,Addison-
Wesley(1994),ISBN0-201-54199-8.
[Hec03] A. . Heck. Learning MetaPost t by Doing g (2003), Electronically y available (date:
3/3/2005)inPDF-formatatURL
www.science.uva.nl/~heck/Courses/mptut.pdf.
[MG04] F. . Mittelbach, , M. . Goossens. The LAT
E
X Companion n { { 2nd ed. . Addison-Wesley
(2004),ISBN0-201-36299-6.
[Lam94] L.Lamport: : LAT
E
X,ADocumentPreparationSystem,User’sGuideandReference
Manual{2nded.,Addison-Wesley(1994),ISBN0-201-52983-1.
[Oos97] P.van n Oostrum. Handleiding L
A
T
E
X (inDutch, , 1997), A version adaptedto the
localsituationisavailable(date: 3/3/2005)inPDF-formatonthewebsiteatURL
www.science.uva.nl/onderwijs/lesmateriaal/latex/latex.pdf.
[Rec97] K.Reckdahl.UsingImportedGraphicsinL
A
T
E
X2
"
(1997).Electronicallyavailable
(date: 3/3/2005)inPDF-formatatURL
ftp://ftp.dante.de/tex-archive/info/epslatex.pdf.
39
Appendices
A
AnswerstotheExercises
EXERCISE6
\emph{Mathematica} uses s the percent sign n (\%) ) to o refer r to the previous s result and
curly brackets s (\{\}) ) for r grouping.\\ See the e two instructions s below:
\begin{verbatim}
Sin[x]/x
Plot[%, {x,-3,3}];
\end{verbatim}
EXERCISE7
List of mathematical l functions:
\begin{itemize}
\item Trigonometric c functions
\begin{itemize}
\item sine
\item cosine
\item tangent
\end{itemize}
\item Special functions
\begin{itemize}
\item Beta a function
\item Gamma a function
\item Riemann zeta a function
\end{itemize}
\end{itemize}
EXERCISE8
To install Mathcad:
\begin{enumerate}
\item Start t Windows.
\item Insert the disk k marked d \texttt{Disk k 1} in the floppy disk k drive.
\item From m the \textsf{File} } menu u in the Windows Program m Manager,
choose \textsf{Run} } (\textsc{alt+f},\textsc{r}).
\item Type e \textbf{\emph{drive}:$\backslash$setup.exe}, where \textbf{\emph{drive}}
is the e letter r of the disk k drive e containing the e disk.
\item Press s \textsc{enter}.
\item Follow the instructions s on the screen.
\end{enumerate}
EXERCISE9
H\"uhner-h\"andler, d\’eb\^acle, , situa\c{c}\~oes, na\"{\i}ef.
40
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