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{\title Original file was misc3.tex}
{\doccomm Created using latex2rtf 2.3.18 r1266 (released May 29, 2020) on Sat May 30 17:35:19 2020
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{\pard\plain\s20\qc\sb240\sa240\b\f0\fs36\sl240\slmult1 \fi0 Getting Started with {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s21\qc\sa120\f0\fs24\sl240\slmult1 \fi360 David R. Wilkins\par
\pard\plain\s21\qc\sa120\f0\fs24\sl240\slmult1 \fi360 2nd Edition\par
\pard\plain\s21\qc\sa120\f0\fs24\sl240\slmult1 \sb60 \fi0  Copyright \'a9 David R. Wilkins 1995\par
\pard\plain\s80\ql\sb240\sa120\keepn\f0\b\fs24\sl240\slmult1 \fi0  Contents\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 \par
{\field{\*\fldinst TOC \\o "1-3" }{\fldrslt }}
\page
\pard\plain\s3\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb300 \fi0 1  Introduction to {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 1.1  What is {\plain L{\up6\fs19 A}T{\dn7 E}X}?\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 {\plain L{\up6\fs19 A}T{\dn7 E}X} is a computer program for typesetting documents. It takes a computer file, prepared according to the rules of {\plain L{\up6\fs19 A}T{\dn7 E}X}{\charscalex50\~}and converts it to a form that may be printed on a high-quality printer, such as a laser writer, to produce a printed document of a quality comparable with good quality books and journals. Simple documents, which do not contain mathematical formulae or tables may be produced very easily: effectively all one has to do is to type the text straight in (though observing certain rules relating to quotation marks and punctuation dashes). Typesetting mathematics is somewhat more complicated, but even here {\plain L{\up6\fs19 A}T{\dn7 E}X} is comparatively straightforward to use when one considers the complexity of some of the formulae that it has to produce and the large number of mathematical symbols which it has to produce.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} is one of a number of \lquote dialects\rquote  of {\plain T{\dn7 E}X}, all based on the version of {\plain T{\dn7 E}X} created by D. E. Knuth which is known as Plain {\plain T{\dn7 E}X}. {\plain L{\up6\fs19 A}T{\dn7 E}X} (created by L. B. Lamport) is one of these \lquote dialects\rquote . It is particularly suited to the production of long articles and books, since it has facilities for the automatic numbering of chapters, sections, theorems, equations etc., and also has facilities for cross-referencing. It is probably one of the most suitable version of {\plain L{\up6\fs19 A}T{\dn7 E}X} for beginners to use.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 1.2  A Typical {\plain L{\up6\fs19 A}T{\dn7 E}X} Input File\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 In order to produce a document using {\plain L{\up6\fs19 A}T{\dn7 E}X}, we must first create a suitable {\i input file} on the computer. We apply the {\plain L{\up6\fs19 A}T{\dn7 E}X} program to the input file and then use the printer to print out the so-called \lquote DVI\rquote  file produced by the {\plain L{\up6\fs19 A}T{\dn7 E}X} program (after first using another program to translate the \lquote DVI\rquote  file into a form that the printer can understand). Here is an example of a typical {\plain L{\up6\fs19 A}T{\dn7 E}X} input file: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper,12pt]\{article\}
\par \\begin\{document\}
\par 
\par The foundations of the rigorous study of \\textit\{analysis\}
\par were laid in the nineteenth century, notably by the
\par mathematicians Cauchy and Weierstrass. Central to the
\par study of this subject are the formal definitions of
\par \\textit\{limits\} and \\textit\{continuity\}.
\par 
\par Let $D$ be a subset of $\\bf R$ and let
\par $f \\colon D \\to \\textbf\{R\}$ be a real-valued function on
\par $D$. The function $f$ is said to be \\textit\{continuous\} on
\par $D$ if, for all $\\epsilon > 0$ and for all $x \\in D$,
\par there exists some $\\delta > 0$ (which may depend on $x$)
\par such that if $y \\in D$ satisfies
\par \\[ |y - x| < \\delta \\]
\par then
\par \\[ |f(y) - f(x)| < \\epsilon. \\]
\par 
\par One may readily verify that if $f$ and $g$ are continuous
\par functions on $D$ then the functions $f+g$, $f-g$ and
\par $f.g$ are continuous. If in addition $g$ is everywhere
\par non-zero then $f/g$ is continuous.
\par 
\par \\end\{document\}
\par 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 When we apply {\plain L{\up6\fs19 A}T{\dn7 E}X} to these paragraphs we produce the text {\par
\fs24 \pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The foundations of the rigorous study of {\i analysis} were laid in the nineteenth century, notably by the mathematicians Cauchy and Weierstrass. Central to the study of this subject are the formal definitions of {\i limits} and {\i continuity}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 Let {{\i D}} be a subset of {\i0\scaps0\b R} and let {{\i f}: {\i D}{\u8594*}{\b R}} be a real-valued function on {{\i D}}. The function {{\i f}} is said to be {\i continuous} on {{\i D}} if, for all {{\u949*}>0} and for all {{\i x}{\f5\u8712*}{\i D}}, there exists some {{\u948*}>0} (which may depend on {{\i x}}) such that if {{\i y}{\f5\u8712*}{\i D}} satisfies \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
|{\i y}\u8722?{\i x}|<{\u948*}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 then \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
|{\i f}({\i y})\u8722?{\i f}({\i x})|<{\u949*}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 One may readily verify that if {{\i f}} and {{\i g}} are continuous functions on {{\i D}} then the functions {{\i f}+{\i g}}, {{\i f}\u8722?{\i g}} and {{\i f}.{\i g}} are continuous. If in addition {{\i g}} is everywhere non-zero then {{\i f}/{\i g}} is continuous. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 This example illustrates various features of {\plain L{\up6\fs19 A}T{\dn7 E}X}. Note that the lines {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper,12pt]\{article\}
\par \\begin\{document\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 are placed at the beginning of the input file. These are followed by the main body of the text, followed by the concluding line {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\end\{document\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Note also that, although most characters occurring in this file have their usual meaning, yet there are special characters such as {\b0\i0\scaps0\f3 \\}, {\b0\i0\scaps0\f3 $}, {\b0\i0\scaps0\f3 \{} and {\b0\i0\scaps0\f3 \}} which have special meanings within {\plain L{\up6\fs19 A}T{\dn7 E}X}. Note in particular that there are sequences of characters which begin with a \lquote backslash\rquote  {\b0\i0\scaps0\f3 \\} which are used to produce mathematical symbols and Greek letters and to accomplish tasks such as changing fonts. These sequences of characters are known as {\i control sequences}.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 1.3  Characters and Control Sequences\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 We now describe in more detail some of the features of {\plain L{\up6\fs19 A}T{\dn7 E}X} illustrated in the above example.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Most characters on the keyboard, such as letters and numbers, have their usual meaning. However the characters {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\ \{ \} $ ^ _ % ~ # &
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 are used for special purposes within {\plain L{\up6\fs19 A}T{\dn7 E}X}. Thus typing one of these characters will not produce the corresponding character in the final document. Of course these characters are very rarely used in ordinary text, and there are methods of producing them when they are required in the final document.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In order to typeset a mathematical document it is necessary to produce a considerable number of special mathematical symbols. One also needs to be able to change fonts. Also mathematical documents often contain arrays of numbers or symbols (matrices) and other complicated expressions. These are produced in {\plain L{\up6\fs19 A}T{\dn7 E}X} using {\i control sequences}. Most control sequences consist of a backslash {\b0\i0\scaps0\f3 \\} followed by a string of (upper or lower case) letters. For example, {\b0\i0\scaps0\f3 \\alpha}, {\b0\i0\scaps0\f3 \\textit} and {\b0\i0\scaps0\f3 \\sum} are control sequences.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In the example above we used the control sequences {\b0\i0\scaps0\f3 \\textit} and {\b0\i0\scaps0\f3 \\textbf} to change the font to {\i italic} and {\b boldface} respectively. Also we used the control sequences {\b0\i0\scaps0\f3 \\to}, {\b0\i0\scaps0\f3 \\in}, {\b0\i0\scaps0\f3 \\delta} and {\b0\i0\scaps0\f3 \\epsilon} to produce the mathematical symbols {{\u8594*}} and {{\f5\u8712*}} and the Greek letters {{\u948*}} and {{\u949*}}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 There is another variety of control sequence which consists of a backslash followed by a {\i single} character that is not a letter. Examples of control sequences of this sort are {\b0\i0\scaps0\f3 \\\{}, {\b0\i0\scaps0\f3 \\"} and {\b0\i0\scaps0\f3 \\$}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The special characters {\b0\i0\scaps0\f3 \{} and {\b0\i0\scaps0\f3 \}} are used for {\i grouping} purposes. Everything enclosed within matching pair of such brackets is treated as a single unit. We have applied these brackets in the example above whenever we changed fonts. We shall see other instances where one needs to use {\b0\i0\scaps0\f3 \{} and {\b0\i0\scaps0\f3 \}} in {\plain L{\up6\fs19 A}T{\dn7 E}X} to group words and symbols together (e.g., when we need to produce superscripts and subscripts which contain more than one symbol).\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The special character {\b0\i0\scaps0\f3 $} is used when one is changing from ordinary text to a mathematical expression and when one is changing back to ordinary text. Thus we used {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par for all $\\epsilon > 0$ and for all $x \\in D$,
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 to produce the phrase {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 for all {{\u949*}>0} and for all {{\i x}{\f5\u8712*}{\i D}}, \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 in the example given above. Note also that we used {\b0\i0\scaps0\f3 \\[} and {\b0\i0\scaps0\f3 \\]} in the example above to mark the beginning and end respectively of a mathematical formula that is displayed on a separate line.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The remaining special characters {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par ^ _ % ~ # &
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 have special purposes within {\plain L{\up6\fs19 A}T{\dn7 E}X} that we shall discuss later.\par
\pard\plain\s3\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 2  Producing Simple Documents using {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 2.1  Producing a {\plain L{\up6\fs19 A}T{\dn7 E}X} Input File\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 We describe the structure of a typical {\plain L{\up6\fs19 A}T{\dn7 E}X} input file.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The first line of the input file should consist of a {\b0\i0\scaps0\f3 \\documentclass} command. The recommended such {\b0\i0\scaps0\f3 \\documentclass} command for mathematical articles and similar documents has the form {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper,12pt]\{article\}
\par \par
}}(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 You do not have to worry about what this command means when first learning to use {\plain L{\up6\fs19 A}T{\dn7 E}X}: its effect is to ensure that the final document is correctly positioned on A4 size paper and that the text is of a size that is easy to read.) There are variants of this {\b0\i0\scaps0\f3 \\documentclass} command which are appropriate for letters or for books.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The {\b0\i0\scaps0\f3 documentstyle} command may be followed by certain other optional commands, such as the {\b0\i0\scaps0\f3 \\pagestyle} command. It is not necessary to find out about these commands when first learning to use {\plain L{\up6\fs19 A}T{\dn7 E}X}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 After the {\b0\i0\scaps0\f3 \\documentclass} command and these other optional commands, we place the command {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{document\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 This command is then followed by the main body of the text, in the format prescribed by the rules of {\plain L{\up6\fs19 A}T{\dn7 E}X}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Finally, we end the input file with a line containing the command {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\end\{document\}
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 2.2  Producing Ordinary Text using {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 To produce a simple document using {\plain L{\up6\fs19 A}T{\dn7 E}X} one should create a {\plain L{\up6\fs19 A}T{\dn7 E}X} input file, beginning with a {\b0\i0\scaps0\f3 \\documentclass} command and the {\b0\i0\scaps0\f3 \\begin\{document\}} command, as described above. The input file should end with the {\b0\i0\scaps0\f3 \\end\{document\}} command, and the text of the document should be sandwiched between the {\b0\i0\scaps0\f3 \\begin\{document\}} and {\b0\i0\scaps0\f3 \\end\{document\}} commands in the manner described below.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 If one merely wishes to type in ordinary text, without complicated mathematical formulae or special effects such as font changes, then one merely has to type it in as it is, leaving a completely blank line between successive paragraphs. You do not have to worry about paragraph indentation: {\plain L{\up6\fs19 A}T{\dn7 E}X} will automatically indent all paragraphs with the exception of the first paragraph of a new section (unless you take special action to override the conventions adopted by {\plain L{\up6\fs19 A}T{\dn7 E}X}) \par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 For example, suppose that we wish to create a document containing the following paragraphs: {\par
\fs24 \pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 If one merely wishes to type in ordinary text, without complicated mathematical formulae or special effects such as font changes, then one merely has to type it in as it is, leaving a completely blank line between successive paragraphs.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 You do not have to worry about paragraph indentation: all paragraphs will be indented with the exception of the first paragraph of a new section. One must take care to distinguish between the \lquote left quote\rquote  and the \lquote right quote\rquote  on the computer terminal. Also, one should use two \lquote single quote\rquote  characters in succession if one requires \ldblquote double quotes\rdblquote . One should never use the (undirected) \lquote double quote\rquote  character on the computer terminal, since the computer is unable to tell whether it is a \lquote left quote\rquote  or a \lquote right quote\rquote . One also has to take care with dashes: a single dash is used for hyphenation, whereas three dashes in succession are required to produce a dash of the sort used for punctuation\emdash such as the one used in this sentence. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 To create this document using {\plain L{\up6\fs19 A}T{\dn7 E}X} we use the following input file: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper,12pt]\{article\}
\par \\begin\{document\}
\par 
\par If one merely wishes to type in ordinary text, without
\par complicated mathematical formulae or special effects such
\par as font changes, then one merely has to type it in as it
\par is, leaving a completely blank line between successive
\par paragraphs.
\par 
\par You do not have to worry about paragraph indentation:
\par all paragraphs will be indented with the exception of
\par the first paragraph of a new section.
\par 
\par One must take care to distinguish between the `left quote'
\par and the `right quote' on the computer terminal.  Also, one
\par should use two `single quote' characters in succession if
\par one requires ``double quotes''.  One should never use the
\par (undirected) `double quote' character on the computer
\par terminal, since the computer is unable to tell whether it
\par is a `left quote' or a `right quote'.  One also has to
\par take care with dashes: a single dash is used for
\par hyphenation, whereas three dashes in succession are required
\par to produce a dash of the sort used for punctuation---such as
\par the one used in this sentence.
\par 
\par \\end\{document\}
\par 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Having created the input file, one then has to run it through the {\plain L{\up6\fs19 A}T{\dn7 E}X} program and then print it out the resulting output file (known as a \lquote DVI\rquote  file).\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 2.3  Blank Spaces and Carriage Returns in the Input File\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 {\plain L{\up6\fs19 A}T{\dn7 E}X} treats the carriage return at the end of a line as though it were a blank space. Similarly {\plain L{\up6\fs19 A}T{\dn7 E}X} treats tab characters as blank spaces. Moreover, {\plain L{\up6\fs19 A}T{\dn7 E}X} regards a sequence of blank spaces as though it were a single space, and similarly it will ignore blank spaces at the beginning or end of a line in the input file. Thus, for example, if we type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par This is
\par     a
\par         silly
\par   example   of   a
\par file with many spaces.
\par 
\par 
\par                    This is the beginning
\par of a new paragraph.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 then we obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 This is a silly example of a file with many spaces.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 This is the beginning of a new paragraph. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 It follows immediately from this that one will obtain the same results whether one types one space or two spaces after a full stop: {\plain L{\up6\fs19 A}T{\dn7 E}X} does not distinguish between the two cases.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Any spaces which follow a control sequence will be ignored by {\plain L{\up6\fs19 A}T{\dn7 E}X}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 If you really need a blank space in the final document following whatever is produced by the control sequence, then you must precede this blank by a {\i backslash} {\b0\i0\scaps0\f3 \\}. Thus in order to obtain the sentence {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} is a very powerful computer typesetting program. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 we must type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\LaTeX\\ is a very powerful computer typesetting program.
