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\begin{document}
\begin{figure}[t] % Quotation from Knuth
\font\ssq=cmss10
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\raggedleft
{\ssi Word-smithing is a much greater percentage\\
of what I am supposed to be doing in life \\
than I would ever have thought.} \\
{\ssq DONALD KNUTH \cite[\rm p. 54]{knuth}}
\end{figure}
\pagestyle{myheadings}
\markboth{\sc MIT Undergraduate Journal of Mathematics}{\sc
Writing a Math Phase Two Paper}
\title{Writing a Math Phase Two Paper}
\author{{\sc Steven L. Kleiman} \\
with the collaboration of {\sc Glenn P. Tesler}}
\def\today{\copyright
February 2, 2005} %% Disable to get the date of compilation
%% \date{\copyright\today}
\maketitle
\begin{abstract}
We discuss the kind of writing that's appropriate in a paper submitted
to the Math department to complete Phase Two of MIT's writing
requirement. First, we review the general purpose of the requirement
and the specific way of completing it for the Math department. Then we
consider the writing itself: the organization into sections, the use
of language, and the presentation of mathematics. Finally, we give a
short example of mathematical writing. \end{abstract}
\section{Introduction}\label{sec-intro}
MIT established the writing requirement to ensure that its graduates can
write both a good general essay and a good technical report.
Correspondingly, the requirement has two phases, which engage students
at the beginning and toward the end of their undergraduate careers. The
requirement is governed by an institute committee, the Committee on the
Writing Requirement (CWR). The requirement is administered by the
Office of the Dean of Students and Undergraduate Education, which works
in cooperation with the individual departments on Phase Two. The
general information given here about the requirement is taken from the
MIT {\it Bulletin\/} and the CWR's brochure \cite{cwr}, which are the
official sources.
To complete Phase One, students must achieve a suitable score on the
College Board Achievement Test or Advanced Placement Examination, pass
the Freshman Essay Evaluation, pass an appropriate writing subject in
Course 21 and be certified by the instructor, or write a satisfactory
five page paper for any MIT subject, Wellesley exchange subject, or UROP
activity. In level, format, and style, a Phase One paper should be like
a magazine article for an informed, but general, readership. Papers are
judged on their logical structure, language and tone, technical
accuracy, and mechanics (grammar, spelling, and punctuation) by the
instructor of the subject and by evaluators for the Office of the Dean
of Students and Undergraduate Education. A paper judged not acceptable
may be revised and resubmitted twice. Students must complete Phase One
by the middle of their third semester at the Institute.
To complete Phase Two, students must receive a grade of B or better for
the quality of writing in a cooperative subject approved by the
student's major department, receive a grade of B or better in one of
several advanced classes in technical writing, or write a satisfactory
ten-page paper for any MIT subject or UROP activity approved by the
major department. A student with two majors needs only complete the
requirement in one department. In level, format, and style, a Phase
Two paper should be like a formal professional report. Thus a term
paper or laboratory report may have to be reworked substantially before
it is acceptable as a Phase Two paper. The paper is judged by its
supervisor primarily for the technical content and by departmental
evaluators primarily for the quality of the writing. Students must
complete Phase Two by the end of registration day of their last
semester; otherwise, they must petition their departments and the CWR.
Petitions for permission to enroll in a writing subject are routinely
approved; petitions to submit a late paper are approved only when there
are exceptional circumstances.
In the Department of Mathematics, there is no cooperative subject, and
most students write a paper to satisfy Phase Two. These students may
also receive three units of credit by signing up for 18.098, Independent
Activities. Each year in the spring, the department collects the papers,
and publishes them here in the {\it MIT Undergraduate Journal of
Mathematics.}
A Phase II paper normally begins as a term paper for a mathematics
class, but every paper must have an MIT supervisor and include some
technical mathematics. When the student and the supervisor feel the
paper is ready, the student picks up a cover sheet, which is available
in the Undergraduate Mathematics Office, Room 2--108. The student fills
out the top, and gives the sheet to the supervisor, who must vouch for
the paper's technical accuracy. The student then submits the paper and
the cover sheet to the departmental coordinator. The paper must be
submitted by the start of IAP if the student intends to graduate the
following June.
After a paper is submitted, the math department's coordinator reads it
for the quality of the writing, and determines whether or not the paper
is acceptable as it stands. If the paper needs improvement (most do),
then the coordinator and the department's Writing TA discuss the paper.
The TA contacts the student and sets up an appointment to discuss the
areas requiring further work. The student submits further revisions to
the TA, and when the paper is ready, it is resubmitted to the
coordinator. Often, the coordinator works directly with the student.
Thus, not only is the paper improved, but, more importantly, the student
learns how to write better. The process is tutorial.
