% ----------------------------------------------------------------------

\chapter{Preface}
\label{preface}

I never meant to write this book.  It emerged unheralded, unexpectedly.

The book began in 1983 when a high-school classmate challenged me to
prove the Pythagorean theorem on the spot.  I lost the dare, but looking
the proof up later I recorded it on loose leaves, adding to it the
derivations of a few other theorems of interest to me.  From such a
kernel the notes grew over time, until family and friends suggested that
the notes might make the material for the book you hold.

The book is neither a tutorial on the one hand nor a bald reference on
the other.  The book is rather a \emph{study reference.}  In this book,
you can look up some particular result directly, or you can begin on
page one and read---with toil and commensurate profit---straight through
to the end of the last chapter.

The book as a whole surveys the general mathematical methods common to
engineering, architecture, chemistry and physics.  As such, the book
serves as a marshal or guide.  It concisely arrays and ambitiously
reprises the mathematics of the scientist and the engineer, deriving the
mathematics it reprises, filling gaps in one's knowledge while extending
one's mathematical reach.

Its focus on derivations is what principally distinguishes this book
from the few others%
\footnote{%
  Other books of the class include
  \cite{Courant/Hilbert}\cite{Jeffreys/Jeffreys}\cite{Arfken/Weber}.
}
of its class.  No result is presented here but that it is justified in a
style engineers, scientists and other applied mathematicians will
recognize---not indeed the high style of the professional mathematician,
which serves other needs; but the long-established style of
applications.

\subsubsection*{Plan}

Following its introduction in chapter~\ref{intro} the book comes in three
parts.  The first part begins with a brief review of classical algebra
and geometry and develops thence the \emph{calculus} of a single complex
variable, this calculus being the axle as it were about which higher
mathematics turns.  The second part laboriously constructs the broadly
useful mathematics of \emph{matrices} and \emph{vectors,} without which
so many modern applications (to the fresh incredulity of each generation
of college students) remain analytically intractable---the jewel of this
second part being the \emph{eigenvalue} of chapter~\ref{eigen}.  The third
and final part, the most interesting but also the most advanced,
introduces the mathematics of the \emph{Fourier transform, probability}
and the \emph{wave equation}---each of which is enhanced by the use of
\emph{special functions,} the third part's unifying theme.

Thus, the book's overall plan, though extensive enough to take several
hundred pages to execute, is straightforward enough to describe in a
single sentence.  The plan is to derive as many mathematical results,
useful to scientists, engineers and the like, as possible in a coherent
train, recording and presenting the derivations together in an orderly
manner in a single volume.  What constitutes ``useful'' or ``orderly''
is a matter of perspective and judgment, of course.  My own peculiar,
heterogeneous background in military service, building construction,
electrical engineering, electromagnetic analysis and software
development, my nativity, residence and citizenship in the United
States, undoubtedly bias the selection and presentation to some degree.
How other authors go about writing their books, I do not know, but I
suppose that what is true for me is true for many of them also: we begin
by organizing notes for our own use, then observe that the same notes
might prove useful to others, and then undertake to revise the notes and
to bring them into a form which actually is useful to others.  Whether
this book succeeds in the last point is for the reader to judge.

\subsubsection*{Notation}

The book deviates from---or cautiously improves, if you will endorse the
characterization---the conventional notation of applied mathematics in
one conspicuous respect which, I think, requires some defense here.  The
book employs hexadecimal numerals.

Why not decimal only?  There is nothing wrong with decimal numerals as
such.  I am for them, whether in the Roman or the Arabic style.
Decimal numerals are well in history and anthropology (man
has ten fingers), finance and accounting (dollars, cents, pounds,
shillings, pence: the base hardly matters), law and engineering (the
physical units are arbitrary anyway); but they are merely serviceable in
mathematical theory, never aesthetic.  Custom is not always defaced, but
sometimes adorned, by the respectful attendance of a prudent
discrimination.  It is in this spirit alone that hexadecimal numerals
are given place here.