\par \par
}}(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Here the control sequence {\b0\i0\scaps0\f3 TeX} is used to produce the {\plain L{\up6\fs19 A}T{\dn7 E}X} logo.)\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In general, preceding a blank space by a backslash forces {\plain L{\up6\fs19 A}T{\dn7 E}X} to include the blank space in the final document.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 As a general rule, you should never put a blank space after a left parenthesis or before a right parenthesis. If you were to put a blank space in these places, then you run the risk that {\plain L{\up6\fs19 A}T{\dn7 E}X} might start a new line immediately after the left parenthesis or before the right parenthesis, leaving the parenthesis marooned at the beginning or end of a line.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 2.4  Quotation Marks and Dashes\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Single quotation marks are produced in {\plain L{\up6\fs19 A}T{\dn7 E}X} using {\b0\i0\scaps0\f3 `} and {\b0\i0\scaps0\f3 '}. Double quotation marks are produced by typing {\b0\i0\scaps0\f3 ``} and {\b0\i0\scaps0\f3 ''}. (The \lquote undirected double quote character {\b0\i0\scaps0\f3 "} produces double right quotation marks: it should {\i never} be used where left quotation marks are required.)\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} allows you to produce dashes of various length, known as \lquote hyphens\rquote , \lquote en-dashes\rquote  and \lquote em-dashes\rquote . Hyphens are obtained in {\plain L{\up6\fs19 A}T{\dn7 E}X} by typing {\b0\i0\scaps0\f3 -}, en-dashes by typing {\b0\i0\scaps0\f3 --} and em-dashes by typing {\b0\i0\scaps0\f3 ---}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 One normally uses en-dashes when specifying a range of numbers. Thus for example, to specify a range of page numbers, one would type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par on pages 155--219.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Dashes used for punctuating are often typeset as em-dashes, especially in older books. These are obtained by typing {\b0\i0\scaps0\f3 ---}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The dialogue {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 \ldblquote You {\i were} a little grave,\rdblquote  said Alice.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 \ldblquote Well just then I was inventing a new way of getting over a gate\emdash would you like to hear it?\rdblquote \par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 \ldblquote Very much indeed,\rdblquote  Alice said politely.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 \ldblquote I\rquote ll tell you how I came to think of it,\rdblquote  said the Knight. \ldblquote You see, I said to myself \lquote The only difficulty is with the feet: the {\i head} is high enough already.\rquote  Now, first I put my head on the top of the gate\emdash then the head\rquote s high enough\emdash then I stand on my head\emdash then the feet are high enough, you see\emdash then I\rquote m over, you see.\rdblquote  \par
}(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 taken from {\i Alice through the Looking Glass}, by Lewis Carroll) illustrates the use of quotation marks and dashes. It is obtained in {\plain L{\up6\fs19 A}T{\dn7 E}X} from the following input: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par ``You \\emph\{were\} a little grave,'' said Alice.
\par 
\par ``Well just then I was inventing a new way of
\par getting over a gate---would you like to hear it?''
\par 
\par ``Very much indeed,'' Alice said politely.
\par 
\par ``I'll tell you how I came to think of it,'' said
\par the Knight.  ``You see, I said to myself `The only
\par difficulty is with the feet: the \\emph\{head\} is
\par high enough already.'  Now, first I put my head on
\par the top of the gate---then the head's high
\par enough---then I stand on my head---then the feet
\par are high enough, you see---then I'm over, you see.''
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Sometimes you need single quotes immediately following double quotes, or vica versa, as in {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 \ldblquote I regard computer typesetting as being reasonably \lquote straightforward\rquote {\charscalex50\~}\rdblquote  he said. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 The way to typeset this correctly in {\plain L{\up6\fs19 A}T{\dn7 E}X} is to use the control sequence {\b0\i0\scaps0\f3 \\,} between the quotation marks, so as to obtain the necessary amount of separation. The above example is thus produced with the input {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par ``I regard computer typesetting as being reasonably
\par `straightforward'\\,'' he said.
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 2.5  Section Headings in {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Section headings of various sizes are produced (in the {\b article} document style) using the commands {\b0\i0\scaps0\f3 \\section},{\b0\i0\scaps0\f3 \\subsection} and {\b0\i0\scaps0\f3 \\subsubsection} commands. {\plain L{\up6\fs19 A}T{\dn7 E}X} will number the sections and subsections automatically. The title of the section should be surrounded by curly brackets and placed immediately after the relevant command. Thus if we type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\section\{Section Headings\}
\par 
\par We explain in this section how to obtain headings
\par for the various sections and subsections of our
\par document.
\par 
\par \\subsection\{Headings in the `article' Document Style\}
\par 
\par In the `article' style, the document may be divided up
\par into sections, subsections and subsubsections, and each
\par can be given a title, printed in a boldface font,
\par simply by issuing the appropriate command.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 then the title of the section and that of the subsection will be printed in a large boldface font, and will be numbered accordingly.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Other document styles (such as the {\b book} and {\b letter} styles) have other \lquote sectioning\rquote  commands available (for example, the {\b book} style has a {\b0\i0\scaps0\f3 \\chapter} command for beginning a new chapter).\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Sometimes one wishes to suppress the automatic numbering provided by {\plain L{\up6\fs19 A}T{\dn7 E}X}. This can be done by placing an asterisk before the title of the section or subsection. Thus, for example, the section numbers in the above example could be suppressed by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\section*\{Section Headings\}
\par 
\par We explain in this section how to obtain headings
\par for the various sections and subsections of our
\par document.
\par 
\par \\subsection*\{Headings in the `article' Document Style\}
\par 
\par In the `article' style, the document may be divided up
\par into sections, subsections and subsubsections, and each
\par can be given a title, printed in a boldface font,
\par simply by issuing the appropriate command.
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 2.6  Changing Fonts in Text Mode\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 {\plain L{\up6\fs19 A}T{\dn7 E}X} has numerous commands for changing the typestyle. The most useful of these is {\b0\i0\scaps0\f3 \\emph\{}{\i text}{\b0\i0\scaps0\f3 \}} which {\i emphasizes} some piece of text, setting it usually in an {\i italic font} (unless the surrounding text is already italicized). Thus for example, the text {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The basic results and techniques of {\i Calculus} were discovered and developed by {\i Newton} and {\i Leibniz}, though many of the basic ideas can be traced to earlier work of {\i Cavalieri}, {\i Fermat}, {\i Barrow} and others. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The basic results and techniques of \\emph\{Calculus\}
\par were discovered and developed by \\emph\{Newton\}
\par and \\emph\{Leibniz\}, though many of the basic ideas
\par can be traced to earlier work of \\emph\{Cavalieri\},
\par \\emph\{Fermat\}, \\emph\{Barrow\} and others.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Another useful font-changing command is {\b0\i0\scaps0\f3 \\textbf\{}{\i text}{\b0\i0\scaps0\f3 \}}, which typesets the specified portion of text in {\b boldface}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 A {\i font family} or {\i typeface} in {\plain L{\up6\fs19 A}T{\dn7 E}X} consists of a collection of related fonts characterized by {\i size}, {\i shape} and {\i series}. The font families available in {\plain L{\up6\fs19 A}T{\dn7 E}X} include {\f0 roman}, {\f2 sans serif} and {\f3 typewriter}: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\f0 Roman is normally the default family and includes {\i0\b0\scaps0 upright}, {\i italic}, {\i slanted}, {\scaps small caps} and {\b boldface} fonts of various sizes.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\f2 There is a sans serif family with {\i0\b0\scaps0 upright}, {\i slanted} and {\b boldface} fonts of various sizes.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\f3 There is a typewriter family with {\i0\b0\scaps0 upright}, {\i italic}, {\i slanted}\line  and {\scaps small caps} fonts of various sizes.} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 The sizes of fonts used in {\plain L{\up6\fs19 A}T{\dn7 E}X} are can be determined and changed by means of the control sequences {\b0\i0\scaps0\f3 \\tiny}, {\b0\i0\scaps0\f3 \\scriptsize}, {\b0\i0\scaps0\f3 \\footnotesize}, {\b0\i0\scaps0\f3 \\small}, {\b0\i0\scaps0\f3 \\normalsize}, {\b0\i0\scaps0\f3 \\large}, {\b0\i0\scaps0\f3 \\Large}, {\b0\i0\scaps0\f3 \\LARGE}, {\b0\i0\scaps0\f3 \\huge} and {\b0\i0\scaps0\f3 \\HUGE}: {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\fs12 This text is {\f3 tiny}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs17 This text is {\f3 scriptsize}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs19 This text is {\f3 footnotesize}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs22 This text is {\f3 small}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs24 This text is {\f3 normalsize}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs29 This text is {\f3 large}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs34 This text is {\f3 Large}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs41 This text is {\f3 LARGE}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs48 This text is {\f3 huge}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi360 {\fs60 This text is {\f3 Huge}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb180 \fi360 The {\i shape} of a font can be {\i0\b0\scaps0 upright}, {\i italic}, {\i slanted} or {\scaps small caps}: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\i0\b0\scaps0 The LaTeX command} {\b0\i0\scaps0\f3 \\textup\{}{\i text}{\b0\i0\scaps0\f3 \}} {\i0\b0\scaps0 typesets the specified text with an upright shape: this is normally the default shape.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\i The LaTeX command} {\b0\i0\scaps0\f3 \\textit\{}{\i text}{\b0\i0\scaps0\f3 \}} {\i typesets the specified text with an italic shape.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\i The LaTeX command} {\b0\i0\scaps0\f3 \\textsl\{}{\i text}{\b0\i0\scaps0\f3 \}} {\i typesets the specified text with a slanted shape: slanted text is similar to italic.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\scaps The LaTeX command} {\b0\i0\scaps0\f3 \\textsc\{}{\i text}{\b0\i0\scaps0\f3 \}} {\scaps typesets the specified text with a small caps shape in which all letters are capitals (with uppercase letters taller than lowercase letters).} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 The {\i series} of a font can be {\b0 medium} (the default) or {\b boldface}: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\i0\b0\scaps0 The LaTeX command} {\b0\i0\scaps0\f3 \\textmd\{}{\i text}{\b0\i0\scaps0\f3 \}} {\b0 typesets the specified text with a medium series font.} \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\b The LaTeX command} {\b0\i0\scaps0\f3 \\textbf\{}{\i text}{\b0\i0\scaps0\f3 \}} {\b typesets the specified text with a boldface series font.} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 If the necessary fonts are available, one can combine changes to the size, shape and series of a font, for example producing {\b {\i boldface slanted text}} by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\textbf\{\\textsl\{boldface slanted text\}\}.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 There are in {\plain L{\up6\fs19 A}T{\dn7 E}X} font declarations corresponding to the the font-changing commands described above. When included in the {\plain L{\up6\fs19 A}T{\dn7 E}X} input such declarations determine the type-style of the subsequent text (till the next font declaration or the end of the current \lquote group\rquote  delimited by curly brackets or by appropriate {\b0\i0\scaps0\f3 \\begin} and {\b0\i0\scaps0\f3 \\end} commands). Here is a list of font-changing commands and declarations in text mode: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 \par
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\i Command}}\cell}
{\pard\intbl\ql {{\i Declaration}}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textrm}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\rmfamily}}\cell}
{\pard\intbl\ql {{\f0 Roman family}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textsf}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\sffamily}}\cell}
{\pard\intbl\ql {{\f2 Sans serif family}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\texttt}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\ttfamily}}\cell}
{\pard\intbl\ql {{\f3 Typewriter family}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textup}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\upshape}}\cell}
{\pard\intbl\ql {{\i0\scaps0 Upright shape}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textit}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\itshape}}\cell}
{\pard\intbl\ql {{\scaps0\i Italic shape}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textsl}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\slshape}}\cell}
{\pard\intbl\ql {{\scaps0\i Slanted shape}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textsc}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\scshape}}\cell}
{\pard\intbl\ql {{\scaps Small caps shape}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textmd}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\mdseries}}\cell}
{\pard\intbl\ql {{\b0 Medium series}}\cell}
\row}
{\trowd\cellx1614\cellx3661\cellx6898
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\textbf}}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\bfseries}}\cell}
{\pard\intbl\ql {{\b Boldface series}}\cell}
\row}
} \par
}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 2.7  Accents used in Text\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 There are a variety of control sequences for producing accents. For example, the control sequence {\b0\i0\scaps0\f3 \\'\{o\}} produces an acute accent on the letter {\b0\i0\scaps0\f3 o}. Thus typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Se\\'\{a\}n \\'\{O\} Cinn\\'\{e\}ide.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 produces {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Se\'e1n \'d3 Cinn\'e9ide. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Similarly we use the control sequence {\b0\i0\scaps0\f3 \\`} to produce the grave accent in \lquote alg\'e8bre\rquote  and we use {\b0\i0\scaps0\f3 \\"} to produce the umlaut in \lquote Universit\'e4t\rquote . The accents provided by {\plain L{\up6\fs19 A}T{\dn7 E}X} include the following: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 \par
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\'\{e\}}}\cell}
{\pard\intbl\ql {\'e9}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 math\\'\{e\}matique} yields \lquote math\'e9matique\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\`\{e\}}}\cell}
{\pard\intbl\ql {\'e8}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 alg\\`\{e\}bre} yields \lquote alg\'e8bre\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\^\{e\}}}\cell}
{\pard\intbl\ql {\'ea}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 h\\^\{o\}te} yields \lquote h\'f4te\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\"\{o\}}}\cell}
{\pard\intbl\ql {\'f6}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 H\\"\{o\}lder} yields \lquote H\'f6lder\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\~\{n\}}}\cell}
{\pard\intbl\ql {\'f1}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 ma\\~\{n\}ana} yields \lquote ma\'f1ana\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\=\{o\}}}\cell}
{\pard\intbl\ql {\u333o}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\.\{o\}}}\cell}
{\pard\intbl\ql {\u559o}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\u\{o\}}}\cell}
{\pard\intbl\ql {\u335o}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\v\{c\}}}\cell}
{\pard\intbl\ql {\u269c}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 \\v\{C\}ech} yields \lquote \u268Cech\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\H\{o\}}}\cell}
{\pard\intbl\ql {\u337o}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\t\{oo\}}}\cell}
{\pard\intbl\ql {{oo}}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\c\{c\}}}\cell}
{\pard\intbl\ql {\'e7}\cell}
{\pard\intbl\ql {e.g., {\b0\i0\scaps0\f3 gar\\c\{c\}on} yields \lquote gar\'e7on\rquote }\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\d\{o\}}}\cell}
{\pard\intbl\ql {\u7885o}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1259\cellx1894\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\b\{o\}}}\cell}
{\pard\intbl\ql {\u817_o}\cell}
{\pard\intbl\ql {}\cell}
\row}
} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 These accents are for use in ordinary text. They cannot be used within mathematical formulae, since different control sequences are used to produce accents within mathematics.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The control sequences {\b0\i0\scaps0\f3 \\i} and {\b0\i0\scaps0\f3 \\j} produce dotless \lquote i\rquote  and \lquote j\rquote . These are required when placing an accent on the letter. Thus \'ed is produced by typing {\b0\i0\scaps0\f3 \\'\{\\i\}}.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 2.8  Active Characters and Special Symbols in Text\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 The \lquote active characters\rquote  {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par # $ % & \\ ^ _ \{ \} ~
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 have special purposes within {\plain L{\up6\fs19 A}T{\dn7 E}X}. Thus they cannot be produced in the final document simply by typing them directly. On the rare occasions when one needs to use the special characters {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 # $ % & _ \{ \} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 in the final document, they can be produced by typing the control sequences {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\# \\$ \\% \\& \\_ \\\{ \\\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 respectively. However the characters {\b0\i0\scaps0\f3 \\}, {\b0\i0\scaps0\f3 ^} and {\b0\i0\scaps0\f3 ~} cannot be produced simply by preceding them with a backslash. They can however be produced using {\b0\i0\scaps0\f3 \\char92} (in the {\b0\i0\scaps0\f3 \\texttt} font only), {\b0\i0\scaps0\f3 \\char94} and {\b0\i0\scaps0\f3 \\char126} respectively. (The decimal numbers 92, 94 and 126 are the ASCII codes of these characters.)\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Other special symbols can be introduced into text using the appropriate control sequences: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 \par
{\trowd\clbrdrb\brdrs\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\clbrdrb\brdrs\cellx6899
{\pard\intbl\ql {{\i Symbol}}\cell}
{\pard\intbl\ql {{\i Control Sequence}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql { \'9c, \'8c}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\oe, \\OE}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'e6, \'c6}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\ae, \\AE}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'e5, \'c5}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\aa, \\AA}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'f8, \'d8}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\o, \\O}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\u323?, \u322?}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\l, \\L}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'df}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\ss}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'bf }\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 ?`}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'a1 }\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 !`}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'86}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\dag}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'87}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\ddag}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'a7}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\S}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'b6}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\P}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'a9}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\copyright}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\'a3}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\pounds}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\u305i}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\i}}\cell}
\row}
{\trowd\clbrdrr\brdrs\cellx2744\clbrdrl\brdrs\cellx6899
{\pard\intbl\ql {\u567j}\cell}
{\pard\intbl\ql {{\b0\i0\scaps0\f3 \\j}}\cell}
\row}
} \par
}\pard\plain\s3\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb300 \fi0 3  Producing Mathematical Formulae using {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 3.1  Mathematics Mode\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 In order to obtain a mathematical formula using {\plain L{\up6\fs19 A}T{\dn7 E}X}, one must enter {\i mathematics mode} before the formula and leave it afterwards. Mathematical formulae can occur either embedded in text or else displayed between lines of text. When a formula occurs within the text of a paragraph one should place a {\b0\i0\scaps0\f3 $} sign before and after the formula, in order to enter and leave mathematics mode. Thus to obtain a sentence like {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Let {{\i f}} be the function defined by {{\i f}({\i x})=3{\i x}+7}, and let {{\i a}} be a positive real number. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one should type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Let $f$ be the function defined by $f(x) = 3x + 7$, and
\par let $a$ be a positive real number.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 In particular, note that even mathematical expressions consisting of a single character, like {{\i f}} and {{\i a}} in the example above, are placed within {\b0\i0\scaps0\f3 $} signs. This is to ensure that they are set in italic type, as is customary in mathematical typesetting.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} also allows you to use {\b0\i0\scaps0\f3 \\(} and {\b0\i0\scaps0\f3 \\)} to mark the beginning and the end respectively of a mathematical formula embedded in text. Thus {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Let {{\i f}} be the function defined by {{\i f}({\i x})=3{\i x}+7}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 may be produced by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Let \\( f \\) be the function defined by \\( f(x) = 3x + 7 \\).