This paper is a primer on mathematical writing, especially the
writing of short papers. Indeed, this paper itself is intended to be a
model of format, language, and style. Mathematical writing is
primarily a craft, which any student of mathematics can learn. Its aim
is to inform efficiently. Its basic principles are discussed and
illustrated here. Some of these principles are simple matters of
common sense; others are conventions that have evolved from experience.
None need be followed slavishly, but none should be broken
thoughtlessly. When one is broken, the break may stand out like a sore
thumb---just as unconventional spelling does. However, the writing
itself should fade into the background, leaving the information to be
conveyed out front. Abiding by these principles will not cramp
anyone's style; there's plenty of room for individual variation. The
various principles themselves are discussed more fully in a number of
works, including the following works on which this primer is based:
Alley's down-to-earth book \cite{alley}, Flanders' article
\cite{flanders} and Gillman's manual \cite{gillman} for authors of
articles for MAA journals, the notes \cite{knuth} to Knuth's Stanford
course on mathematical writing, and Munkres' brief manual of style
\cite{munkres}.
In Section~\ref{sec-org}, we discuss the normal way a short mathematical
paper is broken into sections. We consider the purpose and content of
the individual sections: the abstract, the introduction, the several
sections of the main discussion, the conclusion (which is rare in a
mathematical work), the appendix, and the list of references. In
Section~\ref{sec-lang}, we deal with ``language,'' that is, the choice
of words and symbols, and the structuring of sentences and
paragraphs. We consider seven goals of language: precision, clarity,
familiarity, forthrightness, conciseness, fluidity, and imagery. We
discuss the meaning of these goals and how best to meet them.
Sections~\ref{sec-org} and~\ref{sec-lang} are based mainly on Alley's
book \cite{alley}. In Section~\ref{sec-math}, we deal with a number of
special problems that arise in writing mathematics, such as the
treatment of formulas, the presentation of theorems and proofs, and the
use of symbols. The material is drawn from all five sources cited
above. In Section~\ref{sec-ex}, we give an illustrative sample of
mathematical writing. We treat the two fundamental theorems of calculus,
for the most part paraphrasing the treatment in Apostol's book
\cite[pp. 202--204]{apostol}; we state and prove the theorems, and
explain their significance. Finally, in the appendix, we deal with the
use of such terms as lemma, proposition, and definition, which are
common in treatments of advanced mathematics.
\section{Organization}\label{sec-org}
Most short technical papers are divided up into about a half-dozen
sections, which are numbered and titled. (The pages too should be
numbered for easy reference.) Most papers have an abstract, an
introduction, a number of sections of discussion, and a list of
references, but no formal table of contents or index. On occasion,
papers have appendices, which give special detailed information or
provide necessary general background to secondary audiences. Normally,
the abstract is three-to-six lines long; the list of references has
three-to-nine entries; and each remaining section fills one-to-three
pages.
In some fields, papers routinely have a conclusion. This section is not
present simply to balance the introduction and to close the
paper. Rather, the conclusion discusses the results from an overall
perspective, brings together the loose ends, and makes recommendations
for further research. In mathematics, these issues are almost always
treated in the introduction, where they reach more readers; so a
conclusion is rare.
Sectioning involves more than merely dividing up the material; you have
to decide what to put where, what to leave out, and what to emphasize.
If you make the wrong decisions, you will lose your readers. There is no
simple formula for deciding, because the decisions depend heavily on the
subject and the audience. However, you must structure your paper in a
way that is easy for your readers to follow, and you must emphasize the
key results.
The {\it title} is very important. If it is unclear or misleading, then
it will not attract all the intended readers. A strong title identifies
the general area of the subject and its most distinctive features. A
strong title contains no secondary details and no symbols. A strong
title is {\it concise}\,---\,rather short and to the point.
The {\it abstract} is the most important section. First it identifies
the subject; it repeats words and phrases from the title to corroborate
a reader's first impression, and it gives details that didn't fit into
the title. Then it lays out the central issues, and summarizes the
discussion to come. The abstract includes {\it no general background\/}
material and preferably no symbols. It just summarizes the contents.
The abstract allows readers to decide quickly about reading on.
Although many will decide to stop there, the potentially interested will
continue. The goal is not to entice all, but to inform the interested
efficiently. Remember, readers are busy. They have to decide quickly
whether your paper is worth their time. They have to decide whether the
subject matter is of interest to them, and whether the presentation will
bog them down. A well-written abstract will increase the readership.
The {\it introduction} is where readers settle into the ``story,'' and
often make the final decision about reading the whole paper. Start
strong; don't waste words or time. Your readers have just read your
title and abstract, and they've gained a general idea of your subject
and treatment. However, they are probably still wondering what exactly
your subject is and how you'll present it. A strong introduction
answers these questions with clarity and precision, but in nontechnical
terms. It identifies the subject precisely, and instills interest in it
by giving details that did not fit into the title or abstract, such as
how the subject arose and where it is headed, how it relates to other
subjects and why it is important. A strong introduction touches on all
the significant points, and no more. A strong introduction gives enough
background material for understanding the paper as a whole, and no
more. Put background material pertinent to a particular section in that
section, weaving it unobtrusively into the text. A strong introduction
discusses the relevant literature, citing a good survey or two.