Admittedly, one might judge the last to be more excuse than cause.  Yet,
though a dreary train of sophists down the years, impatient of
experience, eager to innovate, has indisputably abused such causes---in
ways which the mature reader of a certain cast of mind will find all too
familiar---such causes would hardly merit abuse did they not sometimes
hide a latent measure of justice.  It is to the justice, or at least to
the aesthetic, rather than to the sophistry that I affect to appeal
here.

There unfortunately really is no gradual way to bridge the gap to
hexadecimal (shifting to base eleven, thence to twelve, etc., is no
use).  If one wishes to reach hexadecimal ground then one must leap.
Thirty years of keeping my own private notes in hex have persuaded me
that the leap justifies the risk.  In other matters, by contrast, the
book leaps seldom.  The book in general walks a tolerably conventional
applied mathematical line.

\subsubsection*{Audience}

Besides those who have opened this book only to look up some particular
result (a numerous and honorable clan, but likely not reading this
preface), the book's readers arrive in two principle corps.  First come
the engineers, architects, chemists and physicists who seek ideas
toward, and logic to back, their analytical modeling of physical
systems.  Second come those ambitious students of calculus that want a
broader, demand a deeper, and venture a terser treatment of the
discipline than calculus textbooks usually afford.

There are also some others.  In a third corps come the economist and his
brethren, who may find the book a little long on physics and,
comparatively, slightly short on statistics, but still edifying perhaps.
Whether a few students of pure mathematics make a fourth corps, hunting
sketches to elaborate, remains to be seen.

\subsubsection*{Publication}

The book belongs to the emerging tradition of open-source software where
at the time of this writing it fills a void.  Nevertheless it is a
\emph{book,} not a program.  Lore among open-source developers holds
that open development inherently leads to superior work.  Well, maybe.
Often it does in fact.  Personally with regard to my own work, I should
rather not make too many claims.  It would be vain to deny that
professional editing and formal peer review, neither of which the book
enjoys, had substantial value.  On the other hand, it does not do to
despise the \emph{amateur} (literally, one who does for the love of it:
not such a bad motive, after all\footnote{The expression is derived from
an observation I seem to recall George~F.\ Will making.}) on principle,
either---unless one would on the same principle despise a Socrates, a
Washington, an Einstein or a Debian Developer.%
\footnote{%
  \cite{Debian}
}
Open source has a spirit to it which leads readers to be more
generous with their feedback than ever could be the case with a
traditional, proprietary book.  Such readers, among whom a surprising
concentration of talent and expertise are found, enrich the work freely.
This has value, too.

The book's open-source publication implies that it can neither go out of
print nor grow hard to find.  If desired you could, if expedient you
should, copy, archive and distribute the book yourself, without further
permission than the book's license already grants%
\footnote{%
  \cite{GPL}
}%
---though as a courtesy to your own readers and to this writer you might
publish the book's electronic address, \emph{derivations.org,} along
with the book.  Naturally, merely to read the book will quite satisfy
most readers, but the point is this:  the authors of other works can
refer \emph{their} readers hither without thereby setting their readers
off on a quest for some obscure or costly quarry.

\subsubsection*{Edition}

The book is extensively footnoted.  Some of the footnotes unremarkably
cite sources but many are discursive in nature, offering nonessential
material which, though edifying, coheres insufficiently well to join the
book's main narrative.  The footnote is an imperfect messenger, of
course.  Catching the reader's eye, it can break the flow of otherwise
good prose.  Modern publishing promotes various alternatives to the
footnote---numbered examples, sidebars, special fonts, colored inks,
etc\@.  Some of these are merely trendy.  Others, like numbered
examples, really do help the right kind of book; but for this book the
humble footnote, long sanctioned by an earlier era of publishing,
extensively employed by such sages as Gibbon~\cite{Gibbon} and
Shirer~\cite{Shirer}, seems the most able messenger.  In this book it
shall have many messages to bear.

In typical science/engineering style, the book numbers its sections,
tables, figures and formulas, but not its theorems, the last of which it
generally sets in italic type.  Within the book, a theorem is referenced
by the number of the section that states it.