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 However this use of {\b0\i0\scaps0\f3 \\(} ... {\b0\i0\scaps0\f3 \\)} is only permitted in {\plain L{\up6\fs19 A}T{\dn7 E}X}: other dialects of TeX such as Plain {\plain T{\dn7 E}X} and AmSTeX use {\b0\i0\scaps0\f3 $} ... {\b0\i0\scaps0\f3 $}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In order to obtain a mathematical formula or equation which is displayed on a line by itself, one places {\b0\i0\scaps0\f3 \\[} before and {\b0\i0\scaps0\f3 \\]} after the formula. Thus to obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 If {{\i f}({\i x})=3{\i x}+7} and {{\i g}({\i x})={\i x}+4} then \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\i f}({\i x})+{\i g}({\i x})=4{\i x}+11\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 and \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\i f}({\i x}){\i g}({\i x})=3{\i x}\\s\\up6({\fs20 2})+19{\i x}+28.}}{\fldrslt }}
\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one would type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par If $f(x) = 3x + 7$ and $g(x) = x + 4$ then
\par \\[ f(x) + g(x) = 4x + 11 \\] 
\par and
\par \\[ f(x)g(x) = 3x^2 + 19x +28. \\]
\par \par
}}(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Here the character {\b0\i0\scaps0\f3 ^} is used to obtain a superscript.)\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} provides facilities for the automatic numbering of displayed equations. If you want an numbered equation then you use {\b0\i0\scaps0\f3 \\begin\{equation\}} and {\b0\i0\scaps0\f3 \\end\{equation\}} instead of using {\b0\i0\scaps0\f3 \\[} and {\b0\i0\scaps0\f3 \\]} . Thus {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par If $f(x) = 3x + 7$ and $g(x) = x + 4$ then
\par \\begin\{equation\}
\par f(x) + g(x) = 4x + 11
\par \\end\{equation\}
\par and
\par \\begin\{equation\}
\par f(x)g(x) = 3x^2 + 19x +28.
\par \\end\{equation\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 produces {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 If {{\i f}({\i x})=3{\i x}+7} and {{\i g}({\i x})={\i x}+4} then {\par
\pard\plain\s12\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\tqr\tx6900\sl240\slmult1 \li512\ri512\fi0 \tab
{\i f}({\i x})+{\i g}({\i x})=4{\i x}+11\tab{\b0 (1)}\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 and {\par
\pard\plain\s12\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\tqr\tx6900\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\i f}({\i x}){\i g}({\i x})=3{\i x}\\s\\up6({\fs20 2})+19{\i x}+28.}}{\fldrslt }}
\tab{\b0 (2)}\par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 3.2  Characters in Mathematics Mode\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 All the characters on the keyboard have their standard meaning in mathematics mode, with the exception of the characters {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par       # $ % & ~ _ ^ \\ \{ \} '
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Letters are set in italic type. In mathematics mode the character {\b0\i0\scaps0\f3 '} has a special meaning: typing {\b0\i0\scaps0\f3 $u' + v''$} produces {{\i u}'+{\i v}''} When in mathematics mode the spaces you type between letters and other symbols do not affect the spacing of the final result, since {\plain L{\up6\fs19 A}T{\dn7 E}X} determines the spacing of characters in formulae by its own internal rules. Thus {\b0\i0\scaps0\f3 $u v + w = x$} and {\b0\i0\scaps0\f3 $uv+w=x$} both produce {{\i u}{\i v}+{\i w}={\i x}} You can also type carriage returns where necessary in your input file (e.g., if you are typing in a complicated formula with many Greek characters and funny symbols) and this will have no effect on the final result if you are in mathematics mode.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 To obtain the characters {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par       #   $   %   &   _   \{   \}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 in mathematics mode, one should type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par       \\#   \\$   \\%   \\&   \\_   \\\{   \\\} .
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 To obtain {\b0\i0\scaps0\f3 \\} in mathematics mode, one may type {\b0\i0\scaps0\f3 \\backslash}.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.3  Superscripts and Subscripts\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Subscripts and superscripts are obtained using the special characters {\b0\i0\scaps0\f3 _} and {\b0\i0\scaps0\f3 ^} respectively. Thus the identity \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i d}{\i s}\\s\\up6({\fs20 2})={\i d}{\i x}\\s\\up({\fs20 2,1})+{\i d}{\i x}\\s\\up({\fs20 2,2})+{\i d}{\i x}\\s\\up({\fs20 2,3})\u8722?{\i c}\\s\\up6({\fs20 2}){\i d}{\i t}\\s\\up6({\fs20 2})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 It can also be obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ ds^2 = dx^2_1 + dx^2_2 + dx^2_3 - c^2 dt^2 \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 since, when a superscript is to appear above a subscript, it is immaterial whether the superscript or subscript is the first to be specified.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Where more than one character occurs in a superscript or subscript, the characters involved should be enclosed in curly brackets. For example, the polynomial {{\field{\*\fldinst{ EQ {\i x}\\s\\up6({\fs20 17})\u8722?1}}{\fldrslt }}
} is obtained by typing {\b0\i0\scaps0\f3 $x^\{17\} - 1$}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 One may not type expressions such as {\b0\i0\scaps0\f3 $s^n^j$} since this is ambiguous and could be interpreted either as {{\field{\*\fldinst{ EQ {\i s}\\s\\up6({\fs20 {\i n}{\i j}})}}{\fldrslt }}
} or as {{\field{\*\fldinst{ EQ {\i s}\\s\\up6({\fs20 {\i n}\\s\\up5({\fs16 {\i j}})})}}{\fldrslt }}
} The first of these alternatives is obtained by typing {\b0\i0\scaps0\f3 $s^\{n j\}$}, the second by typing {\b0\i0\scaps0\f3 $s^\{n^j\}$}. A similar remark applies to subscripts. Note that one can obtain in this way double superscripts (where a superscript is placed on a superscript) and double subscripts.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 It is sometimes necessary to obtain expressions in which the horizontal ordering of the subscripts is significant. One can use an \lquote empty group\rquote  {\b0\i0\scaps0\f3 \{\}} to separate superscripts and subscripts that must follow one another. For example, the identity \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i R}\\s\\do6({\fs20 {\i i}}){}\\s\\up6({\fs20 {\i j}}){}\\s\\do6({\fs20 {\i k}{\i l}})={\i g}\\s\\up6({\fs20 {\i j}{\i m}}){\i R}\\s\\do6({\fs20 {\i i}{\i m}{\i k}{\i l}})=\u8722?{\i g}\\s\\up6({\fs20 {\i j}{\i m}}){\i R}\\s\\do6({\fs20 {\i m}{\i i}{\i k}{\i l}})=\u8722?{\i R}\\s\\up6({\fs20 {\i j}}){}\\s\\do6({\fs20 {\i i}{\i k}{\i l}})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 can be obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ R_i\{\}^j\{\}_\{kl\} = g^\{jm\} R_\{imkl\}
\par    = - g^\{jm\} R_\{mikl\} = - R^j\{\}_\{ikl\} \\]
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 3.4  Greek Letters\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Greek letters are produced in mathematics mode by preceding the name of the letter by a backslash {\b0\i0\scaps0\f3 \\}. Thus to obtain the formula {{\field{\*\fldinst{ EQ {\i A}={\u960*}{\i r}\\s\\up6({\fs20 2})}}{\fldrslt }}
} one types {\b0\i0\scaps0\f3 A = \\pi r^2}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Here are the control sequences for the standard forms of the lowercase Greek letters:-\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u945*}} {\f3 \u92?alpha}\cell}
{\pard\intbl\ql {{\u953*}} {\f3 \u92?iota}\cell}
{\pard\intbl\ql {{\u1009*}} {\f3 \u92?rho}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u946*}} {\f3 \u92?beta}\cell}
{\pard\intbl\ql {{\u954*}} {\f3 \u92?kappa}\cell}
{\pard\intbl\ql {{\u963*}} {\f3 \u92?sigma}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u947*}} {\f3 \u92?gamma}\cell}
{\pard\intbl\ql {{\u955*}} {\f3 \u92?lambda}\cell}
{\pard\intbl\ql {{\u964*}} {\f3 \u92?tau}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u948*}} {\f3 \u92?delta}\cell}
{\pard\intbl\ql {{\u956*}} {\f3 \u92?mu}\cell}
{\pard\intbl\ql {{\u8593*}} {\f3 \u92?upsilon}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u949*}} {\f3 \u92?epsilon}\cell}
{\pard\intbl\ql {{\u957*}} {\f3 \u92?nu}\cell}
{\pard\intbl\ql {{\u966*}} {\f3 \u92?phi}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u950*}} {\f3 \u92?zeta}\cell}
{\pard\intbl\ql {{\u958*}} {\f3 \u92?xi}\cell}
{\pard\intbl\ql {{\u967*}} {\f3 \u92?chi}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u951*}} {\f3 \u92?eta}\cell}
{\pard\intbl\ql {      }{{\i o}}{      }{\f3 o}\cell}
{\pard\intbl\ql {{\u968*}} {\f3 \u92?psi}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u920*}} {\f3 \u92?theta}\cell}
{\pard\intbl\ql {{\u960*}} {\f3 \u92?pi}\cell}
{\pard\intbl\ql {{\u969*}} {\f3 \u92?omega}\cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 There is no special command for omicron: just use {\b0\i0\scaps0\f3 o}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Some Greek letters occur in variant forms. The variant forms are obtained by preceding the name of the Greek letter by \lquote var\rquote . The following table lists the usual form of these letters and the variant forms:- {\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u949*}} {\f3 \u92?epsilon}\cell}
{\pard\intbl\ql {{\u949*}} {\f3 \u92?varepsilon}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u920*}} {\f3 \u92?theta}\cell}
{\pard\intbl\ql {{\u977*}} {\f3 \u92?vartheta}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u960*}} {\f3 \u92?pi}\cell}
{\pard\intbl\ql {{\u982*}} {\f3 \u92?varpi}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u1009*}} {\f3 \u92?rho}\cell}
{\pard\intbl\ql {{\u1009*}} {\f3 \u92?varrho}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u963*}} {\f3 \u92?sigma}\cell}
{\pard\intbl\ql {{\u962*}} {\f3 \u92?varsigma}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u966*}} {\f3 \u92?phi}\cell}
{\pard\intbl\ql {{\u981*}} {\f3 \u92?varphi}\cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Upper case Greek letters are obtained by making the first character of the name upper case. Here are the control sequence for the uppercase letters:\emdash \par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u915*}} {\f3 \u92?Gamma}\cell}
{\pard\intbl\ql {{\u926*}} {\f3 \u92?Xi}\cell}
{\pard\intbl\ql {{\u934*}} {\f3 \u92?Phi}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u916*}} {\f3 \u92?Delta}\cell}
{\pard\intbl\ql {{\u928*}} {\f3 \u92?Pi}\cell}
{\pard\intbl\ql {{\u936*}} {\f3 \u92?Psi}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u964*}} {\f3 \u92?Theta}\cell}
{\pard\intbl\ql {{\u931*}} {\f3 \u92?Sigma}\cell}
{\pard\intbl\ql {{\u937*}} {\f3 \u92?Omega}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u923*}} {\f3 \u92?Lambda}\cell}
{\pard\intbl\ql {{\u933*}} {\f3 \u92?Upsilon}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 3.5  Mathematical Symbols\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 There are numerous mathematical symbols that can be used in mathematics mode. These are obtained by typing an appropriate control sequence.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Miscellaneous Symbols:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8501*}} {\f3 \u92?aleph}\cell}
{\pard\intbl\ql {'} {\f3 \u92?prime}\cell}
{\pard\intbl\ql {{\f5\u8704*}} {\f3 \u92?forall}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u295*}} {\f3 \u92?hbar}\cell}
{\pard\intbl\ql {{\u8709*}} {\f3 \u92?emptyset}\cell}
{\pard\intbl\ql {{\f5\u8707*}} {\f3 \u92?exists}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u305*}} {\f3 \u92?imath}\cell}
{\pard\intbl\ql {{\f5\u8711*}} {\f3 \u92?nabla}\cell}
{\pard\intbl\ql {{\u8800*}} {\f3 \u92?neg}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u567*}} {\f3 \u92?jmath}\cell}
{\pard\intbl\ql {{\u8730*}} {\f3 \u92?surd}\cell}
{\pard\intbl\ql {{\f5\u9837*}} {\f3 \u92?flat}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8467*}} {\f3 \u92?ell}\cell}
{\pard\intbl\ql {{\u8890*}} {\f3 \u92?top}\cell}
{\pard\intbl\ql {{\u9838*}} {\f3 \u92?natural}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8472*}} {\f3 \u92?wp}\cell}
{\pard\intbl\ql {{\f5\u8869*}} {\f3 \u92?bot}\cell}
{\pard\intbl\ql {{\u9839*}} {\f3 \u92?sharp}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8476*}} {\f3 \u92?Re}\cell}
{\pard\intbl\ql {|} {\f3 \u92?\emdash }\cell}
{\pard\intbl\ql {{\u9827*}} {\f3 \u92?clubsuit}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8465*}} {\f3 \u92?Im}\cell}
{\pard\intbl\ql {{\f5\u8736*}} {\f3 \u92?angle}\cell}
{\pard\intbl\ql {{\u9830*}} {\f3 \u92?diamondsuit}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8706*}} {\f3 \u92?partial}\cell}
{\pard\intbl\ql {{\u9653*}} {\f3 \u92?triangle}\cell}
{\pard\intbl\ql {{\u9829*}} {\f3 \u92?heartsuit}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8734*}} {\f3 \u92?infty}\cell}
{\pard\intbl\ql {\\} {\f3 \u92?backslash}\cell}
{\pard\intbl\ql {{\u9824*}} {\f3 \u92?spadesuit}\cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 \ldblquote Large\rdblquote  Operators:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ  \\i \\su(,, )}}{\fldrslt }}
} {\f3 \u92?sum}\cell}
{\pard\intbl\ql {{\u8899*}} {\f3 \u92?bigcap}\cell}
{\pard\intbl\ql {{\f5\u10752*}} {\f3 \u92?bigodot}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ  \\i \\pr(,, )}}{\fldrslt }}
} {\f3 \u92?prod}\cell}
{\pard\intbl\ql {{\u8898*}} {\f3 \u92?bigcup}\cell}
{\pard\intbl\ql {{\f5\u10754*}} {\f3 \u92?bigotimes}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8720*}} {\f3 \u92?coprod}\cell}
{\pard\intbl\ql {{\f5\u10758*}} {\f3 \u92?bigsqcup}\cell}
{\pard\intbl\ql {{\f5\u10753*}} {\f3 \u92?bigoplus}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ  \\i \\in(,, )}}{\fldrslt }}
} {\f3 \u92?int}\cell}
{\pard\intbl\ql {{\u8897*}} {\f3 \u92?bigvee}\cell}
{\pard\intbl\ql {{\f5\u10756*}} {\f3 \u92?biguplus}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8750*}} {\f3 \u92?oint}\cell}
{\pard\intbl\ql {{\u8896*}} {\f3 \u92?bigwedge}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Binary Operations:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u177*}} {\f3 \u92?pm}\cell}
{\pard\intbl\ql {{\u8745*}} {\f3 \u92?cap}\cell}
{\pard\intbl\ql {{\f5\u8744*}} {\f3 \u92?vee}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8723*}} {\f3 \u92?mp}\cell}
{\pard\intbl\ql {{\f5\u8746*}} {\f3 \u92?cup}\cell}
{\pard\intbl\ql {{\f5\u8743*}} {\f3 \u92?wedge}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8726*}} {\f3 \u92?setminus}\cell}
{\pard\intbl\ql {{\u8846*}} {\f3 \u92?uplus}\cell}
{\pard\intbl\ql {{\f5\u8853*}} {\f3 \u92?oplus}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8901*}} {\f3 \u92?cdot}\cell}
