Recommended Books in the
Mathematical Sciences
Views expressed here and the
recommendations here, are those of J. M. Cargal and do not reflect
the views of any organizations or journals to which he is
associated. (Other views are incorrect.) This site does
not take money from publishers, authors, or their agents. It is
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Write to jmcargal@gmail.com
James M. Cargal
 This is the most recent photograph of James M. Cargal (used
with permission).

 Edition 1.53, September 1, 2013. One book each on Information Theory, Matroids (in section on linear algebra) and General Physics.

 Edition 1.52 April 1, 2012. Three books added on real analysis. One on advanced calculus. Two on combinatorics. One on group theory.
Edition 1.5 October 14, 2011: An essay:

 Elements of Boolean Algebra (22 pages) Note that there is also a chapter on Boolean Algebra in the Lectures on algorithms, number theory, probability and other stuff link below.
Edition 1.49 January 26, 2009:
One book on General Advanced Mathematics. One book on General
Applied Mathematics. Three books added to Combinatorics ‒
two on Fibonacci numbers (the other is very strong on
Fibonacci numbers as well). One book on evolution.
Edition 1.4 (Jan 19, 2006): Due to the efforts of Bob Hofacker I have
added ISBN numbers to most books here. However, these
are here only as an aid. It is easy to switch them around or
have the wrong edition. Also added here are two books on
Abstract Algebra and one on Logic.
Edition 1.31 (June 7, 2003): Cargal's lecture on The
EOQ Formula for manufacturing (added to section on
Inventory).
Aitions in 1.3 (Jan 22, 2003) : Two books in Number
Theory. Also a new section: Lectures
on algorithms, number theory, probability and other stuff.
Site Created December 1998.
Copyright © 19982012
You
can copy, but with proper attribution.
Top
Principles_of_Learning_a_Mathematical_Discipline
Principles of
Learning Calculus
Calculus Pedagogy
Principles
of Teaching and Learning Mathematics
Study it Twice
Ask Questions
Two Books for
Undergraduates in the Mathematical Sciences
PreCalculus Algebra
Trigonometry
Calculus
Linear Algebra
Multivariable Calculus
Differential Equations
(ODE's and PDE's)
Difference Equations
Dynamical Systems and Chaos
Real Analysis
Infinitesimal
Calculus (modern theory of infinitesimals)
Complex Analysis
Vector
Calculus, Tensors, Differential Forms
General Applied Math
General Mathematics
General Advanced Mathematics
General Computer Science
Combinatorics (including Graph
Theory)
Numerical Analysis
Fourier Analysis
Number Theory
Abstract Algebra
Geometry
Topology
Set Theory
Logic and Abstract Automata
Foundations
Algorithms
Coding and
Information Theory
Probability
Fuzzy Stuff
(logic and set theory)
Statistics
Operations Research (and
linear, nonlinear, integer programming, and simulation)
Game Theory
Stochastic Processes (and
Queueing)
Inventory Theory and Scheduling
Investment Theory
General Physics
Mechanics
Fluid Mechanics
Thermodynamics and Statistical
Mechanics
Electricity and Electromagnetism
Quantum Mechanics
Relativity
Waves
Evolution
Philosophy
Science Studies
Lectures
on algorithms, number theory, probability and other stuff
Related Sites for Mathematical
Resources
Principles of Learning a Mathematical Discipline
If you have not had the
prerequisites in the last two years, retake a prerequisite. The
belief that it will come back quickly has scuttled thousands of
careers.

Study every day – if you study
less than three days a week, you are wasting your time completely.
 Break up your study: do problems,
rest and let it sink in, do problems; work in a comfortable
environment.
 Remember, even if you are able to
survive by cramming for exams, the math you learn will only go into
short term memory. Eventually, you will reach a level where you can
no longer survive by cramming, and your study habits will kill you.
Principles of
Learning Calculus
If you have not had precalc for two years or more, retake
precalc!