Finally, a strong introduction describes the organization of the paper,
making explicit references to the section numbers. It summarizes the
contents in more detail than the abstract, and it says what can be found
in each section. It gives a {\it road map}, which indicates the route to
be followed and the prominent features along the way. This road map is
essentially a table of contents in a paragraph of prose. It is always
placed at the end of the introduction to ease the transition into the
next section.
The {\it body} discusses the various aspects of the subject
individually. In writing the body, your hardest job is developing a
strategy for parceling out the information. Every paper requires its own
strategy, which must be worked out by trial and error. There are,
however, a few guidelines. First, present the material in small
digestible portions. Second, don't jump haphazardly from one
detail to another, and don't illogically make some details specific and
others generic. Third, try to follow a sequential path through the
subject. If such a path doesn't exist, simply break the subject down
into logical units, and present them in the order most conducive to
understanding. If the units are independent, then order them according
to their importance to the primary audience.
There are three main reasons for dividing the body into sections:
(1)~the division indicates the strategy of your presentation; (2)~it
allows readers to quickly and easily find the information that interests
them; and (3)~it gives readers restful white space, allowing them to
stop and reflect on what was said. Make the introduction and the several
sections of the body roughly equal in length. When you title a section,
strive for conciseness, precision, and clarity; then readers will have
an easier time jumping to a particular topic. Don't simply insert a
title, as is often done in newspaper articles, to recapture interest;
rather, wind down the discussion in the first section in preparation for
a break, and then restart the discussion in the next section, after the
title. When you refer to Section~3, remember to capitalize the word
``Section''; it is considered a proper name. Don't subsection a short
paper; the breaks would make the flow too choppy.
Accent each main point via stylistic repetition, illustration, or
language. Stylistic repetition is the selective repetition of something
important; for example, you should talk about the important points once
in the abstract, a second time in the introduction, and a third time in
the body. When appropriate, repeat an important point in a figure or
diagram. Finally, accent an important point with a linguistic device:
italics, boldface, or quotation marks; a one-sentence paragraph; or a
short sentence at the end of a long paragraph. In particular, set a
technical term in italics or boldface---or enclose it in quotation marks
if it is only moderately technical---once, at the time it is being
defined. Do not underline when italics or boldface is available. Use
headings such as {\bf Table~1-1}, {\bf Figure~1-2}, and {\bf
Theorem~5-2}, and refer to them as Table~1-1, Figure~1-2, and
Theorem~5-2; note that the references are capitalized and set in
roman. When you employ linguistic devices, be consistent: always use the
same device for the same job.
The list of {\it references} contains bibliographical information about
each source cited. The style of the list is different in technical and
nontechnical writing; so is the style of citation. In fact, there are
several different styles used in technical writing, but they are
relatively minor variations of each other. The style used in this paper
is commonly used in contemporary mathematical writing.
The citation is treated somewhat like a parenthetical remark within a
sentence, but the reason for the citation must be immediately apparent.
Footnotes are not used; neither are the abbreviations ``loc.~cit.,''
``op.~cit.,'' and ``ibid.'' The reference key, trafitionally a numeral,
is enclosed in square brackets. Within the brackets and after the
reference key, place---as a service---specific page numbers, section
numbers, or equation numbers, preceded by a comma; see Gillman's book
\cite[p. 9]{gillman}. The reason for the citation must be
immediately apparent, and governs its placement, for example, after a
mention of an author's name or work. If the citation comes at the end
of a sentence, put the period after the citation, {\it not\/} before the
brackets or inside them. In the list of reference, give the full page
numbers of each article appearing in a journal, a proceedings volume, or
other collection; do not give the numbers of the particular pages cited
in the text.
\section{Language}\label{sec-lang}
In the subject of writing, the word ``language''
means the choice of words and symbols, and their arrangement in
phrases. It means the structuring of sentences and paragraphs, and the
use of examples and analogies. When you write, watch your language.
When it falters, your readers stumble; if they stumble too often,
they'll lose their patience and stop reading. Write, rewrite, then
rewrite again, improving your language as you go; there is no short
cut!
Alley \cite[pp. 25--130]{alley} identifies seven goals of language:
two primary goals---precision and clarity---and five secondary
goals---familiarity, forthrightness, conciseness, fluidity, and
imagery. These goals often reinforce one another. For example,
clarity and forthrightness promote conciseness; precision and
familiarity promote clarity. We will now consider these goals
individually.