The book subjoins an alphabetical index as a standard convenience.  Even
so, the canny reader will avoid using the index (of this and other
books), which alone of the book's pages is not to be regarded as a
proper part of the book.  Such a reader will tend rather to consult the
book's table of contents which is a proper part.

The book includes a bibliography listing works to which I have referred
while writing.  However, mathematics by its nature promotes queer
bibliographies, for its methods and truths are established by derivation
rather than by authority.  A mathematical bibliography may thus less appeal
to the works it lists than simply afford those works due credit.

Of course, not every point in the book is backed by a bibliographic
citation of any kind.  Some of the book consists of common mathematical
knowledge or even of proofs I have worked out with my own pencil from
various ideas gleaned---who knows from where?---over the years.  The
latter proofs are perhaps original or semi-original from my personal
point of view but it is unlikely that many if any of them are truly new.
To the initiated, the mathematics itself often tends to suggest the form
of the proof: if to me, then surely also to others who came before; and
even where a proof is new the idea proven is probably not.

\subsubsection*{Philosophy}

Speaking of ideas and proofs:  an idea is one thing, but what precisely
constitutes a \emph{proof?}

%\index{Kant, Immanuel (1724--1804)}
%\index{Weierstrass, Karl Wilhelm Theodor (1815--1897)}
%\index{Hilbert, David (1862--1943)}
%\index{Lewis, C.~S.\ (1898--1963)}
Modern pure mathematics tends to make one shy of the question.  To me at
least, a mathematical proof remains what an earlier era once
unsuspectingly held it to be:  it remains a morally convincing appeal to
man's faculty of logic, geometry and number (though I study not to focus
the reader's attention, unprofitably to the book's purpose, on any such
metadefinition).  Neither in this book nor elsewhere do I wish to
deconstruct, reconstruct, lay bare, replace, supersede or explain away
the faculty so named.  Indeed, towering figures like Kant, Weierstrass
and Hilbert notwithstanding, C.~S.\ Lewis (1898--1963) speaks for me
when he writes:
\begin{quote}
  You cannot go on ``seeing through'' things for ever.  The whole point
  of seeing through something is to see something through it.  It is
  good that the window should be transparent, because the street or
  garden beyond it is opaque.  How if you saw through the garden too?
  It is no use trying to ``see through'' first principles.  If you see
  through everything, then everything is transparent.  But a wholly
  transparent world is an invisible world.  To ``see through'' all
  things is the same as not to see.~\cite[chapter~3]{Lewis:abolition}
\end{quote}
Such are my sympathies.

%\index{Plato (428--348~B.C.)}
%\index{Weyl, Hermann (1885--1955)}
%\index{G\"odel, Kurt Friedrich (1906--1978)}
%\index{Frege, Friedrich Ludwig Gottlob (1848--1925)}
%\index{Courant, Richard (1888--1972)}
Would the Kantian era in which we live countenance it, the book should
sooner merely have let pure mathematics' abstract foundations lie
undisturbed.  However, since the era probably will not countenance it,
chapter~\ref{intro} engages the question briefly but soberly, after
which other chapters touch the question as necessary.  When
philosophically put to it, the book tends less to follow Kant,
Weierstrass or Hilbert in spirit than Plato, Frege, Weyl and G\"odel%
\footnote{%
  Readers who know the subject well may note the omission of the
  important name of Richard Dedekind (1831--1916) from these two lists.
  However, in which of the two would you name Dedekind?  It is no easy
  question---nor is it a question this book will tackle.  As respectable
  as Dedekind is, this book does not especially follow him,
  anyway.~\cite{Sieg/Schlimm}\cite{Toader}

  One could further mention several other respectable names---%
  Georg Cantor's (1845--1918),
  Bertrand Russell's (1872--1970) and
  L.~E.~J.\ Brouwer's (1881--1966),
  for instance, after the name of the great
  Carl Friedrich Gauss (1777-1855)%
  ---and one could bring the lists generally more up to date, but we
  will leave the matter there.