Do at least two hours of calculus a day

Get another calculus book (bookstores are constantly closing out
university books, selling perfectly good texts for $5 or $7). A
second perspective always seems to help

Get a study aida book of the type: "calculus for absolute
morons"

Never miss class

Do not split the sequence. That is, do not take calc I at one school
and calc II at another. Probably your second teacher will use a
different approach from your first, when you have difficulty
changing horses midstream, your second teacher will blame it on your
first teacher having done an inferior job.
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Calculus Pedagogy
Most people come out of the calculus sequence with superficial
knowledge of the subject. However, the students who survive
with a superficial knowledge have always been the norm. Merely by
surviving, they have shown they are the good students. The really
good students will acquire a deeper knowledge of calculus with time
and continued study. Those that don't are not using calculus and it
is not clear why they needed to take it in the first place. Calculus, like basic algebra, is partly a course in technique. That is another reason to do all of your homework. There is technique and there is substance, and these things reinforce one another.

Deltaepsilon proofs in the initial sequence are generally a waste
and are abusive. They take time away from learning concepts that the
students can handle (and need). The time to learn deltaepsilon
proofs is in the first analysis course. Some students who could not
understand such proofs at all during the initial sequence actually
find them quite easy when they return to the subject.
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Principles
of Teaching and Learning Mathematics
People like to go from simple models and examples to abstraction
later. This is the normal way to learn.

There is nothing wrong to learning the syntax of the area before the
theory.

Too much motivation can be as bad as too little.

As you learn concepts, let them digest; play with them and study
them some more before moving on to the next concept.

When you get into a new area, there is something to be said for
starting with the most elementary works. For example, even if you
have a Ph.D. in physics, if you are trying to learn number theory
but have no knowledge of the subject go ahead and start with the
most elementary texts available. You are likely to find that you
will penetrate the deeper works more ably than if you had started
off with deeper works.
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Study it Twice!
A basic principle is this: most serious students of mathematics
start to achieve depth in any given area the second time they study
it. If it has been three or four years since you had the calculus
sequence, go back and study your old text; you might be surprised by
how different (and easier) it seems (and how interesting). Often if
one comes back to a discipline after a sixmonth layoff (from that
discipline, not from math) it seems so different and much easier
than it was before. Things that went over your head the first time
now seem obvious.
A similar trick that is not for everyone and that I do not
necessarily recommend has worked for me. When studying a new area it
sometimes works to read two books simultaneously. That is: read a
chapter of one and then of the other. Pace the books so that you
read the same material at roughly the same time. The two different
viewpoints will reinforce each other in a manner that makes the
effort worthwhile.
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Ask Questions!
Serious students ask questions. Half or more of all
questions are stupid. Good students are willing to ask stupid
questions. Generally, willingness to ask stupid questions is a
sign of intelligence.
Two Books for
Undergraduates in the Mathematical Sciences
Jan Gullberg was a Swedish surgeon.
When his son decided to major in engineering, Dr. Gullberg sat down
and wrote a book containing all the elementary mathematics he felt
every beginning engineer should know (or at least have at his
disposal). He then produced the book in cameraready English. The
result is almost a masterpiece. It is the most readable reference
around. Every freshman and sophomore in the mathematical sciences
should have this book. It covers most calculus and everything up to
calculus, including basic algebra, and solutions of cubic and
quartic polynomials. It covers some linear algebra, quite a bit of
geometry, trigonometry, and some complex analysis and differential
equations, and more. A great book:
 Gullberg, Jan. Mathematics From the Birth of
Numbers. Norton. 1997. 1093pp. 039304002X
 There are loads of books at many levels on mathematics for
engineers and/or scientists. The following book is as friendly as
any, and is well written. In many ways it is a companion to Gullberg
in that it starts primarily where Gullberg leaves off. (There is
some overlap, primarily basic calculus, but I for one don't think
that is a bad thing.) It covers much of the mathematics an engineer
might see in the last year as an undergraduate. Not only are there
the usual topics but topics one usually doesn't see in such a book,
such as group theory.
 K. F. Riley, Hobson, M. P., Bence, N.
J. Mathematics Methods for Physics and Engineering.
Cambridge. 1997. 1008pp. 0521890675