Being precise means using the right word. However, finding the right
word can be difficult. Consult a dictionary, not a thesaurus, because
the dictionary explains the differences among words. For example, the
{\it American Heritage Dictionary\/} is a good choice, because it has
many notes on usage. Consult a book on usage, such as {\it Webster's
Dictionary of English Usage}. Always consider a word's connotations
(associated meanings) along with its denotations (explicit meanings);
the wrong connotations can trip up your readers by suggesting
unintended ideas. For example, the word ``adequate'' means enough for
what is required, but it gives you the feeling that there's not quite
enough; its connotation is the exact opposite of its denotation.
Strong writing does not require using synonyms, contrary to popular
belief. Indeed, by repeating a word, you often strengthen the bond
between two thoughts. Moreover, few words are exact synonyms, and
often, using an exact synonym adds nothing to the discussion.
Being precise means giving specific and concrete details. Without the
details, readers stop and wonder needlessly. On the other hand, readers
remember by means of the details. Being precise does not mean giving all
the details, but giving the informative details. Giving the wrong
details or giving the right ones at the wrong time makes the writing
boring and hard to follow. Being specific does not mean eradicating
general statements. General statements are important, particularly in
summaries. However, specific examples, illustrations, and analogies add
meaning to the general statements.
Being clear means using no wrong words. An ambiguous phrase or sentence
will disrupt the continuity and diminish the authority of an entire
section. A common mistake is to use overly complex prose. Don't string
adjectives together, especially if they are really nouns. Many high
quality pure mathematics original research journal article sentences
illustrate this problem.
Keep your sentences simple and to the point. Avoid long subjects. A
sentence in which a lot goes on between the noun and the verb is hard to
read. But a sentence is easy to read when little goes on between the
noun and the verb. Need to express a complex idea? Then use several
short sentences. Readers are thus led to stop and reflect. However,
you do need some longer sentences to keep your writing from sounding
choppy and to provide variety and emphasis.
A pronoun normally refers to the first preceding noun. However,
sometimes it refers broadly to a preceding phrase, topic, or idea. This
should be avoided. Make sure the reference is immediately clear,
especially with ``it,'' ``this,'' and ``which.'' Consider repeating the
antecedent or summarizing it.
It is common to use a plural pronoun such as ``their'' to refer back to
a singular, but indefinite, antecedent such as ``reader.'' This usage is
still considered unacceptable in formal writing; reformulate your
sentence if necessary.
The pronouns ``that'' and ``which'' are not always interchangeable.
Either may be used to introduce a restrictive clause, but use ``that''
ordinarily. Only ``which'' may be used to introduce a descriptive
clause, and the clause must be set off with commas. In their classic
guide to style \cite[p. 47]{SandW}, Strunk and White recommend
``which-hunting.''
Punctuation is used to eliminate ambiguities in language, and to ease
the flow of the text. Learn how to punctuate properly. Develop the
habit of consulting a handbook like {\it The Chicago Manual of Style.}
When punctuation is optional, use it if it promotes clarity, but strive
for consistency through out the paper. Here are a few rules.
Use periods only to end sentences. (A complete sentence within
parentheses should begin with a capital letter and end with a
punctuation mark, unless the sentence is part of another and would end
with a period.) Avoid abbreviations that require periods; for example,
write ``MIT'' instead of ``M.I.T.'' and use ``that is'' instead of
``i.e.''
Always use commas to separate three or more items in a list and to set
off contrasted elements (they often begin with ``but'' or ``not''). Most
of the time, use a comma after an introductory word, phrase, or
clause.
Use colons to introduce lists, explanations, and displays, but not
lemmas, theorems, and corollaries. Do not use colons in continuing
statements: if a statement is stopped at the colon, then the
introductory words should form a complete sentence. For example, don't
write, ``Use colons to introduce: lists, explanations, and displays.''
Use a semicolon to join two sentences to indicate that they are closely
linked in content; however, if you insert a conjunction, not an adverb,
then use a comma.
Use a dash as a comma of extra strength---but use it sparingly---it
carries a hint of emotion. Place closing quotation marks ('') after
commas and periods; it is a matter of appearance, not logic. Enclose
incidental material in parentheses; generally, footnotes and endnotes
are discouraged in technical reports. Don't use the apostrophe to form
the plurals of one or more digits and letters used as nouns, except to
avoid confusion. For example, write this: the early 1970s, many YMCAs,
several PhD's, the $x$'s and $y$'s.
To inform, you must use language familiar to your readers. Define
unfamiliar words, and familiar words used in unfamiliar ways. If the
definition is short, then include it in the same sentence, preceding it
by ``or'' or setting it off by commas or parentheses. If the
definition is complex or technical, then expand it in a sentence or
two. Do not use words like ``capability,'' ``utilize,'' and
``implement''; they offer no precision, clarity, or continuity and
smack of pseudo-intellectualism. Beware of words like ``interface'';
they are precise in some contexts, yet imprecise and pretentious in
others.