{\pard\intbl\ql {{\u8851*}} {\f3 \u92?sqcap}\cell}
{\pard\intbl\ql {{\f5\u8854*}} {\f3 \u92?ominus}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u215*}} {\f3 \u92?times}\cell}
{\pard\intbl\ql {{\u8852*}} {\f3 \u92?sqcup}\cell}
{\pard\intbl\ql {{\f5\u8855*}} {\f3 \u92?otimes}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8727*}} {\f3 \u92?ast}\cell}
{\pard\intbl\ql {{\u9667*}} {\f3 \u92?triangleleft}\cell}
{\pard\intbl\ql {{\f5\u8856*}} {\f3 \u92?oslash}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8902*}} {\f3 \u92?star}\cell}
{\pard\intbl\ql {{\u9657*}} {\f3 \u92?triangleright}\cell}
{\pard\intbl\ql {{\f5\u8857*}} {\f3 \u92?odot}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8900*}} {\f3 \u92?diamond}\cell}
{\pard\intbl\ql {{\u8768*}} {\f3 \u92?wr}\cell}
{\pard\intbl\ql {\'86} {\f3 \u92?dagger}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8728*}} {\f3 \u92?circ}\cell}
{\pard\intbl\ql {{\u9675*}} {\f3 \u92?bigcirc}\cell}
{\pard\intbl\ql {\'87} {\f3 \u92?ddagger}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8226*}} {\f3 \u92?bullet}\cell}
{\pard\intbl\ql {{\f5\u9651*}} {\f3 \u92?bigtriangleup}\cell}
{\pard\intbl\ql {{\f5\u8720*}} {\f3 \u92?amalg}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u247*}} {\f3 \u92?div}\cell}
{\pard\intbl\ql {{\f5\u9661*}} {\f3 \u92?bigtriangledown}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Relations:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8804*}} {\f3 \u92?leq}\cell}
{\pard\intbl\ql {{\u8805*}} {\f3 \u92?geq}\cell}
{\pard\intbl\ql {{\u8801*}} {\f3 \u92?equiv}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8826*}} {\f3 \u92?prec}\cell}
{\pard\intbl\ql {{\u8827*}} {\f3 \u92?succ}\cell}
{\pard\intbl\ql {{\f5\u8764*}} {\f3 \u92?sim}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u10927*}} {\f3 \u92?preceq}\cell}
{\pard\intbl\ql {{\f5\u10928*}} {\f3 \u92?succeq}\cell}
{\pard\intbl\ql {{\u8771*}} {\f3 \u92?simeq}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8810*}} {\f3 \u92?ll}\cell}
{\pard\intbl\ql {{\f5\u8811*}} {\f3 \u92?gg}\cell}
{\pard\intbl\ql {{\u8781*}} {\f3 \u92?asymp}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8834*}} {\f3 \u92?subset}\cell}
{\pard\intbl\ql {{\f5\u8835*}} {\f3 \u92?supset}\cell}
{\pard\intbl\ql {{\u8776*}} {\f3 \u92?approx}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8838*}} {\f3 \u92?subseteq}\cell}
{\pard\intbl\ql {{\f5\u8839*}} {\f3 \u92?supseteq}\cell}
{\pard\intbl\ql {{\u8773*}} {\f3 \u92?cong}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8849*}} {\f3 \u92?sqsubseteq}\cell}
{\pard\intbl\ql {{\u8850*}} {\f3 \u92?sqsupseteq}\cell}
{\pard\intbl\ql {{\u8904*}} {\f3 \u92?bowtie}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8712*}} {\f3 \u92?in}\cell}
{\pard\intbl\ql {{\f5\u8715*}} {\f3 \u92?ni}\cell}
{\pard\intbl\ql {{\f5\u8733*}} {\f3 \u92?propto}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\u8866*}} {\f3 \u92?vdash}\cell}
{\pard\intbl\ql {{\u8867*}} {\f3 \u92?dashv}\cell}
{\pard\intbl\ql {{\u8871*}} {\f3 \u92?models}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8995*}} {\f3 \u92?smile}\cell}
{\pard\intbl\ql {|} {\f3 \u92?mid}\cell}
{\pard\intbl\ql {{\u8784*}} {\f3 \u92?doteq}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {{\f5\u8994*}} {\f3 \u92?frown}\cell}
{\pard\intbl\ql {{\f5\u8741*}} {\f3 \u92?parallel}\cell}
{\pard\intbl\ql {{\f5\u8869*}} {\f3 \u92?perp}\cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Negated Relations:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8814?}{      }{\f3 \u92?not<}\cell}
{\pard\intbl\ql  {      }{\u8815?}{      }{\f3 \u92?not>}\cell}
{\pard\intbl\ql  {      }{\u8800?}{      }{\f3 \u92?not=}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8816?} {\f3 \u92?not\u92?leq}\cell}
{\pard\intbl\ql {\u8817?} {\f3 \u92?not\u92?geq}\cell}
{\pard\intbl\ql {\u8802?} {\f3 \u92?not\u92?equiv}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8832?} {\f3 \u92?not\u92?prec}\cell}
{\pard\intbl\ql {\u8833?} {\f3 \u92?not\u92?succ}\cell}
{\pard\intbl\ql {\u8769?} {\f3 \u92?not\u92?sim}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8928?} {\f3 \u92?not\u92?preceq}\cell}
{\pard\intbl\ql {\u8929?} {\f3 \u92?not\u92?succeq}\cell}
{\pard\intbl\ql {\u8772?} {\f3 \u92?not\u92?simeq}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8836?} {\f3 \u92?not\u92?subset}\cell}
{\pard\intbl\ql {\u8837?} {\f3 \u92?not\u92?supset}\cell}
{\pard\intbl\ql {\u8777?} {\f3 \u92?not\u92?approx}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8840?} {\f3 \u92?not\u92?subseteq}\cell}
{\pard\intbl\ql {\u8841?} {\f3 \u92?not\u92?supseteq}\cell}
{\pard\intbl\ql {\u8775?} {\f3 \u92?not\u92?cong}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\u8930?} {\f3 \u92?not\u92?sqsubseteq}\cell}
{\pard\intbl\ql {\u8931?} {\f3 \u92?not\u92?sqsupseteq}\cell}
{\pard\intbl\ql {\u8813?} {\f3 \u92?not\u92?asymp}\cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Arrows:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8592*}}}{\fldrslt }}
} {\f3 \u92?leftarrow}\cell}
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8594*}}}{\fldrslt }}
} {\f3 \u92?rightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u10229*}} {\f3 \u92?longleftarrow}\cell}
{\pard\intbl\ql {{\f5\u10230*}} {\f3 \u92?longrightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8656*}} {\f3 \u92?Leftarrow}\cell}
{\pard\intbl\ql {{\f5\u8658*}} {\f3 \u92?Rightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u10232*}} {\f3 \u92?Longleftarrow}\cell}
{\pard\intbl\ql {{\f5\u10233*}} {\f3 \u92?Longrightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8596*}}}{\fldrslt }}
} {\f3 \u92?leftrightarrow}\cell}
{\pard\intbl\ql {{\f5\u8660*}} {\f3 \u92?Leftrightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u10231*}} {\f3 \u92?longleftrightarrow}\cell}
{\pard\intbl\ql {{\f5\u10234*}} {\f3 \u92?Longleftrightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8617*}} {\f3 \u92?hookleftarrow}\cell}
{\pard\intbl\ql {{\u8618*}} {\f3 \u92?hookrightarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8636*}}}{\fldrslt }}
} {\f3 \u92?leftharpoonup}\cell}
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8640*}}}{\fldrslt }}
} {\f3 \u92?rightharpoonup}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8637*}}}{\fldrslt }}
} {\f3 \u92?leftharpoondown}\cell}
{\pard\intbl\ql {{\field{\*\fldinst{ EQ {\u8641*}}}{\fldrslt }}
} {\f3 \u92?rightharpoondown}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8593*}} {\f3 \u92?uparrow}\cell}
{\pard\intbl\ql {{\u8595*}} {\f3 \u92?downarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8657*}} {\f3 \u92?Uparrow}\cell}
{\pard\intbl\ql {{\u8659*}} {\f3 \u92?Downarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8597*}} {\f3 \u92?updownarrow}\cell}
{\pard\intbl\ql {{\u8661*}} {\f3 \u92?Updownarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8599*}} {\f3 \u92?nearrow}\cell}
{\pard\intbl\ql {{\f5\u8598*}} {\f3 \u92?nwarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8600*}} {\f3 \u92?searrow}\cell}
{\pard\intbl\ql {{\f5\u8601*}} {\f3 \u92?swarrow}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8614*}} {\f3 \u92?mapsto}\cell}
{\pard\intbl\ql {{\f5\u10236*}} {\f3 \u92?longmapsto}\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ  \\o ({\up5 {\u8636*}},{\dn5 {\u8641*}}) }}{\fldrslt }}
} {\f3 \u92?rightleftharpoons}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Openings:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {[} {\f3 \u92?lbrack}\cell}
{\pard\intbl\ql {{\f5\u8970*}} {\f3 \u92?lfloor}\cell}
{\pard\intbl\ql {{\f5\u8968*}} {\f3 \u92?lceil}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\{} {\f3 \u92?lbrace}\cell}
{\pard\intbl\ql {{\f5\u9001*}} {\f3 \u92?langle}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Closings:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {]} {\f3 \u92?rbrack}\cell}
{\pard\intbl\ql {{\f5\u8971*}} {\f3 \u92?rfloor}\cell}
{\pard\intbl\ql {{\f5\u8969*}} {\f3 \u92?rceil}\cell}
\row}
{\trowd\cellx2300\cellx4600\cellx6900
{\pard\intbl\ql {\}} {\f3 \u92?rbrace}\cell}
{\pard\intbl\ql {{\f5\u9002*}} {\f3 \u92?rangle}\cell}
{\pard\intbl\ql \cell}
\row}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Alternative Names:\par
{\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
\par
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {\u8800?}{      }{      } {\f3 \u92?ne} or {\f3 \u92?neq}\cell}
{\pard\intbl\ql (same as {\f3 \u92?not=})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8804*}} {\f3 \u92?le}\cell}
{\pard\intbl\ql (same as {\f3 \u92?leq})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8805*}} {\f3 \u92?ge}\cell}
{\pard\intbl\ql (same as {\f3 \u92?geq})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {\{}{      }{      }{\f3 \u92?\u123?}\cell}
{\pard\intbl\ql (same as {\f3 \u92?lbrace})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {\}}{      }{      }{\f3 \u92?\u125?}\cell}
{\pard\intbl\ql (same as {\f3 \u92?lbrace})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8594*}} {\f3 \u92?to}\cell}
{\pard\intbl\ql (same as {\f3 \u92?rightarrow})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u8592*}} {\f3 \u92?gets}\cell}
{\pard\intbl\ql (same as {\f3 \u92?leftarrow})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8715*}} {\f3 \u92?owns}\cell}
{\pard\intbl\ql (same as {\f3 \u92?ni})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8743*}} {\f3 \u92?land}\cell}
{\pard\intbl\ql (same as {\f3 \u92?wedge})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8744*}} {\f3 \u92?lor}\cell}
{\pard\intbl\ql (same as {\f3 \u92?vee})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\u172*}} {\f3 \u92?lnot}\cell}
{\pard\intbl\ql (same as {\f3 \u92?neg})\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {|} {\f3 \u92?vert}\cell}
{\pard\intbl\ql {      }(same as {\f3 \emdash })\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u8741*}} {\f3 \u92?Vert}\cell}
{\pard\intbl\ql {      }(same as {\f3 \u92?\emdash })\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {{\f5\u10234* }} {\f3 \u92?iff}\cell}
{\pard\intbl\ql {      }(same as {\f3 \u92?Longleftrightarrow}, but with\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql \cell}
{\pard\intbl\ql {      } extra space at each end)\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql {: } {\f3 \u92?colon}\cell}
{\pard\intbl\ql {      }(same as {\f3 :}, but with less space around it and\cell}
\row}
{\trowd\cellx3450\cellx6900
{\pard\intbl\ql \cell}
{\pard\intbl\ql  {      } less likelihood of a line break after it)\cell}
\row}
}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 3.6  Changing Fonts in Mathematics Mode\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 {\i (The following applies to {\plain L{\up6\fs19 A}T{\dn7 E}X}2{{\u949*}}, a recent version of {\plain L{\up6\fs19 A}T{\dn7 E}X}. It does not apply to older versions of {\plain L{\up6\fs19 A}T{\dn7 E}X}.)}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The \lquote math italic\rquote  font is automatically used in mathematics mode unless you explicitly change the font. The rules for changing the font in mathematics mode are rather different to those applying when typesetting ordinary text. In mathematics mode any change only applies to the single character or symbol that follows (or to any text enclosed within curly brackets immediately following the control sequence). Also, to change a character to the roman or boldface font, the control sequences {\b0\i0\scaps0\f3 \\mathrm} and {\b0\i0\scaps0\f3 \\mathbf} must be used (rather than {\b0\i0\scaps0\f3 \\textrm} and {\b0\i0\scaps0\f3 \\textbf}).\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The following example illustrates the use of boldface in mathematical formulae. To obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Let {{\b u}},{{\b v}} and {{\b w}} be three vectors in {{\field{\*\fldinst{ EQ {{\b R}}\\s\\up6({\fs20 3})}}{\fldrslt }}
}. The volume\~{{\i V}} of the parallelepiped with corners at the points {{\b 0}}, {{\b u}}, {{\b v}}, {{\b w}}, {{\b u}+{\b v}}, {{\b u}+{\b w}}, {{\b v}+{\b w}} and {{\b u}+{\b v}+{\b w}} is given by the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\i V}=({\b u}{\u215*}{\b v}){\u8901*}{\b w}.\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one could type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Let $\\mathbf\{u\}$,$\\mathbf\{v\}$ and $\\mathbf\{w\}$ be three
\par vectors in $\{\\mathbf R\}^3$. The volume~$V$ of the
\par parallelepiped with corners at the points
\par $\\mathbf\{0\}$, $\\mathbf\{u\}$, $\\mathbf\{v\}$,
\par $\\mathbf\{w\}$, $\\mathbf\{u\}+\\mathbf\{v\}$,
\par $\\mathbf\{u\}+\\mathbf\{w\}$, $\\mathbf\{v\}+\\mathbf\{w\}$
\par and $\\mathbf\{u\}+\\mathbf\{v\}+\\mathbf\{w\}$
\par is given by the formula
\par \\[ V = (\\mathbf\{u\} \\times \\mathbf\{v\}) \\cdot \\mathbf\{w\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 There is also a \lquote calligraphic\rquote  font available in mathematics mode. This is obtained using the control sequence {\b0\i0\scaps0\f3 \\cal}. {\i This font can only be used for uppercase letters.} These calligraphic letters have the form \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{{\i A}}{{\i B}}{{\i C}}{{\i D}}{{\i E}}{{\i F}}{{\i G}}{{\i H}}{{\i I}}{{\i J}}{{\i K}}{{\i L}}{{\i M}}{{\i N}}{{\i O}}{{\i P}}{{\i Q}}{{\i R}}{{\i S}}{{\i T}}{{\i U}}{{\i V}}{{\i W}}{{\i X}}{{\i Y}}{{\i Z}}.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.7  Standard Functions (sin, cos etc.)\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 The names of certain standard functions and abbreviations are obtained by typing a backlash {\b0\i0\scaps0\f3 \\} before the name. For example, one obtains \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\plain cos}({\u920*}+{\u966*})={\plain cos}{\u920*}{\plain cos}{\u966*}\u8722?{\plain sin}{\u920*}{\plain sin}{\u966*}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\cos(\\theta + \\phi) = \\cos \\theta \\cos \\phi
\par       - \\sin \\theta \\sin \\phi \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 The following standard functions are represented by control sequences defined in {\plain L{\up6\fs19 A}T{\dn7 E}X}: \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
\par
{\trowd\cellx862\cellx1725\cellx2587\cellx3450\cellx4312\cellx5175\cellx6037\cellx6900
{\pard\intbl\ql {\\}{\f3 arccos}\cell}
{\pard\intbl\ql {\\}{\f3 cos}\cell}
{\pard\intbl\ql {\\}{\f3 csc}\cell}
{\pard\intbl\ql {\\}{\f3 exp}\cell}
{\pard\intbl\ql {\\}{\f3 ker}\cell}
{\pard\intbl\ql {\\}{\f3 limsup}\cell}
{\pard\intbl\ql {\\}{\f3 min}\cell}
{\pard\intbl\ql {\\}{\f3 sinh}\cell}
\row}
{\trowd\cellx862\cellx1725\cellx2587\cellx3450\cellx4312\cellx5175\cellx6037\cellx6900
{\pard\intbl\ql {\\}{\f3 arcsin}\cell}
{\pard\intbl\ql {\\}{\f3 cosh}\cell}
{\pard\intbl\ql {\\}{\f3 deg}\cell}
{\pard\intbl\ql {\\}{\f3 gcd}\cell}
{\pard\intbl\ql {\\}{\f3 lg}\cell}