  To date the names listed:
  Plato (428--348~B.C.);
  Immanuel Kant (1724--1804);
  Karl Weierstrass (1815--1897);
  Gottlob Frege (1848--1925);
  David Hilbert (1862--1943);
  Hermann Weyl (1885--1955);
  Richard Courant (1888--1972);
  Kurt G\"odel (1906--1978).
}
(though there remains the peculiar matter of ``the Courant-Hilbert-Shilov
perspective,'' of which more will be said).  That is in spirit.  In
method, with little further apology, the book follows the time-honored
practice of the working scientist and engineer.

% diagn: review this new subsubsection
\subsubsection*{Reliance}

I hope that the book harbors no more errors than other books of the
kind do.  I hope that the book harbors fewer.  Having revised the
book's manuscript (or the notes from which the manuscript is
drawn) over a period of~30 years, I believe that the book's results are
correct in the main.

Nevertheless, the book gives reasons the reader can evaluate.  The book
details steps the reader can check.  The book illuminates patterns the
reader can study.  The book teaches principles the reader can absorb.
To look up a result in the book without evaluating, checking, studying
or absorbing might not always be an unreasonable risk to run when stakes
are small and time is short, but application of the book's results must
in every case remain the responsibility of the applicationist.

\subsubsection*{A curiosity}

To my surprise, the later parts of the book, which treat more advanced
topics, have been the easier to write.  By contrast, to explain the mere
basics of single-variable calculus and complex numbers in the book's
part~\ref{part10} has been unexpectedly harder.  An extra effort has
been required there (which however I have not omitted to make).

The matter need not concern the reader, yet it is curious, is it not?
One never quite knows which parts of a book will prove the more
difficult to write until one has sat down to write them.

%\subsection*{Completion}
%
%For~35 years, 1983 to 2018---and even for the last~12 of those years, from
%2006 when I first granted the public access to the book---I cautiously labeled
%the book a prepublished draft.  From Jan.~15, 2018, however, I have given the
%book as a proper edition.  Version 1.1 (there was no 1.0) was released on that
%date.
%
%Applied mathematics is unbounded.  Therefore, in some sense, \emph{Derivations
%of Applied Mathematics} may never be complete.  I still mean to add to the book
%from time to time.  Nevertheless, by~2018, the book---having been more
%thoroughly revised, more carefully checked, and more widely read than a
%commercially published proprietary book will typically have been at \emph{its}
%initial publication---seemed complete enough for its intended reader's purpose.
%The book's part~III still wants more sections and chapters, if I ever find time
%to write them, but such a want should no longer bar the present publication.

\subsubsection*{Acknowledgements}

The style and insight of Prof.\ G.~S.\ Brown~\cite{Brown-conversation}
and of Mr.\ D.~E.\ Davis~\cite{DEDavis-conversation} implicitly touch
the book in many places and in many ways.

Steady encouragement from my wife and children contribute to the book in
ways only an author can properly appreciate.

% diagn: review this paragraph
For a preface to state that the book's shortcomings remain the author's
own seems \emph{de rigueur.}  The statement remains true, too, though,
so I am glad for this chance to state it.

More and much earlier than to Prof.~Brown, to Mr.~Davis,
or to my wife and children,
the book owes a debt to my mother and, separately, to my father, without
either of whom the book would never have come to be.  Admittedly, any
author of any book might say as much in a certain respect, but it is no
office of a mathematical book's preface to burden readers with an
author's expressions of filial piety.  No, it is in entirely another
respect that I lay the matter here.  My mother taught me at her own hand
most of the mathematics I ever learned as a child, patiently building a
foundation that cannot but be said to undergird the whole book today.
My father generously financed much of my formal education but---more than
this---one would have had to grow up with my brother and me in my
father's home to appreciate the grand sweep of the man's curiosity, the
general depth of his knowledge, the profound wisdom of his practicality
and the enduring example of his love of excellence.

% The next, if uncommented, is an especially bad break.
% bad break
%\pagebreak

May the book deserve such a heritage.

\nopagebreak

\noindent\\
THB