Jargon is vocabulary particular to a certain group, and it consists of
abbreviations and slang terms. Jargon is not inherently bad. Indeed,
it is useful in internal memos and reports. However, jargon alienates
external readers and may even mislead them. So beware. Clich\'es are
figurative expressions that have been overused and have taken on
undesirable connotations. Most are imprecise and unclear. Avoid them,
or be laughed at. In addition, avoid numerals because they slow down
the reading. Write numbers out if they can be expressed in one or two
words and are used as adjectives, unless they are accompanied by units,
a percentage sign, or a monetary sign. For instance, write, ``The
equation has two roots,'' and ``One root is 2.'' Don't
begin a sentence with a numeral or a symbol; reformulate the sentence
if necessary.
Be forthright: write in an unhesitating, straightforward, and friendly
style, ridding your language of needless and bewildering formality. Be
wary of awkward and inefficient passive constructions. Often the passive
voice is used simply to avoid the first person. However, the pronoun
``we'' is now generally considered acceptable in contexts where it means
the author and reader together, or less often, the author with the
reader looking on. Still, ``we'' should not be used as a formal
equivalent of ``I,'' and ``I'' should be used rarely, if at all.
For instance, don't write, ``By solving the equation, it is found that
the roots are real.'' Instead write, ``Solving the equation, we find the
roots are real,'' or ``Solving the equation yields {\it real\/} roots.''
It is acceptable, but less desirable, to write, ``Solving the equation,
one finds the roots are real''. The personal pronoun ``one'' is a sign
of formality; save ``one'' for use as a number. Beware of dangling
participles. It is wrong to write, ``Solving the equation, the roots are
real,'' because ``the roots'' cannot solve the equation.
Concise writing is vigorous; wordy writing is tedious. Conciseness
comes from reducing sentences to their simplest forms. For instance,
don't write, ``In order to find the solution of the equation, we can use
one of two alternative methods.'' Instead, write, ``To solve the
equation, we can use one of two methods,'' thus eliminating empty words
(``in order), reducing fat phrases (``to find the solution of''), and
eliminating needless repetition (``alternative''). If it goes without
saying, don't say it! Concise writing is simple and efficient, thus
beautiful.
The flow of a paper is disturbed by weak transitions between sentences
and paragraphs. To smooth out the flow, start a sentence where the
preceding one left off. Use connective words and phrases. Avoid gaps in
the logic, and give ample details. Don't take needless jumps when
deriving equations. Use parallel wording when discussing parallel
concepts. Don't raise questions implicitly, and leave them unanswered.
Pay attention to the tense, voice, and mode of verbs; prefer the active
present indicative.
Some papers stagnate because they lack variety. The sentences begin
the same way, run the same length, and are of the same type. The
paragraphs have the same length and structure. Don't worry about
varying your sentences and paragraphs at first; wait until you polish
your writing. Remember though, if you have to choose between fluidity
and clarity, then you must choose clarity.
The very structure of a sentence conveys meaning. Readers expect the
stress to lie at the beginning and end. They take a breath at the
beginning, but will run out of breath before the end if the structure
is too complex, for instance, if the subject is too far from the verb.
Most people think and remember images, not abstractions, and images are
clarified by illustrations. Illustrations also provide pauses, so
complex ideas can soak in. Moreover, illustrations can make a paper
more palatable and less forbiding. However, the use of illustrations
can be overdone; it must fit the audience and the subject.
Illustrations cannot stand alone; they must be introduced in the text.
Assign them titles, like Figure 5-1 or Table 5-1, for reference. Assign
them captions that tell, independently of the text, what they are and
how they differ from one another, without being overly specific. In
addition, clearly label the parts of your illustrations: label the axes
of graphs with words, not symbols; identify any unusual symbols of your
diagrams in the text. Don't put too much information into one
illustration, because papers without white space tire readers. For the
same reason, use adequate borders. Smooth the transitions between your
words and pictures. First, match the information in your text and
illustrations. Second, place the illustrations closely after---never
before---their first mention in the text.
\section{Mathematics}\label{sec-math}
Mathematical writing tends to involve many abstract symbols and formal
arguments, and they present special problems. To help you understand
these problems and deal with them in your writing, here are some
comments and guidlines.
Formulas are difficult to read because readers have to stop and work
through the meaning of each term. Don't merely list a sequence of
formulas with no discernible goal, but give a running commentary.
Define terms as they are introduced, state any assumptions about their
validity, and give examples to provide a feeling for them. Similarly,
motivate and explain formal statements. Don't simply call a statement
``important,'' ``interesting,'' or ``remarkable,'' but explain why it is
so.