{\pard\intbl\ql {\\}{\f3 ln}\cell}
{\pard\intbl\ql {\\}{\f3 Pr}\cell}
{\pard\intbl\ql {\\}{\f3 sup}\cell}
\row}
{\trowd\cellx862\cellx1725\cellx2587\cellx3450\cellx4312\cellx5175\cellx6037\cellx6900
{\pard\intbl\ql {\\}{\f3 arctan}\cell}
{\pard\intbl\ql {\\}{\f3 cot}\cell}
{\pard\intbl\ql {\\}{\f3 det}\cell}
{\pard\intbl\ql {\\}{\f3 hom}\cell}
{\pard\intbl\ql {\\}{\f3 lim}\cell}
{\pard\intbl\ql {\\}{\f3 log}\cell}
{\pard\intbl\ql {\\}{\f3 sec}\cell}
{\pard\intbl\ql {\\}{\f3 tan}\cell}
\row}
{\trowd\cellx862\cellx1725\cellx2587\cellx3450\cellx4312\cellx5175\cellx6037\cellx6900
{\pard\intbl\ql {\\}{\f3 arg}\cell}
{\pard\intbl\ql {\\}{\f3 coth}\cell}
{\pard\intbl\ql {\\}{\f3 dim}\cell}
{\pard\intbl\ql {\\}{\f3 inf}\cell}
{\pard\intbl\ql {\\}{\f3 liminf}\cell}
{\pard\intbl\ql {\\}{\f3 max}\cell}
{\pard\intbl\ql {\\}{\f3 sin}\cell}
{\pard\intbl\ql {\\}{\f3 tanh}\cell}
\row}
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Names of functions and other abbreviations not in this list can be obtained by converting to the roman font. Thus one obtains {{\f0 {\i c}{\i o}{\i s}{\i e}{\i c}}{\i A}} by typing {\b0\i0\scaps0\f3 $\\mathrm\{cosec\} A$}. Note that if one were to type simply {\b0\i0\scaps0\f3 $cosec A$} one would obtain {{\i c}{\i o}{\i s}{\i e}{\i c}{\i A}}, because {\plain L{\up6\fs19 A}T{\dn7 E}X} has treated {\b0\i0\scaps0\f3 cosec A} as the product of six quantities {{\i c}}, {{\i o}}, {{\i s}}, {{\i e}}, {{\i c}} and {{\i A}} and typeset the formula accordingly.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.8  Text Embedded in Displayed Equations\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Text can be embedded in displayed equations (in {\plain L{\up6\fs19 A}T{\dn7 E}X}) by using {\b0\i0\scaps0\f3 \\mbox\{}{\i embedded text}{\b0\i0\scaps0\f3 \}}. For example, one obtains \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i M}\\s\\up6({\fs20 {\f5\u8869*}})=\{{\i f}{\f5\u8712*}{\i V}':{\i f}({\i m})=0{\~for\~all\~}{\i m}{\f5\u8712*}{\i M}\}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ M^\\bot = \\\{ f \\in V' : f(m) = 0 \\mbox\{ for all \} m \\in M \\\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Note the blank spaces before and after the words \lquote for all\rquote  in the above example. Had we typed {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ M^\\bot = \\\{ f \\in V' : f(m) = 0 \\mbox\{for all\} m \\in M \\\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 we would have obtained \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i M}\\s\\up6({\fs20 {\f5\u8869*}})=\{{\i f}{\f5\u8712*}{\i V}':{\i f}({\i m})=0{for\~all}{\i m}{\f5\u8712*}{\i M}\}.}}{\fldrslt }}
\par
(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In Plain {\plain T{\dn7 E}X} one should use {\b0\i0\scaps0\f3 \\hbox} in place of {\b0\i0\scaps0\f3 \\mbox}.)\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.9  Fractions and Roots\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Fractions of the form \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\F({{\i numerator}},{{\i denominator}})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 are obtained in {\plain L{\up6\fs19 A}T{\dn7 E}X} using the construction {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\frac\{}{\i numerator}{\b0\i0\scaps0\f3 \}\{}{\i denominator}{\b0\i0\scaps0\f3 \}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 For example, to obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The function {{\i f}} is given by \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\i f}({\i x})=2{\i x}+ \\F({\i x}\u8722?7,{\i x}\\s\\up6({\fs20 2})+4)}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 for all real numbers {{\i x}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one would type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The function $f$ is given by
\par \\[ f(x) = 2x + \\frac\{x - 7\}\{x^2 + 4\}\\] 
\par for all real numbers $x$.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 To obtain square roots one uses the control sequence {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\sqrt\{}{\i expression}{\b0\i0\scaps0\f3 \}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 For example, to obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The roots of a quadratic polynomial {{\field{\*\fldinst{ EQ {\i a}{\i x}\\s\\up6({\fs20 2})+{\i b}{\i x}+{\i c}}}{\fldrslt }}
} with {{\i a}{\u8800*}0} are given by the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\F(\u8722?{\i b}{\u177*} \\R(,{\i b}\\s\\up6({\fs20 2})\u8722?4{\i a}{\i c}),2{\i a})}}{\fldrslt }}
\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one would type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The roots of a quadratic polynomial $a x^2 + bx + c$ with
\par $a \\neq 0$ are given by the formula
\par \\[ \\frac\{-b \\pm \\sqrt\{b^2 - 4ac\}\}\{2a\} \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 In {\plain L{\up6\fs19 A}T{\dn7 E}X}, an {{\i n}}th root is produced using {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\sqrt[n]\{}{\i expression}{\b0\i0\scaps0\f3 \}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 For example, to obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The roots of a cubic polynomial of the form {{\field{\*\fldinst{ EQ {\i x}\\s\\up6({\fs20 3})\u8722?3{\i p}{\i x}\u8722?2{\i q}}}{\fldrslt }}
} are given by the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\R(3,{\i q}+ \\R(,{\i q}\\s\\up6({\fs20 2})\u8722?{\i p}\\s\\up6({\fs20 3})))+ \\R(3,{\i q}\u8722? \\R(,{\i q}\\s\\up6({\fs20 2})\u8722?{\i p}\\s\\up6({\fs20 3})))}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 where the values of the two cube roots must are chosen so as to ensure that their product is equal to {{\i p}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 in {\plain L{\up6\fs19 A}T{\dn7 E}X}, one would type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The roots of a cubic polynomial of the form $x^3 - 3px - 2q$
\par are given by the formula
\par \\[ \\sqrt[3]\{q + \\sqrt\{ q^2 - p^3 \}\}
\par   + \\sqrt[3]\{q - \\sqrt\{ q^2 - p^3 \}\} \\]
\par where the values of the two cube roots must are chosen
\par so as to ensure that their product is equal to $p$.
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 3.10  Ellipsis (i.e., \lquote three dots\rquote )\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Ellipsis (three dots) is produced in mathematics mode using the control sequences {\b0\i0\scaps0\f3 \\ldots} (for dots aligned with tbe baseline of text), and {\b0\i0\scaps0\f3 \\cdots} (for dots aligned with the centreline of mathematical formulae). Thus the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i f}({\i x}\\s\\do6({\fs20 1})\\,{\i x}\\s\\do6({\fs20 2})\\,\u8230?\\,{\i x}\\s\\do6({\fs20 {\i n}}))={\i x}\\s\\up({\fs20 2,1})+{\i x}\\s\\up({\fs20 2,2})+{\u8943*}+{\i x}\\s\\up({\fs20 2,{\i n}})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ f(x_1, x_2,\\ldots, x_n) = x_1^2 + x_2^2 + \\cdots + x_n^2 \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Similarly the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\F(1\u8722?{\i x}\\s\\up6({\fs20 {\i n}+1}),1\u8722?{\i x})=1+{\i x}+{\i x}\\s\\up6({\fs20 2})+{\u8943*}+{\i x}\\s\\up6({\fs20 {\i n}})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is produced using {\b0\i0\scaps0\f3 \\cdots}, by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\frac\{1 - x^\{n+1\}\}\{1 - x\} = 1 + x + x^2 + \\cdots + x^n \\]
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 3.11  Accents in Mathematics Mode\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 There are various control sequences for producing underlining, overlining and various accents in mathematics mode. The following table lists these control sequences, applying them to the letter\~{{\i a}}: {\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ }}{\fldrslt }}
\par
{\trowd\cellx6900
{\pard\intbl\ql {{\ul {\i a}}} {      }{\f3 \u92?underline\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {{\field{\*\fldinst{ EQ  \\x\\to( {\i a}) }}{\fldrslt }}
} {      }{\f3 \u92?overline\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\'e2} {      }{\f3 \u92?hat\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\u462a} {      }{\f3 \u92?check\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\'e3} {      }{\f3 \u92?tilde\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\'e1} {      }{\f3 \u92?acute\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\'e0} {      }{\f3 \u92?grave\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\u551a} {      }{\f3 \u92?dot\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\'e4} {      }{\f3 \u92?ddot\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\u259a} {      }{\f3 \u92?breve\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {\u257a} {      }{\f3 \u92?bar\u123?a\u125?}\cell}
\row}
{\trowd\cellx6900
{\pard\intbl\ql {{\f5 \u8407-}{\i a}} {      }{\f3 \u92?vec\u123?a\u125?}\cell}
\row}
{\field{\*\fldinst{ EQ }}{\fldrslt }}
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 It should be borne in mind that when a character is underlined in a mathematical manuscript then it is normally typeset in bold face without any underlining. Underlining is used very rarely in print.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The control sequences such as {\b0\i0\scaps0\f3 \\'} and {\b0\i0\scaps0\f3 \\"}, used to produce accents in ordinary text, may not be used in mathematics mode.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.12  Brackets and Norms\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 The frequently used left delimiters include (, [ and \{, which are obtained by typing {\b0\i0\scaps0\f3 (}, {\b0\i0\scaps0\f3 [} and {\b0\i0\scaps0\f3 \\\{} respectively. The corresponding right delimiters are of course obtained by typing {\b0\i0\scaps0\f3 )}, {\b0\i0\scaps0\f3 ]} and {\b0\i0\scaps0\f3 \\\}}. In addition {|} and {|} are used as both left and right delimiters, and are obtained by typing {\b0\i0\scaps0\f3 |} and {\b0\i0\scaps0\f3 \\|} respectively. For example, we obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Let {{\i X}} be a Banach space and let {{\i f}: {\i B}{\u8594*}{\b R}} be a bounded linear functional on {{\i X}}. The {\i norm} of {{\i f}}, denoted by {|{\i f}|}, is defined by \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
|{\i f}|={\plain inf}\{{\i K}{\f5\u8712*}[0,+{\u8734*}):|{\i f}({\i x})|{\u8804*}{\i K}|{\i x}|{\~for\~all\~}{\i x}{\f5\u8712*}{\i X}\}.\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Let $X$ be a Banach space and let $f \\colon B \\to \\textbf\{R\}$
\par be a bounded linear functional on $X$. The \\textit\{norm\} of
\par $f$, denoted by $\\|f\\|$, is defined by
\par \\[ \\|f\\| = \\inf \\\{ K \\in [0,+\\infty) :
\par           |f(x)| \\leq K \\|x\\| \\mbox\{ for all \} x \\in X \\\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Larger delimiters are sometimes required which have the appropriate height to match the size of the subformula which they enclose. Consider, for instance, the problem of typesetting the following formula: \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ {\i f}({\i x}\\,{\i y}\\,{\i z})=3{\i y}\\s\\up6({\fs20 2}){\i z} \\b (3+ \\F(7{\i x}+5,1+{\i y}\\s\\up6({\fs20 2}))).}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 The way to type the large parentheses is to type {\b0\i0\scaps0\f3 \\left(} for the left parenthesis and {\b0\i0\scaps0\f3 \\right)} for the right parenthesis, and let {\plain L{\up6\fs19 A}T{\dn7 E}X} do the rest of the work for you. Thus the above formula was obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ f(x,y,z) = 3y^2 z \\left( 3 + \\frac\{7x+5\}\{1 + y^2\} \\right).\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 If you type a delimiter which is preceded by {\b0\i0\scaps0\f3 \\left} then {\plain L{\up6\fs19 A}T{\dn7 E}X} will search for a corresponding delimiter preceded by {\b0\i0\scaps0\f3 \\right} and calculate the size of the delimiters required to enclose the intervening subformula. One is allowed to balance a {\b0\i0\scaps0\f3 \\left(} with a {\b0\i0\scaps0\f3 \\right]} (say) if one desires: there is no reason why the enclosing delimiters have to have the same shape. One may also nest pairs of delimiters within one another: by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\left| 4 x^3 + \\left( x + \\frac\{42\}\{1+x^4\} \\right) \\right|.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 we obtain \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\b \\lc\\|\\rc\\| (4{\i x}\\s\\up6({\fs20 3})+ \\b ({\i x}+ \\F(42,1+{\i x}\\s\\up6({\fs20 4})))).}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 By typing {\b0\i0\scaps0\f3 \\left.} and {\b0\i0\scaps0\f3 \\right.} one obtains {\i null delimiters} which are completely invisible. Consider, for example, the problem of typesetting \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\b \\rc\\| ( \\F({\i d}{\i u},{\i d}{\i x}))\\s\\do17({\fs20 {\i x}=0}).}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 We wish to make the vertical bar big enough to match the derivative preceding it. To do this, we suppose that the derivative is enclosed by delimiters, where the left delimiter is invisible and the right delimiter is the vertical line. The invisible delimiter is produced using {\b0\i0\scaps0\f3 \\left.} and thus the whole formula is produced by typing \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\b \\rc\\| ( \\F({\i d}{\i u},{\i d}{\i x}))\\s\\do17({\fs20 {\i x}=0}).}}{\fldrslt }}
\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.13  Multiline Formulae in {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Consider the problem of typesetting the formula {\par
{\pard\plain\s15\ql\sb120\sa120\keep\widctlpar\f0\tqr\tx3105\tqc\tx3450\tql\tx3795\sl240\slmult1 \sb60 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\plain cos}2{\u920*}}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ =}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ {\plain cos}\\s\\up6({\fs20 2}){\u920*}\u8722?{\plain sin}\\s\\up6({\fs20 2}){\u920*}}}{\fldrslt }}
\par
\pard\plain\s15\ql\sb120\sa120\keep\widctlpar\f0\tqr\tx3105\tqc\tx3450\tql\tx3795\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ }}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ =}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ 2{\plain cos}\\s\\up6({\fs20 2}){\u920*}\u8722?1.}}{\fldrslt }}
\par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 It is necessary to ensure that the = signs are aligned with one another. In {\plain L{\up6\fs19 A}T{\dn7 E}X}, such a formula is typeset using the {\b0\i0\scaps0\f3 eqnarray*} environment. The above example was obtained by typing the lines {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{eqnarray*\}
\par \\cos 2\\theta & = & \\cos^2 \\theta - \\sin^2 \\theta \\\\
\par              & = & 2 \\cos^2 \\theta - 1.