Display an important formula by centering it on a line by itself, and
give a reference number in the margin if you need to refer to it. Also
display any formula that's more than a quarter of a line long, that
would be broken badly between lines, or that sticks out into the margin.
Punctuate the display with commas, a period, and so forth as you would
if you had not displayed it; see Section 5 for some examples. Keep in
mind that the display is not a figure, but an integral part of the
sentence, and therefore needs punctuation.
Be clear about the status of every assertion; indicate whether it is a
conjecture, the previous theorem, or the next corollary. If it is not a
standard result and you omit its proof, then give a precise reference,
in the text just before the statement. Tell whether the omitted proof is
hard or easy to help readers decide whether to try to work it out for
themselves. If the theorem has a name, use it: say ``by the First
Fundamental Theorem,'' not just ``by Theorem~\ref{thm-1stfund}.'' State
a theorem before proving it. Keep the statement concise; put definitions
and discussion elsewhere.
Prefer a conceptual proof to a computational one; ideas are easier to
communicate, understand, and remember. Omit the details of purely
routine computations and arguments---ones with no unexpected tricks and
no new ideas. Beware of any proof by contradiction; often there's a
simpler direct argument. Finally, when the proof has ended, say so
outright. For instance, say, ``The proof is now complete,'' or use the
Halmos symbol $\square$. In addition, surround the proof---and the
statement as well---with some extra white space. (These matters are
usually now handled by a \LaTeX\ style file.)
\newlength{\taglen}\settowidth{\taglen}{MVery bad:M}
\newcounter{listctr}
\newenvironment{compare}{
\par
\begin{list}{}{
\setlength{\labelwidth}{\taglen}
\setlength{\parsep}{0in}
\setlength{\itemsep}{0in}
\setlength{\partopsep}{0in}
\setlength{\topsep}{0pt}
\setlength{\itemsep}{0in}
\setlength{\leftmargin}{.8in}
}}{\end{list}}
\newenvironment{myenumerate}%
{\begin{list}%
{\arabic{listctr}.}{\parsep 3pt\partopsep 3pt\topsep 3pt\itemsep
0pt plus 0.5pt\parskip 0pt\listparindent 0pt\usecounter{listctr}}}
{\end{list}}
\def\bad{\item[Bad:]}
\def\good{\item[Good:]}
\def\verybad{\item[Very bad:]}
Here are some more guidelines:
\begin{myenumerate}
\item Separate symbols in different formulas with words.
\begin{compare}
\bad Consider $S_q$, $q=1,\dots,n$.
\good Consider $S_q$ for $q=1,\dots,n$.
\end{compare}
\item Don't use such symbols as $\exists$, $\forall$, $\wedge$,
$\Rightarrow$, $\approx$, $=$, $>$ in text; replace them by
words. They may, of course, be used in formulas placed in text.
\begin{compare}
\bad Let $S$ be the set of all numbers of absolute value $<1$.
\good Let $S$ be the set of all numbers of absolute value less than 1.
\good Let $S$ be the set of all numbers $x$ such that $|x|<1$.
\end{compare}
\item Don't start a sentence with a symbol.
\begin{compare}
\bad $ax^2+bx+c=0$ has real roots if $b^2-4ac\ge0$.
\good The quadratic equation $ax^2+bx+c=0$ has real roots if $b^2-4ac\ge0$.
\end{compare}
\item Beware of using symbols to convey too much information all at once.
\begin{compare}
\verybad If $\Delta=b^2-4ac \ge0$, then the roots are real.
\bad If $\Delta=b^2-4ac$ is nonnegative, then the roots are real.
\good Set $\Delta=b^2-4ac$. If $\Delta\ge0$, then the roots are real.
\end{compare}
\item If you introduce a condition with ``if,'' then introduce the
conclusion with ``then.''
\begin{compare}
\verybad If $\Delta\ge0$, $ax^2+bx+c=0$ has real roots.
\bad If $\Delta\ge0$, the roots are real.
\good If $\Delta\ge0$, then the roots are real.
\end{compare}
\item Don't set off by commas any symbol or formula used in text in
apposition to a noun.
\begin{compare}
\bad If the discriminant, $\Delta$, is nonnegative, then the roots are real.
\good If the discriminant $\Delta$ is nonnegative, then the roots are real.
\end{compare}
\item Use consistent notation. Don't say ``$A_j$ where $1\le j\le
n$'' one place and ``$A_k$ where $1\le k\le n$'' another place.
\item Keep the notation simple. For example, don't write ``$x_i$ is
an element of X'' if ``$x$ is an element of X'' will do.
\item Precede a theorem, algorithm, and the like with a complete sentence.
\begin{compare}
\bad We now have the following \\
{\bf Theorem 4-1.} $H(x)$ is continuous.
\good We can now prove the following result. \\
{\bf Theorem 4-1.} Let $H(x)$ be the function defined by
Formula (4-1). Then $H(x)$ is continuous.
\end{compare}
\end{myenumerate}
\section{Example}\label{sec-ex}
As an example of mathematical writing, we discuss the two fundamental
theorems of calculus. Our discussion is based on that in Apostol's
book \cite[pp. 202--207]{apostol}. The First Fundamental Theorem says
that the process of differentiation reverses that of integration.
This statement is remarkable because the two processes appear to be so
different: differentiation gives us the slope of a curve; integration,
the area under the curve. Here is a precise statement of the theorem.
\begin{theorem}[First Fundamental Theorem of Calculus]\label{thm-1stfund}
Let $f$ be a function defined and continuous on the closed
interval $[a,\,b]$ and let $c$ be in $[a,\,b]$. Then for each $x$ in
the open interval $(a,\,b)$, we have
$$\frac{d}{dx}\int_c^x f(t)\,dt = f(x).$$
\end{theorem}
\begin{proof}
Take a positive number $h$ such that $x+h\le b$. Then
$$\int_c^{x+h}f(t)\,dt - \int_c^xf(t)\,dt = \int_x^{x+h}f(t)\,dt.$$
By hypothesis, $f$ is continuous. Hence there is some $z$ in
$[x,\,x+h]$ for which this last integral is equal to $h\,f(z)$ by the
Mean Value Theorem for integrals \cite[p. 154]{apostol}, which is not
hard to derive from the Intermediate Value Theorem. The setup is
shown in Figure~\ref{fig-1stfund}; the Mean Value Theorem says that
the area under the graph of $f$ is equal to the area of the
rectangle. Therefore,
$$\frac1h\biggl(\int_c^{x+h}f(t)\,dt - \int_c^xf(t)\,dt\biggr)
=f(z) .$$
Now, $x\le z \le x+h$. Hence, as $h$ approaches $0$, the difference
quotient on the left approaches $f(x)$. A similar argument holds for
negative $h$. Thus the derivative of the integral exists and is equal
to $f(x)$.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{figure.ps} %% dvi output
%% \includegraphics[width=0.75\textwidth]{figure.pdf} %% pdf output
\caption{Geometric setup of the proof of the First Fundamental
Theorem.}\label{fig-1stfund}
\end{figure}
The First Fundamental Theorem says that, given a continuous function
$f$, there exists a function $F$, namely, $F(x)=\int_c^xf(t)\,dt$, whose
derivative is equal to $f$:
$$F'(x) = f(x).$$
Such a function $F$ is called an {\it integral,} or a {\it primitive,}
or an {\it antiderivative,} of $f$. Integrals are not unique: if $F$ is
an integral of $f$, then obviously so is $F+C$ for any constant $C$. On
the other hand, there is no further ambiguity: any two integrals $F$ and
$G$ of $f$ differ by a constant. Indeed, their difference $F-G$ has
vanishing derivative: for every $x$,
\begin{align*}
(F-G)'(x) &= F'(x)-G'(x) \\
&= f(x)-f(x) = 0.
\end{align*}
Therefore, $F-G$ is constant owing to the Mean Value
Theorem for derivatives; see \cite[Thm. 4.7(c), p. 187]{apostol}.
When we combine the First Fundamental Theorem with the fact that an
integral is unique up to an additive constant, we obtain the
following theorem.
\begin{theorem}[Second Fundamental Theorem of Calculus]
Let $f$ be a function defined and continuous
on the open interval $I$, and let $F$ be an integral of $f$ on
$I$. Then for each $c$ and $x$ in $I$,
\begin{equation}\label{eq-2ndfund}
\int_c^x f(t)\,dt = F(x) - F(c).
\end{equation}
\end{theorem}
\begin{proof}
Set $G(x)=\int_c^x f(t)\,dt$. By the First Fundamental
Theorem, $G$ is an integral of $f$. Now, any two integrals differ by a
constant. Hence $G(x)-F(x)=C$ for some constant $C$. Taking $x=c$
yields $-F(c)=C$ because $G(c)=0$. Thus $G(x)-F(x)=-F(c)$, and
Equation~(\ref{eq-2ndfund}) follows.