\par \\end\{eqnarray*\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Note the use of the special character {\b0\i0\scaps0\f3 &} as an {\i alignment tab}. When the formula is typeset, the part of the second line of the formula beginning with an occurrence of {\b0\i0\scaps0\f3 &} will be placed immediately beneath that part of the first line of the formula which begins with the corresponding occurrence of {\b0\i0\scaps0\f3 &}. Also {\b0\i0\scaps0\f3 \\\\} is used to separate the lines of the formula.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Although we have placed corresponding occurrences of {\b0\i0\scaps0\f3 &} beneath one another in the above example, it is not necessary to do this in the input file. It was done in the above example merely to improve the appearance (and readability) of the input file.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The more complicated example {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 If {{\field{\*\fldinst{ EQ {\i h}{\u8804*} \\F(1,2)|{\u950*}\u8722?{\i z}|}}{\fldrslt }}
} then \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ |{\u950*}\u8722?{\i z}\u8722?{\i h}|{\u8805*} \\F(1,2)|{\u950*}\u8722?{\i z}|}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 and hence {\par
\pard\plain\s15\ql\sb120\sa120\keep\widctlpar\f0\tqr\tx3105\tqc\tx3450\tql\tx3795\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\b \\lc\\|\\rc\\| ( \\F(1,{\u950*}\u8722?{\i z}\u8722?{\i h})\u8722? \\F(1,{\u950*}\u8722?{\i z}))}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ =}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ  \\b \\lc\\|\\rc\\| ( \\F(({\u950*}\u8722?{\i z})\u8722?({\u950*}\u8722?{\i z}\u8722?{\i h}),({\u950*}\u8722?{\i z}\u8722?{\i h})({\u950*}\u8722?{\i z})))}}{\fldrslt }}
\par
\pard\plain\s15\ql\sb120\sa120\keep\widctlpar\f0\tqr\tx3105\tqc\tx3450\tql\tx3795\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ }}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ =}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ  \\b \\lc\\|\\rc\\| ( \\F({\i h},({\u950*}\u8722?{\i z}\u8722?{\i h})({\u950*}\u8722?{\i z})))}}{\fldrslt }}
\par
\pard\plain\s15\ql\sb120\sa120\keep\widctlpar\f0\tqr\tx3105\tqc\tx3450\tql\tx3795\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ }}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ {\u8804*}}}{\fldrslt }}
\tab
{\field{\*\fldinst{ EQ  \\F(2|{\i h}|,|{\u950*}\u8722?{\i z}|\\s\\up6({\fs20 2})).}}{\fldrslt }}
\par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 was obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par If $h \\leq \\frac\{1\}\{2\} |\\zeta - z|$ then
\par \\[ |\\zeta - z - h| \\geq \\frac\{1\}\{2\} |\\zeta - z|\\] 
\par and hence
\par \\begin\{eqnarray*\}
\par \\left| \\frac\{1\}\{\\zeta - z - h\} - \\frac\{1\}\{\\zeta - z\} \\right|
\par & = & \\left|
\par \\frac\{(\\zeta - z) - (\\zeta - z - h)\}\{(\\zeta - z - h)(\\zeta - z)\}
\par \\right| \\\\  & = &
\par \\left| \\frac\{h\}\{(\\zeta - z - h)(\\zeta - z)\} \\right| \\\\
\par   & \\leq & \\frac\{2 |h|\}\{|\\zeta - z|^2\}.
\par \\end\{eqnarray*\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 The asterisk in {\b0\i0\scaps0\f3 eqnarray*} is put there to suppress the automatic equation numbering produced by {\plain L{\up6\fs19 A}T{\dn7 E}X}. If you wish for an automatically numbered multiline formula, you should use {\b0\i0\scaps0\f3 \\begin\{eqnarray\}} and {\b0\i0\scaps0\f3 \\end\{eqnarray\}}.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 3.14  Matrices and other arrays in {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Matrices and other arrays are produced in {\plain L{\up6\fs19 A}T{\dn7 E}X} using the {\b array} environment. For example, suppose that we wish to typeset the following passage: {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The {\i characteristic polynomial} {{\u967*}({\u955*})} of the {3{\u215*}3}\~matrix \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\b ({ \\a \\ac \\co3 ({\i a},{\i b},{\i c},{\i d},{\i e},{\i f},{\i g},{\i h},{\i i})})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 is given by the formula \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\u967*}({\u955*})= \\b \\lc\\|\\rc\\| ({ \\a \\ac \\co3 ({\u955*}\u8722?{\i a},\u8722?{\i b},\u8722?{\i c},\u8722?{\i d},{\u955*}\u8722?{\i e},\u8722?{\i f},\u8722?{\i g},\u8722?{\i h},{\u955*}\u8722?{\i i})}).}}{\fldrslt }}
\par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 This passage is produced by the following input: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The \\emph\{characteristic polynomial\} $\\chi(\\lambda)$ of the
\par $3 \\times 3$~matrix
\par \\[ \\left( \\begin\{array\}\{ccc\}
\par a & b & c \\\\
\par d & e & f \\\\
\par g & h & i \\end\{array\} \\right)\\] 
\par is given by the formula
\par \\[ \\chi(\\lambda) = \\left| \\begin\{array\}\{ccc\}
\par \\lambda - a & -b & -c \\\\
\par -d & \\lambda - e & -f \\\\
\par -g & -h & \\lambda - i \\end\{array\} \\right|.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 First of all, note the use of {\b0\i0\scaps0\f3 \\left} and {\b0\i0\scaps0\f3 \\right} to produce the large delimiters around the arrays. As we have already seen, if we use {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\left)} ... {\b0\i0\scaps0\f3 \\right)} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 then the size of the parentheses is chosen to match the subformula that they enclose. Next note the use of the alignment tab character {\b0\i0\scaps0\f3 &} to separate the entries of the matrix and the use of {\b0\i0\scaps0\f3 \\\\} to separate the rows of the matrix, exactly as in the construction of multiline formulae described above. We begin the array with {\b0\i0\scaps0\f3 \\begin\{array\}} and end it with {\b0\i0\scaps0\f3 \\end\{array\}}. The only thing left to explain, therefore, is the mysterious {\b0\i0\scaps0\f3 \{ccc\}} which occurs immediately after {\b0\i0\scaps0\f3 \\begin\{array\}}. Now each of the {\b0\i0\scaps0\f3 c}\rquote s in {\b0\i0\scaps0\f3 \{ccc\}} represents a column of the matrix and indicates that the entries of the column should be {\i centred}. If the {\b0\i0\scaps0\f3 c} were replaced by {\b0\i0\scaps0\f3 l} then the corresponding column would be typeset with all the entries flush {\i left}, and {\b0\i0\scaps0\f3 r} would produce a column with all entries flush {\i right}. Thus {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\begin\{array\}\{lcr\}
\par \\mbox\{First number\} & x & 8 \\\\
\par \\mbox\{Second number\} & y & 15 \\\\
\par \\mbox\{Sum\} & x + y & 23 \\\\
\par \\mbox\{Difference\} & x - y & -7 \\\\
\par \\mbox\{Product\} & xy & 120 \\end\{array\}\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 produces \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ { \\a \\al \\co3 ({First\~number}\~,\~x\~,\~8\~,{Second\~number}\~,\~y\~,\~15\~,{Sum}\~,\~x\~+\~y\~,\~23\~,{Difference}\~,\~x\~- y , -7 ,{Product} , xy , 120 )}}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 We can use the array environment to produce formulae such as \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ |{\i x}|= \\b \\lc\\\{ ({ \\a \\al \\co2 ({\i x},{if\~{{\i x}{\u8805*}0}};,-x , {if\~{{\i x}<0}}.)})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 Note that both columns of this array are set flush left. Thus we use {\b0\i0\scaps0\f3 \{ll\}} immediately after {\b0\i0\scaps0\f3 \\begin\{array\}}. The large curly bracket is produced using {\b0\i0\scaps0\f3 \\left\\\{}. However this requires a corresponding {\b0\i0\scaps0\f3 \\right} delimiter to match it. We therefore use the {\i null delimiter} {\b0\i0\scaps0\f3 \\right.} discussed earlier. This delimiter is invisible. We can therefore obtain the above formula by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ |x| = \\left\\\{ \\begin\{array\}\{ll\}
\par          x & \\mbox\{if $x \\geq 0$\};\\\\
\par         -x & \\mbox\{if $x < 0$\}.\\end\{array\} \\right. \\] 
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 3.15  Derivatives, Limits, Sums and Integrals\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 The expressions \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\F({\i d}{\i u},{\i d}{\i t}){\~and\~} \\F({\i d}\\s\\up6({\fs20 2}){\i u},{\i d}{\i x}\\s\\up6({\fs20 2}))}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 are obtained in {\plain L{\up6\fs19 A}T{\dn7 E}X} by typing {\b0\i0\scaps0\f3 \\frac\{du\}\{dt\}} and {\b0\i0\scaps0\f3 \\frac\{d^2 u\}\{dx^2\}} respectively. The mathematical symbol {{\u8706*}} is produced using {\b0\i0\scaps0\f3 \\partial}. Thus the Heat Equation \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\F({\u8706*}{\i u},{\u8706*}{\i t})= \\F({\u8706*}\\s\\up6({\fs20 2}){\i u},{\u8706*}{\i x}\\s\\up6({\fs20 2}))+ \\F({\u8706*}\\s\\up6({\fs20 2}){\i u},{\u8706*}{\i y}\\s\\up6({\fs20 2}))+ \\F({\u8706*}\\s\\up6({\fs20 2}){\i u},{\u8706*}{\i z}\\s\\up6({\fs20 2}))}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is obtained in {\plain L{\up6\fs19 A}T{\dn7 E}X} by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[\\frac\{\\partial u\}\{\\partial t\}
\par    = \\frac\{\\partial^2 u\}\{\\partial x^2\}
\par       + \\frac\{\\partial^2 u\}\{\\partial y^2\}
\par       + \\frac\{\\partial^2 u\}\{\\partial z^2\} \\]
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 To obtain mathematical expressions such as \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ \\a\\ac(lim,{\i x}{\u8594*}+{\u8734*}){\\,\~}{\plain inf}\\s\\do6({\fs20 {\i x}>{\i s}}){\~and\~}{\plain sup}\\s\\do6({\fs20 {\i K}})}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 in displayed equations we type {\b0\i0\scaps0\f3 \\lim_\{x \\to +\\infty\}}, {\b0\i0\scaps0\f3 \\inf_\{x > s\}} and {\b0\i0\scaps0\f3 \\sup_K} respectively. Thus to obtain \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ \\a\\ac(lim,{\i x}{\u8594*}0) \\F(3{\i x}\\s\\up6({\fs20 2})+7,{\i x}\\s\\up6({\fs20 2})+1)=3.}}{\fldrslt }}
\par
(\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 in {\plain L{\up6\fs19 A}T{\dn7 E}X}) we type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\lim_\{x \\to 0\} \\frac\{3x^2 +7x^3\}\{x^2 +5x^4\} = 3.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 To obtain a summation sign such as \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\su({\i i}=1,2{\i n}, )}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 we type {\b0\i0\scaps0\f3 sum_\{i=1\}^\{2n\}}. Thus \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\su({\i k}=1,{\i n}, ){\i k}\\s\\up6({\fs20 2})= \\F(1,2){\i n}({\i n}+1).}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\sum_\{k=1\}^n k^2 = \\frac\{1\}\{2\} n (n+1).\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 We now discuss how to obtain {\i integrals} in mathematical documents. A typical integral is the following: \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i ({\i a},{\i b}, ){\i f}({\i x}){\charscalex50\~}{\i d}{\i x}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 This is typeset using {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_a^b f(x)\\,dx.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 The integral sign {{\field{\*\fldinst{ EQ  \\i \\in(,, )}}{\fldrslt }}
} is typeset using the control sequence {\b0\i0\scaps0\f3 \\int}, and the {\i limits of integration} (in this case {{\i a}} and {{\i b}} are treated as a subscript and a superscript on the integral sign.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Most integrals occurring in mathematical documents begin with an integral sign and contain one or more instances of\~{{\i d}} followed by another (Latin or Greek) letter, as in {{\i d}{\i x}}, {{\i d}{\i y}} and {{\i d}{\i t}}. To obtain the correct appearance one should put extra space before the\~{{\i d}}, using {\b0\i0\scaps0\f3 \\,}. Thus \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (0,+{\u8734*}, ){\i x}\\s\\up6({\fs20 {\i n}}){\i e}\\s\\up6({\fs20 \u8722?{\i x}}){\charscalex50\~}{\i d}{\i x}={\i n}!.}}{\fldrslt }}
\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ){\plain cos}{\u920*}{\charscalex50\~}{\i d}{\u920*}={\plain sin}{\u920*}.}}{\fldrslt }}
\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i ({\i x}\\s\\up6({\fs20 2})+{\i y}\\s\\up6({\fs20 2}){\u8804*}{\i R}\\s\\up6({\fs20 2}),, ){\i f}({\i x}\\,{\i y}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}= \\i ({\u920*}=0,2{\u960*}, ) \\i ({\i r}=0,{\i R}, ){\i f}({\i r}{\plain cos}{\u920*}\\,{\i r}{\plain sin}{\u920*}){\i r}{\charscalex50\~}{\i d}{\i r}{\charscalex50\~}{\i d}{\u920*}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 and \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (0,{\i R}, ) \\F(2{\i x}{\charscalex50\~}{\i d}{\i x},1+{\i x}\\s\\up6({\fs20 2}))={\plain log}(1+{\i R}\\s\\up6({\fs20 2})).}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 are obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_0^\{+\\infty\} x^n e^\{-x\} \\,dx = n!.\\] 
\par \par
}}{{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb180 \li512\ri512\fi0 
\par \\[ \\int \\cos \\theta \\,d\\theta = \\sin \\theta.\\] 
\par \par
}}{{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb180 \li512\ri512\fi0 
\par \\[ \\int_\{x^2 + y^2 \\leq R^2\} f(x,y)\\,dx\\,dy
\par    = \\int_\{\\theta=0\}^\{2\\pi\} \\int_\{r=0\}^R
\par       f(r\\cos\\theta,r\\sin\\theta) r\\,dr\\,d\\theta.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 and {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_0^R \\frac\{2x\\,dx\}\{1+x^2\} = \\log(1+R^2).\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 respectively.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In some multiple integrals (i.e., integrals containing more than one integral sign) one finds that {\plain L{\up6\fs19 A}T{\dn7 E}X} puts too much space between the integral signs. The way to improve the appearance of of the integral is to use the control sequence {\b0\i0\scaps0\f3 \\!} to remove a thin strip of unwanted space. Thus, for example, the multiple integral \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (0,1, ) \\i (0,1, ){\i x}\\s\\up6({\fs20 2}){\i y}\\s\\up6({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_0^1 \\! \\int_0^1 x^2 y^2\\,dx\\,dy.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Had we typed {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_0^1 \\int_0^1 x^2 y^2\\,dx\\,dy.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 we would have obtained \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (0,1, ) \\i (0,1, ){\i x}\\s\\up6({\fs20 2}){\i y}\\s\\up6({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 A particularly noteworthy example comes when we are typesetting a multiple integral such as \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ) \\i ({\i D},, ){\i f}({\i x}\\,{\i y}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 Here we use {\b0\i0\scaps0\f3 \\!} three times to obtain suitable spacing between the integral signs. We typeset this integral using {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int \\!\\!\\! \\int_D f(x,y)\\,dx\\,dy.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 Had we typed {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int \\int_D f(x,y)\\,dx\\,dy.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 we would have obtained \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ) \\i \\in(,, )\\s\\do6({\fs20 {\i D}}){\i f}({\i x}\\,{\i y}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The following (reasonably complicated) passage exhibits a number of the features which we have been discussing: {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 In non-relativistic wave mechanics, the wave function {{\u968*}({\b r},{\i t})} of a particle satisfies the {\i Schr\'f6dinger Wave Equation} \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ {\i i}{\u295*} \\F({\u8706*}{\u968*},{\u8706*}{\i t})= \\F(\u8722?{\u295*}\\s\\up6({\fs20 2}),2{\i m}) \\b ( \\F({\u8706*}\\s\\up6({\fs20 2}),{\u8706*}{\i x}\\s\\up6({\fs20 2}))+ \\F({\u8706*}\\s\\up6({\fs20 2}),{\u8706*}{\i y}\\s\\up6({\fs20 2}))+ \\F({\u8706*}\\s\\up6({\fs20 2}),{\u8706*}{\i z}\\s\\up6({\fs20 2}))){\u968*}+{\i V}{\u968*}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 It is customary to normalize the wave equation by demanding that \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ) \\i \\in(,, ) \\i ({\b R}\\s\\up6({\fs20 3}),, ) \\b \\lc\\|\\rc\\| ({\u968*}({\b r}\\,0))\\s\\up17({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}{\charscalex50\~}{\i d}{\i z}=1.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 A simple calculation using the Schr\'f6dinger wave equation shows that \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\F({\i d},{\i d}{\i t}) \\i \\in(,, ) \\i \\in(,, ) \\i ({\b R}\\s\\up6({\fs20 3}),, ) \\b \\lc\\|\\rc\\| ({\u968*}({\b r}\\,{\i t}))\\s\\up17({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}{\charscalex50\~}{\i d}{\i z}=0\\,}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 and hence \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ) \\i \\in(,, ) \\i ({\b R}\\s\\up6({\fs20 3}),, ) \\b \\lc\\|\\rc\\| ({\u968*}({\b r}\\,{\i t}))\\s\\up17({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}{\charscalex50\~}{\i d}{\i z}=1}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 for all times\~{{\i t}}. If we normalize the wave function in this way then, for any (measurable) subset\~{{\i V}} of {{\field{\*\fldinst{ EQ {\b R}\\s\\up6({\fs20 3})}}{\fldrslt }}
} and time\~{{\i t}}, \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \li512\ri512\fi0 \tab
{\field{\*\fldinst{ EQ  \\i \\in(,, ) \\i \\in(,, ) \\i ({\i V},, ) \\b \\lc\\|\\rc\\| ({\u968*}({\b r}\\,{\i t}))\\s\\up17({\fs20 2}){\charscalex50\~}{\i d}{\i x}{\charscalex50\~}{\i d}{\i y}{\charscalex50\~}{\i d}{\i z}}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \li512\ri512\fi0 represents the probability that the particle is to be found within the region\~{{\i V}} at time\~{{\i t}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 One would typeset this in {\plain L{\up6\fs19 A}T{\dn7 E}X} by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par In non-relativistic wave mechanics, the wave function
\par $\\psi(\\mathbf\{r\},t)$ of a particle satisfies the
\par \\textit\{Schr\\"\{o\}dinger Wave Equation\}
\par \\[ i\\hbar\\frac\{\\partial \\psi\}\{\\partial t\}
\par   = \\frac\{-\\hbar^2\}\{2m\} \\left(
\par     \\frac\{\\partial^2\}\{\\partial x^2\}
\par     + \\frac\{\\partial^2\}\{\\partial y^2\}
\par     + \\frac\{\\partial^2\}\{\\partial z^2\}
\par   \\right) \\psi + V \\psi.\\] 
\par It is customary to normalize the wave equation by
\par demanding that
\par \\[ \\int \\!\\!\\! \\int \\!\\!\\! \\int_\{\\textbf\{R\}^3\}
\par       \\left| \\psi(\\mathbf\{r\},0) \\right|^2\\,dx\\,dy\\,dz = 1.\\] 
\par A simple calculation using the Schr\\"\{o\}dinger wave
\par equation shows that
\par \\[ \\frac\{d\}\{dt\} \\int \\!\\!\\! \\int \\!\\!\\! \\int_\{\\textbf\{R\}^3\}
\par       \\left| \\psi(\\mathbf\{r\},t) \\right|^2\\,dx\\,dy\\,dz = 0,\\] 
\par and hence
\par \\[ \\int \\!\\!\\! \\int \\!\\!\\! \\int_\{\\textbf\{R\}^3\}
\par       \\left| \\psi(\\mathbf\{r\},t) \\right|^2\\,dx\\,dy\\,dz = 1\\] 
\par for all times~$t$. If we normalize the wave function in this
\par way then, for any (measurable) subset~$V$ of $\\textbf\{R\}^3$
\par and time~$t$,
\par \\[ \\int \\!\\!\\! \\int \\!\\!\\! \\int_V
\par       \\left| \\psi(\\mathbf\{r\},t) \\right|^2\\,dx\\,dy\\,dz\\] 
\par represents the probability that the particle is to be found
\par within the region~$V$ at time~$t$.