\end{proof}
The Second Fundamental Theorem is a powerful statement. It says that we
can compute the value of a definite integral merely by subtracting two
values of any integral of the integrand. In practice, integrals are
often found by reading a differentiation formula in reverse. For
example, the integrals in Table~\ref{tab-integrals} were found this way.
\begin{table}
\caption{A brief table of integrals}\label{tab-integrals}
\setcounter{listctr}{0}
\def\nx#1#2{\stepcounter{listctr}\arabic{listctr}.&\int#1\,dx=#2 \\[2pt] }
$$\begin{array}{ll}
\hline\noalign{\medskip}
\nx {x^a}{\tfrac{x^{a+1}}{a+1} +C,
\hbox{\rm\ if }a\not=-1}
\nx {x^{-1}}{\ln x +C}
\nx {\sin x}{-\cos x +C}
\nx {\cos x}{\sin x + C}
\nx {e^x}{e^x + C} \noalign{\medskip}
\hline
\end{array}$$
\end{table}
The notation in the table is standard \cite[p. 178]{TandF}: the equation
$$\int f(x)\,dx =F(x)+C$$
is read, ``The integral of $f(x)\,dx$ is equal to $F(x)$ plus $C$.''
A longer table of integrals is found on the endpapers of the
calculus textbook \cite{TandF}.
\appendix
\section*{Appendix. Advanced mathematics}
In many treatments of advanced mathematics, the key results are stated
formally as theorems, propositions, corollaries, and lemmas. However,
these four terms are often used carelessly, robbing them of some useful
information they have to convey: the nature of the result.
A {\it theorem\/} is a major result, one of the main goals of the work.
Use the term ``theorem'' sparingly. Call a minor result a {\it
proposition\/} if it is of independent interest. Call a minor result a
{\it corollary\/} if it follows with relatively little proof from a
theorem, a proposition, or another corollary. Sometimes a result could
properly be called either a proposition or a corollary. If so, then
call it a proposition if it is relatively important, and call it a
corollary if it is relatively unimportant. Call a subsidiary statement
a {\it lemma\/} if it is used in the proof of a theorem, a proposition,
or another lemma. Thus a lemma never has a corollary, although a lemma
may be used, on occasion, in deriving a corollary. Normally, a lemma
is stated and proved before it is used.
The terms ``definition'' and ``remark'' are also often abused. A
formal {\it definition\/} should simply introduce some terminology or
notation; there should be no accompanying discussion of the new terms
or symbols. It is traditional to use ``if'' instead of ``if and only
if''; for example, a matrix is called {\it symmetric\/} if it is equal
to its transpose. A formal {\it remark\/} should be a brief comment
made in passing; the main discussion should be logically independent of
the content of the remark. Often it is better to weave definitions and
remarks into the general discussion rather than setting them apart
formally.
Typographically, the statements of theorems, propositions, corollaries,
and lemmas are traditionally set in italics, and the headings
themselves are set in boldface or in caps and small caps ({\bf Theorem}
or {\sc Theorem}, and so forth). The texts of definitions and remarks
are set as ordinary text; so are the texts of proofs, examples, and the
like. These headings are traditionally set in italics, boldface, or
small caps. (There is also a tradition of treating definitions
typographically like theorems, but this tradition is less common today
and less desirable.) All these formal statements and texts are usually
set off from the rest of the discussion by putting some extra white
space before and after them.
Assign sequential reference numbers to these headings, irrespective of
their different natures, and use a hierarchical scheme whose first
component is the section number. Thus ``Corollary 3-6'' refers to the
prominent statement in the sixth subsection of Section 3, and indicates
that the statement is a corollary. If the statement is the second
corollary of the third proposition in the paper, then it may seem more
logical to name the statement ``Corollary 2,'' but doing so may make the
statement considerably more difficult to locate.
\begin{thebibliography}{9}
%% Use 99 in place of 9 if there are between 10 and 99 \bibitem{}s.
\bibitem{alley}
Alley, M., ``The Craft of Scientific Writing,'' Prentice-Hall, 1987.
\bibitem{apostol}
Apostol, T. M., ``Calculus,'' Volume I, Second Edition, Blaisdell,
1967.
\bibitem{cwr}
Committee on the Writing Requirement, ``Guide to the MIT Writing
Requirement,'' Undergraduate Academic Affairs, Room 20B--140, MIT, 1993.
\bibitem{flanders}
Flanders, H., {\it Manual for\/ {\rm Monthly} Authors}, Amer. Math.
Monthly {\bf 78} (1971), 1--10.
\bibitem{gillman}
Gillman, L., ``Writing Mathematics Well,'' Math Association of America,
1987.
\bibitem{knuth}
Knuth, D. E., Larrabee, T., and Roberts, P. M., ``Mathematical
writing,'' MAA Notes Series {\bf 14}, Math Association of America, 1989.
\bibitem{munkres}
Munkres, J. R., ``Manual of style for mathematical writing,''
Undergraduate Mathematics Office, Room 2--108, MIT, 1986.
\bibitem{SandW}
Strunk Jr., W., and White, E. B., ``The Elements of Style,'' Macmillan
Paperbacks Edition, 1962.
\bibitem{TandF}
Thomas, G. B., and Finney, R. L., ``Calculus and Analytic Geometry,''
Fifth edition, Addison-Wesley, 1982.
\end{thebibliography}
\end{document}