\par \par
}}\pard\plain\s3\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb360 \fi0 4  Further Features of {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb180 \fi0 4.1  Producing White Space in {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 To produce (horizontal) blank space within a paragraph, use {\b0\i0\scaps0\f3 \\hspace}, followed by the length of the blank space enclosed within curly brackets. The length of the skip should be expressed in a unit recognized by {\plain L{\up6\fs19 A}T{\dn7 E}X}. These recognized units are given in the following table: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 \par
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 pt}}\cell}
{\pard\intbl\ql {point}\cell}
{\pard\intbl\ql {(1 in = 72.27 pt)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 pc}}\cell}
{\pard\intbl\ql {pica}\cell}
{\pard\intbl\ql {(1 pc = 12 pt)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 in}}\cell}
{\pard\intbl\ql {inch}\cell}
{\pard\intbl\ql {(1 in = 25.4 mm)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 bp}}\cell}
{\pard\intbl\ql {big point}\cell}
{\pard\intbl\ql {(1 in = 72 bp)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 cm}}\cell}
{\pard\intbl\ql {centimetre}\cell}
{\pard\intbl\ql {(1 cm = 10 mm)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 mm}}\cell}
{\pard\intbl\ql {millimetre}\cell}
{\pard\intbl\ql {}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 dd}}\cell}
{\pard\intbl\ql {didot point}\cell}
{\pard\intbl\ql {(1157 dd = 1238 pt)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 cc}}\cell}
{\pard\intbl\ql {cicero}\cell}
{\pard\intbl\ql {(1 cc = 12 dd)}\cell}
\row}
{\trowd\cellx1828\cellx3813\cellx6898
{\pard\intbl\ql {{\f3 sp}}\cell}
{\pard\intbl\ql {scaled point}\cell}
{\pard\intbl\ql {(65536 sp = 1 pt)}\cell}
\row}
} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Thus to produce a horizontal blank space of 20 mm in the middle of a paragraph one would type {\b0\i0\scaps0\f3 \\hspace\{20 mm\}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 If {\plain L{\up6\fs19 A}T{\dn7 E}X} decides to break between lines at a point in the document where an {\b0\i0\scaps0\f3 \\hspace} is specified, then no white space is produced. To ensure that white space is produced even at points in the document where line breaking takes place, one should replace {\b0\i0\scaps0\f3 \\hspace} by {\b0\i0\scaps0\f3 \\hspace*}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 To produce (vertical) blank space between paragraphs, use {\b0\i0\scaps0\f3 \\vspace}, followed by the length of the blank space enclosed within curly brackets. Thus to obtain {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 This is the first paragraph of some text. It is separated from the second paragraph by a vertical skip of 10 millimetres.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb569 \li512\ri512\fi360 This is the second paragraph. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 one should type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par This is the first paragraph of some text.  It is
\par separated from the second paragraph by a vertical skip of
\par 10 millimetres.
\par 
\par \\vspace\{10 mm\}
\par This is the second paragraph.
\par 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 If {\plain L{\up6\fs19 A}T{\dn7 E}X} decides to introduce at a point in the document where a {\b0\i0\scaps0\f3 \\vspace} is specified, then no white space is produced. To ensure that white space is produced even at points in the document where page breaking takes place, one should replace {\b0\i0\scaps0\f3 \\vspace} by {\b0\i0\scaps0\f3 \\vspace*}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 We now describe certain features of {\plain L{\up6\fs19 A}T{\dn7 E}X} relating to blank spaces and paragraph indentation which will improve the appearance of the final document. Experienced users of {\plain L{\up6\fs19 A}T{\dn7 E}X} will improve the appearance of their documents if they bear these remarks in mind.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 First note that, as a general rule, you should never put a blank space after a left parenthesis or before a right parenthesis. If you were to put a blank space in these places, then you run the risk that {\plain L{\up6\fs19 A}T{\dn7 E}X} might start a new line immediately after the left parenthesis or before the right parenthesis, leaving the parenthesis marooned at the beginning or end of a line.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} has its own rules for deciding the lengths of blank spaces. For instance, {\plain L{\up6\fs19 A}T{\dn7 E}X} will put an extra amount of space after a full stop if it considers that the full stop marks the end of a sentence.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The rule adopted by {\plain L{\up6\fs19 A}T{\dn7 E}X} is to regard a period (full stop) as the end of a sentence if it is preceded by a lowercase letter. If the period is preceded by an uppercase letter then {\plain L{\up6\fs19 A}T{\dn7 E}X} assumes that it is not a full stop but follows the initials of somebody\rquote s name.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 This works very well in most cases. However {\plain L{\up6\fs19 A}T{\dn7 E}X} occasionally gets things wrong. This happens with a number of common abbreviations (as in \lquote Mr. Smith\rquote  or in \lquote etc.\rquote ), and, in particular, in the names of journals given in abbreviated form (e.g., \lquote Proc. Amer. Math. Soc.\rquote ). The way to overcome this problem is to put a backslash before the blank space in question. Thus we should type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Mr.\\ Smith
\par etc.\\ and
\par Proc.\\ Amer.\\ Math.\\ Soc.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} determines itself how to break up a paragraph into lines, and will occasionally hyphenate long words where this is desirable. However it is sometimes necessary to tell {\plain L{\up6\fs19 A}T{\dn7 E}X} not to break at a particular blank space. The special character used for this purpose is {\b0\i0\scaps0\f3 ~}. It represents a blank space at which {\plain L{\up6\fs19 A}T{\dn7 E}X} is not allowed to break between lines. It is often desirable to use {\b0\i0\scaps0\f3 ~} in names where the forenames are represented by initials. Thus to obtain \lquote W. R. Hamilton\rquote  it is best to type {\b0\i0\scaps0\f3 W.~R.~Hamilton}. It is also desirable in phrases like \lquote Example 7\rquote  and \lquote the length\~{{\i l}} of the rod\rquote , obtained by typing {\b0\i0\scaps0\f3 Example~7} and {\b0\i0\scaps0\f3 the length~$l$ of the rod.}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 {\plain L{\up6\fs19 A}T{\dn7 E}X} will automatically indent paragraphs (with the exception of the first paragraph of a new section). One can prevent {\plain L{\up6\fs19 A}T{\dn7 E}X} from indenting a paragraph though by beginning the paragraph with the control sequence {\b0\i0\scaps0\f3 \\noindent}. Thus one obtains {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 This is the beginning of a paragraph which is not indented in the usual way. This has been achieved by placing an appropriate control sequence at the beginning of the paragraph. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\noindent
\par This is the beginning of a paragraph which is not
\par indented in the usual way.  This has been achieved
\par by placing an appropriate control sequence at the
\par beginning of the paragraph.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Conversely, the control sequence {\b0\i0\scaps0\f3 \\indent} forces {\plain L{\up6\fs19 A}T{\dn7 E}X} to indent the paragraph.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 4.2  Lists\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 {\plain L{\up6\fs19 A}T{\dn7 E}X} provides the following list environments: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\b0\i0\scaps0\f3 enumerate} for numbered lists, \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\b0\i0\scaps0\f3 itemize} for un-numbered lists, \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 \bullet\tab
{\b0\i0\scaps0\f3 description} for description lists \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Numbered lists are produced using {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{enumerate\} ... \\end\{enumerate\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 The items in the list should be enclosed between {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\begin\{enumerate\}} and {\b0\i0\scaps0\f3 \\end\{enumerate\}} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 and should each be preceded by the control sequence {\b0\i0\scaps0\f3 \\item} (which will automatically generate the number labelling the item). For example, the text {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 A {\i metric space} {({\i X},{\i d})} consists of a set\~{{\i X}} on which is defined a {\i distance function} which assigns to each pair of points of {{\i X}} a distance between them, and which satisfies the following four axioms: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 1.\tab
{{\i d}({\i x},{\i y}){\u8805*}0} for all points {{\i x}} and {{\i y}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 2.\tab
{{\i d}({\i x},{\i y})={\i d}({\i y},{\i x})} for all points {{\i x}} and {{\i y}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 3.\tab
{{\i d}({\i x},{\i z}){\u8804*}{\i d}({\i x},{\i y})+{\i d}({\i y},{\i z})} for all points {{\i x}}, {{\i y}} and {{\i z}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 4.\tab
{{\i d}({\i x},{\i y})=0} if and only if the points {{\i x}} and {{\i y}} coincide. \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 is generated by {\plain L{\up6\fs19 A}T{\dn7 E}X} from the following input: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par A \\emph\{metric space\} $(X,d)$ consists of a set~$X$ on
\par which is defined a \\emph\{distance function\} which assigns
\par to each pair of points of $X$ a distance between them,
\par and which satisfies the following four axioms:
\par \\begin\{enumerate\}
\par \\item
\par $d(x,y) \\geq 0$ for all points $x$ and $y$ of $X$;
\par \\item
\par $d(x,y) = d(y,x)$ for all points $x$ and $y$ of $X$;
\par \\item
\par $d(x,z) \\leq d(x,y) + d(y,z)$ for all points $x$, $y$
\par and $z$ of $X$;
\par \\item
\par $d(x,y) = 0$ if and only if the points $x$ and $y$
\par coincide.
\par \\end\{enumerate\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Un-numbered lists are produced using {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{itemize\} ... \\end\{itemize\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 If we replace {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\begin\{enumerate\}} and {\b0\i0\scaps0\f3 \\end\{enumerate\}} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 in the above input by {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\begin\{itemize\}} and {\b0\i0\scaps0\f3 \\end\{itemize\}} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 respectively, {\plain L{\up6\fs19 A}T{\dn7 E}X} generates an itemized list in which each item is preceeded by a \lquote bullet\rquote : {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 A {\i metric space} {({\i X},{\i d})} consists of a set\~{{\i X}} on which is defined a {\i distance function} which assigns to each pair of points of {{\i X}} a distance between them, and which satisfies the following four axioms: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 \bullet\tab
{{\i d}({\i x},{\i y}){\u8805*}0} for all points {{\i x}} and {{\i y}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 \bullet\tab
{{\i d}({\i x},{\i y})={\i d}({\i y},{\i x})} for all points {{\i x}} and {{\i y}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 \bullet\tab
{{\i d}({\i x},{\i z}){\u8804*}{\i d}({\i x},{\i y})+{\i d}({\i y},{\i z})} for all points {{\i x}}, {{\i y}} and {{\i z}} of {{\i X}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 \bullet\tab
{{\i d}({\i x},{\i y})=0} if and only if the points {{\i x}} and {{\i y}} coincide. \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 Description lists (for glossaries etc.) are produced using {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{description\} ... \\end\{description\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 The items in the list should be enclosed between {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\begin\{description\}} and {\b0\i0\scaps0\f3 \\end\{description\}} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 and should each be preceded by {\b0\i0\scaps0\f3 \\item[}{\i label}{\b0\i0\scaps0\f3 ]}, where {\i label} is the label to be assigned to each item. For example, the text {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 We now list the definitions of {\i open ball}, {\i open set} and {\i closed set} in a metric space. {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 {\b open ball} The {\i open ball} of radius\~{{\i r}} about any point\~{{\i x}} is the set of all points of the metric space whose distance from {{\i x}} is strictly less than {{\i r}}; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 {\b open set} A subset of a metric space is an {\i open set} if, given any point of the set, some open ball of sufficiently small radius about that point is contained wholly within the set; \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li1112\ri512\fi-300 {\b closed set} A subset of a metric space is a {\i closed set} if its complement is an open set. \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 is generated by {\plain L{\up6\fs19 A}T{\dn7 E}X} from the following input: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par We now list the definitions of \\emph\{open ball\},
\par \\emph\{open set\} and \\emph\{closed set\} in a metric space.
\par \\begin\{description\}
\par \\item[open ball]
\par The \\emph\{open ball\} of radius~$r$ about any point~$x$
\par is the set of all points of the metric space whose
\par distance from $x$ is strictly less than $r$;
\par \\item[open set]
\par A subset of a metric space is an \\emph\{open set\} if,
\par given any point of the set, some open ball of
\par sufficiently small radius about that point is contained
\par wholly within the set;
\par \\item[closed set]
\par A subset of a metric space is a \\emph\{closed set\} if its
\par complement is an open set.
\par \\end\{description\}
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 4.3  Displayed Quotations\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Displayed quotations can be embedded in text using the {\b quote} and {\b quotation} environments {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{quote\} ... \\end\{quote\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 and {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\begin\{quotation\} ... \\end\{quotation\}.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 The {\b quote} environment is recommended for short quotations: the whole quotation is indended in the {\b quote} environment, but the first lines of individual paragraphs are not further indented. The input file {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par Isaac Newton discovered the basic techiques of
\par the differential and integral calculus, and
\par applied them in the study of many problems
\par in mathematical physics.  His main mathematical
\par works are the \\emph\{Principia\} and the \\emph\{Optics\}.
\par He summed up his own estimate of his work as follows:
\par \\begin\{quote\}
\par I do not know what I may appear to the world; but to
\par myself I seem to have been only like a boy, playing
\par on the sea-shore, and diverting myself, in now and
\par then finding a smoother pebble, or a prettier shell
\par than ordinary, whilst the great ocean of truth lay
\par all undiscovered before me.
\par \\end\{quote\}
\par In later years Newton became embroiled in a bitter
\par priority dispute with Leibniz over the discovery
\par of the basic techniques of calculus.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 is typeset by {\plain L{\up6\fs19 A}T{\dn7 E}X} as follows: {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 Isaac Newton discovered the basic techiques of the differential and integral calculus, and applied them in the study of many problems in mathematical physics. His main mathematical works are the {\i Principia} and the {\i Optics}. He summed up his own estimate of his work as follows: {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li1024\ri1024\fi360 I do not know what I may appear to the world; but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble, or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 In later years Newton became embroiled in a bitter priority dispute with Leibniz over the discovery of the basic techniques of calculus. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 For longer quotations one may use the {\b quotation} environment: the whole quotation is indented, and the openings of paragraphs are then further indented in the normal fashion.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 4.4  Tables\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Tables can be produced in {\plain L{\up6\fs19 A}T{\dn7 E}X} using the {\b tabular} environment. For example, the text {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The first five International Congresses of Mathematicians were held in the following cities: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li1024\ri1024\fi0 \par
{\trowd\cellx2740\cellx5744\cellx6900
{\pard\intbl\ql {Chicago}\cell}
{\pard\intbl\ql {U.S.A.}\cell}
{\pard\intbl\ql {1893}\cell}
\row}
{\trowd\cellx2740\cellx5744\cellx6900
{\pard\intbl\ql {Z\'fcrich}\cell}
{\pard\intbl\ql {Switzerland}\cell}
{\pard\intbl\ql {1897}\cell}
\row}
{\trowd\cellx2740\cellx5744\cellx6900
{\pard\intbl\ql {Paris}\cell}
{\pard\intbl\ql {France}\cell}
{\pard\intbl\ql {1900}\cell}
\row}
{\trowd\cellx2740\cellx5744\cellx6900
{\pard\intbl\ql {Heidelberg}\cell}
{\pard\intbl\ql {Germany}\cell}
{\pard\intbl\ql {1904}\cell}
\row}
{\trowd\cellx2740\cellx5744\cellx6900
{\pard\intbl\ql {Rome}\cell}
{\pard\intbl\ql {Italy}\cell}
{\pard\intbl\ql {1908}\cell}
\row}
} \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 is produced in {\plain L{\up6\fs19 A}T{\dn7 E}X}using the following input file: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The first five International Congresses of Mathematicians
\par were held in the following cities:
\par \\begin\{quote\}
\par \\begin\{tabular\}\{lll\}
\par Chicago&U.S.A.&1893\\\\
\par Z\\"\{u\}rich&Switzerland&1897\\\\
\par Paris&France&1900\\\\
\par Heidelberg&Germany&1904\\\\
\par Rome&Italy&1908
\par \\end\{tabular\}
\par \\end\{quote\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 The {\b0\i0\scaps0\f3 \\begin\{tabular\}} command must be followed by a string of characters enclosed within braces which specifies the format of the table. In the above example, the string {\b0\i0\scaps0\f3 \{lll\}} is a format specification for a table with three columns of left-justified text. Within the body of the table the ampersand character\~{\b0\i0\scaps0\f3 &} is used to separate columns of text within each row, and the double backslash\~{\b0\i0\scaps0\f3 \\\\} is used to separate the rows of the table.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The next example shows how to obtain a table with vertical and horizontal lines. The text {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 The group of permutations of a set of {{\i n}}\~elements has order {{\i n}!}, where {{\i n}!}, the factorial of {{\i n}}, is the product of all integers between {1} and {{\i n}}. The following table lists the values of the factorial of each integer\~{{\i n}} between 1 and 10: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li1024\ri1024\fi0 \par
{\trowd\clbrdrl\brdrs\clbrdrt\brdrs\clbrdrb\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrt\brdrs\clbrdrb\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr { {{\i n}}}\cell}
{\pard\intbl\qr {{{\i n}!}}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr { 1}\cell}
{\pard\intbl\qr {1}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {2}\cell}
{\pard\intbl\qr {2}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {3}\cell}
{\pard\intbl\qr {6}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {4}\cell}
{\pard\intbl\qr {24}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {5}\cell}
{\pard\intbl\qr {120}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {6}\cell}
{\pard\intbl\qr {720}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {7}\cell}
{\pard\intbl\qr {5040}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {8}\cell}
{\pard\intbl\qr {40320}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {9}\cell}
{\pard\intbl\qr {362880}\cell}
\row}
{\trowd\clbrdrl\brdrs\clbrdrb\brdrs\clbrdrr\brdrs\cellx4041\clbrdrl\brdrs\clbrdrb\brdrs\clbrdrr\brdrs\cellx6899
{\pard\intbl\qr {10}\cell}
{\pard\intbl\qr {3628800}\cell}
\row}
} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 Note how rapidly the value of {{\i n}!} increases with {{\i n}}. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 is produced in {\plain L{\up6\fs19 A}T{\dn7 E}X}using the following input file: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par The group of permutations of a set of $n$~elements has
\par order $n!$, where $n!$, the factorial of $n$, is the
\par product of all integers between $1$ and $n$.  The
\par following table lists the values of the factorial of each
\par integer~$n$ between 1 and 10:
\par \\begin\{quote\}
\par \\begin\{tabular\}\{|r|r|\}
\par \\hline
\par $n$&$n!$\\\\
\par \\hline
\par 1&1\\\\
\par 2&2\\\\
\par 3&6\\\\
\par 4&24\\\\
\par 5&120\\\\
\par 6&720\\\\
\par 7&5040\\\\
\par 8&40320\\\\
\par 9&362880\\\\
\par 10&3628800\\\\
\par \\hline
\par \\end\{tabular\}
\par \\end\{quote\}
\par Note how rapidly the value of $n!$ increases with $n$.
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 In this example the format specification {\b0\i0\scaps0\f3 \{|r|r|\}} after {\b0\i0\scaps0\f3 \\begin\{tabular\}} specifies that the table should consist of two columns of right-justified text, with vertical lines to the left and to the right of the table, and between columns.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Within the body of the table, the command {\b0\i0\scaps0\f3 \\hline} produces a horizontal line; this command can only be placed between the format specification and the body of the table (to produce a line along the top of the table) or immediately after a row separator (to produce a horizontal line between rows or at the bottom of the table).\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 In a {\b tabular} environment, the format specification after {\b0\i0\scaps0\f3 \\begin\{tabular\}} should consist of one or more of the following, enclosed within braces {\b0\i0\scaps0\f3 \{} and {\b0\i0\scaps0\f3 \}}: {\par
{\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 \par
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 l}}\cell}
{\pard\intbl\ql {specifies a column of left-justified text}\cell}
\row}
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 c}}\cell}
{\pard\intbl\ql {specifies a column of centred text}\cell}
\row}
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 r}}\cell}
{\pard\intbl\ql {specifies a column of right-justified text}\cell}
\row}
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 p\{}{\i width}{\b0\i0\scaps0\f3 \}}}\cell}
{\pard\intbl\ql {specifies a left-justified column of the given width}\cell}
\row}
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 |}}\cell}
{\pard\intbl\ql {inserts a vertical line between columns}\cell}
\row}
{\trowd\cellx2413\cellx6899
{\pard\intbl\ql {{\b0\i0\scaps0\f3 @\{}{\i text}{\b0\i0\scaps0\f3 \}}}\cell}
{\pard\intbl\ql {inserts the given {\i text} between columns}\cell}
\row}
} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 A string {\i str} of characters in the format specification can be repeated {\i num} times using the construction {\b0\i0\scaps0\f3 *\{}{\i num}{\b0\i0\scaps0\f3 \}\{}{\i str}{\b0\i0\scaps0\f3 \}}. For example, a table with 15 columns of right-justified text enclosed within vertical lines can be produced using the format specification {\b0\i0\scaps0\f3 \{|*\{15\}\{r|\}\}}.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 If additional vertical space is required between rows of the table, then this can be produced by specifying the amount of space within square brackets after {\b0\i0\scaps0\f3 \\\\}. For example, on would use {\b0\i0\scaps0\f3 \\\\[6pt]} to separate two rows of the table by 6\~points of blank space.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 A horizontal line in a table from column\~{{\i i}} to column\~{{\i j}} inclusive can be produced using {\b0\i0\scaps0\f3 \\cline\{}{{\i i}}{\b0\i0\scaps0\f3 -}{{\i j}}{\b0\i0\scaps0\f3 \}}. For example {\b0\i0\scaps0\f3 \\cline\{3-5\}} produces a horizontal line spanning columns 3, 4 and 5 of some table.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 A command of the form {\b0\i0\scaps0\f3 \\multicolumn\{}{\i num}{\b0\i0\scaps0\f3 \}\{}{\i fmt}{\b0\i0\scaps0\f3 \}\{}{\i text}{\b0\i0\scaps0\f3 \}} can be used within the body of a table to produce an entry spanning several columns. Here {\i num} specifies the number of columns to be spanned, {\i fmt} specifies the format for the entry (e.g., {\b0\i0\scaps0\f3 l} if the entry is to be left-justified entry, or {\b0\i0\scaps0\f3 c} if the entry is to be centred), and {\i text} is the text of the entry. For example, to span three columns of a table with the words \lquote Year of Entry\rquote  (centred with respect to the three columns), one would use {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\multicolumn\{3\}\{c\}\{Year of entry\}
\par \par
}}\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb240 \fi0 4.5  The Preamble of the {\plain L{\up6\fs19 A}T{\dn7 E}X} Input file\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 We describe the options available in {\plain L{\up6\fs19 A}T{\dn7 E}X} for specifying the overall style of a document.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 A {\plain L{\up6\fs19 A}T{\dn7 E}X} document should begin with a {\b0\i0\scaps0\f3 \\documentclass} command and any text to be printed must be included between {\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \li512\ri512\fi360 {\b0\i0\scaps0\f3 \\begin\{document\}} and {\b0\i0\scaps0\f3 \\end\{document\}} \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 commands. The {\b0\i0\scaps0\f3 \\begin\{document\}} command is sometimes preceded by commands that set the page-style and set up user-defined control sequences.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Here is a typical {\plain L{\up6\fs19 A}T{\dn7 E}X} input file: {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper,12pt]\{article\}
\par \\begin\{document\}
\par 
\par This is the first paragraph of a typical document. It is
\par produced in a `12~point' size. A \\textit\{point\} is a unit
\par of length used by printers. One point is approximately
\par $1/72$~inch. In a `12~point' font the height of the
\par parentheses is 12~points (i.e. about $1/6$~inch) and the
\par letter~`m' is about 12 points long. 
\par 
\par This is the second paragraph of the document. There are
\par also `10 point' and `11 point' styles available in \\LaTeX.
\par The required size is specified in the `documentstyle'
\par command. If no such size is specified then the 10~point
\par size is assumed.
\par 
\par \\end\{document\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi360 The syntax of the {\b0\i0\scaps0\f3 \\documentclass} command is as follows. The command begins with {\b0\i0\scaps0\f3 \\documentclass} and ends with the names of one of the available styles, enclosed in curly brackets. The available styles are {\b0\i0\scaps0\f3 article}, {\b0\i0\scaps0\f3 report}, {\b0\i0\scaps0\f3 book} and {\b0\i0\scaps0\f3 letter}. Between the \ldblquote {\b0\i0\scaps0\f3 \\documentclass}\rdblquote  and the name of the document style, one may place a list of {\i options}. These options are separated by commas and the list of options is enclosed in square brackets (as in the above example). The options available (which are usually the names of certain \lquote style files\rquote ) include the following: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b 11pt} Specifies a size of type known as {\i eleven-point}, which is ten percent larger than the ten-point type normally used. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b 12pt} Specifies a twelve-point type size, which is twenty percent larger than ten-point. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b twocolumn} Produces two-column output. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b a4paper} This ensures that the page is appropriately positioned on A4 size paper. \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi360 Typing simply {\b0\i0\scaps0\f3 \\documentclass\{article\}} will produce a document in ten-point type size. However the printed output will not be nicely positioned on A4 paper, since the default size is intended for a different (American) paper size.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 Pages will be automatically numbered at the bottom of the page, unless you specify otherwise. This can be done using the {\b0\i0\scaps0\f3 \\pagestyle} command. This command should come after the {\b0\i0\scaps0\f3 \\documentclass} command and before the {\b0\i0\scaps0\f3 \\begin\{document\}} command. This command has the syntax {\b0\i0\scaps0\f3 \\pagestyle\{}{\i option}{\b0\i0\scaps0\f3 \}}, where the {\i option} is one of the following: {\par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b plain} The page number is at the foot of the page. This is the default page style for the {\b0\i0\scaps0\f3 article} and {\b0\i0\scaps0\f3 report} document styles. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b empty} No page number is printed. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b headings} The page number (and any other information determined by the document style) is put at the top of the page. \par
\pard\plain\s46\ql\fi-283\li283\lin283\sb0\sa120\widctlpar\tql\tx283\f0\fs24\sl240\slmult1 \sb50 \li600\fi-300 {\b myheadings} Similar to the {\b headings} pagestyle, except that the material to go at the top of the page is determined by {\b0\i0\scaps0\f3 \\markboth} and {\b0\i0\scaps0\f3 \\markright} commands (see the {\plain L{\up6\fs19 A}T{\dn7 E}X} manual). \par
}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 For example, the input file {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\documentclass[a4paper]\{article\}
\par \\pagestyle\{empty\}
\par \\begin\{document\}
\par The main body of the document is placed here.
\par \\end\{document\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 produces a document without page numbers, using the standard ten-point type size.\par
\pard\plain\s4\ql\sb240\sa120\keepn\f0\b\fs32\sl240\slmult1 \sb120 \fi0 4.6  Defining your own Control Sequences in {\plain L{\up6\fs19 A}T{\dn7 E}X}\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb60 \fi0 Suppose that we are producing a paper that makes frequent use of some mathematical expression. For example, suppose that integrals like \par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (\u8722?{\u8734*},+{\u8734*}, ){\i f}({\i x}){\charscalex50\~}{\i d}{\i x}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 occur frequently throughout the text. This formula is obtained by typing {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\int_\{-\\infty\}^\{+\\infty\} f(x)\\,dx.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 It would be nice if we could type {\b0\i0\scaps0\f3 \\inftyint} (say) to obtain the integral sign at the beginning. This can be done using {\b0\i0\scaps0\f3 \\newcommand}. What we do is to place a line with the command {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\newcommand\{\\inftyint\}\{\\int_\{-\\infty\}^\{+\\infty\}\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 near the beginning of the input file (e.g., after the {\b0\i0\scaps0\f3 \\documentclass} command but before the {\b0\i0\scaps0\f3 \\begin\{document\}} command). Then we only have to type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\inftyint f(x)\\,dx.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 to obtain the above formula.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 We can modify this procedure slightly. Suppose that we we defined a new control sequence {\b0\i0\scaps0\f3 \\intwrtx} by putting the line {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\newcommand\{\\intwrtx\}[1]\{\\int_\{-\\infty\}^\{+\\infty\} #1 \\,dx\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 at the beginning of the input file. If we then type the line {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\intwrtx\{f(x)\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 then we obtain\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (\u8722?{\u8734*},+{\u8734*}, ){\i f}({\i x}){\charscalex50\~}{\i d}{\i x}.}}{\fldrslt }}
\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi0 What has happened is that the expression in curly brackets after {\b0\i0\scaps0\f3 \\intwrtx} has been substituted in the expression defining {\b0\i0\scaps0\f3 \\intwrtx}, replacing the {\b0\i0\scaps0\f3 #1} in that expression.\par
\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \fi360 The number 1 inside square brackets in the {\b0\i0\scaps0\f3 \\newcommand} line defining {\b0\i0\scaps0\f3 \\intwrtx} indicates to {\plain L{\up6\fs19 A}T{\dn7 E}X} that it is to expect one expression (in curly brackets) after {\b0\i0\scaps0\f3 \\intwrtx} to substitute for {\b0\i0\scaps0\f3 #1} in the definition of {\b0\i0\scaps0\f3 \\intwrtx}. If we defined a control sequence {\b0\i0\scaps0\f3 \\intwrt} by {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\newcommand\{\\intwrt\}[2]\{\\int_\{-\\infty\}^\{+\\infty\} #2 \\,d #1\}
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 then it would expect two expressions to substitute in for {\b0\i0\scaps0\f3 #1} and {\b0\i0\scaps0\f3 #2} in the definition of {\b0\i0\scaps0\f3 \\intwrt}. Thus if we then type {\par
{\pard\plain\s38\ql\widctlpar\f3\fs24\sl240\slmult1 \sb60 \li512\ri512\fi0 
\par \\[ \\intwrt\{y\}\{f(y)\}.\\] 
\par \par
}}\pard\plain\s0\qj\widctlpar\f0\fs24\sl240\slmult1 \sb120 \fi0 we obtain\par
\pard\plain\s11\ql\sb120\sa120\keep\widctlpar\f0\tqc\tx3450\sl240\slmult1 \fi0 \tab
{\field{\*\fldinst{ EQ  \\i (\u8722?{\u8734*},+{\u8734*}, ){\i f}({\i y}){\charscalex50\~}{\i d}{\i y}.}}{\fldrslt }}
\par
}}
}}}