I might mention that Mathematical Methods for Physicists by
Arfken and Weber ( AP ) has a very
good reputation, but I can't vouch for it personally (since I have
never studied it). It is aimed at the senior level and above.
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PreCalculus Algebra
Most books on algebra are pretty much alike. For self study you
can almost always find decent algebra books for sale at large
bookstores (closing out inventory for various schools). Algebra at
this level is a basic tool, and it is critical to do many problems
until doing them becomes automatic. It is also critical to move on
to calculus with out much delay. For the student who has already
reached calculus I suggest Gullberg as a
reference.

With the preceding in mind I prefer books in the workbook format.

An excellent textbook series is the series by Bittinger published by
AisonWesley.
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Trigonometry
Trig like precalculus algebra and calculus itself tends to be
remarkably similar from one text to another.

A good example of the genre is: Keedy, Mervin L., Marvin Bittinger.
Trigonometry: Triangles and Functions. AisonWesley.
0201133326

There is an excellent treatment of trig in
Gullberg .

There is a recent (1998) book about trig for the serious student.
This is a much needed book and has my highest recommendation:
 Maor, Eli. Trigonometric Delights. Princeton
University. 0691057540
 There are many short fascinating articles on trigonometry in:
 Apostol, Tom M., et al. Selected Papers on Precalculus.
MAA
0883852055
 There is a treatment of trig that is informative but it is a
little more sophisticated than the usual text and is in Stillwell's
words at the calculus level.

Stillwell, John. Numbers and Geometry.
SV . 1998. 0387982892
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Calculus
First, see
Principle of Learning
Calculus.
Regular
Calculus Texts
 There are books on elementary calculus that are great when you
have already had the sequence. These are books for the serious
student of elementary calculus. The MAA series below is great
reading. Every student of the calculus should have both volumes.
 Apostol, Tom, et al. A Century of Calculus.2 Volumes.
MAA .
0471000051 and 0471000078
 A book that is about calculus but falls short of analysis is:
 Klambauer, Gabriel. Aspects of Calculus. SV
. 1986. 0387962743
 The following book is simply a great book covering basic
calculus. It could work as a supplement to the text for either
the teacher or the student. It is one of the first books in a
long time to make significant use of infinitesimals without using
nonstandard analysis (although Comenetz is clearly familiar with
it). I think many engineers and physicists would love
this book.
 Comenetz, Michael. Calculus: The Elements.
World Scientific. 2002. 9810249047
 See also Bressoud .
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Linear Algebra
Matroids
 There are a few giid books on matroids. However, the best introduction might be (beside Hassler Whitney's original paper — which is very readable) the following:
 Gordon, Gary and Jennifer McNulty. Matroids: A Geometric Introduction. Cambridge. 2012. 9780521145688
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Multivariable Calculus
See Vector
Calculus, Tensors, and Differential Forms. Also see
Courant and John..
Most standard calculus texts have a section on multivariable
calculus and many sell these sections as separate texts as an
option. For example the Harvard Calculus Consortium mentioned
in Calculus sell their multivariable volume
separately.

The most informal treatment is the
second half of a series. This is a great
book for the student in third semester calculus to have on the side.
 Adams, Colin, Abigail Thompson and Joel Hass. How to Ace the
Rest of Calculus: the Streetwise Guide. Freeman. 2001.
0716741741
 Another very friendly text is:
 Beatrous, Frank and Caspar Curjel. Multivariate Calculus:
A Geometric Approach. 2002. PH.
0130304379
 Often texts in advanced calculus concentrate on multivariable
calculus. A particularly good example is:
 Kaplan, Wilfred. Advanced Calculus, 3^{rd} ed.
AW . 0201799375
 A nice introductory book:
 Dineen, Seán. Functions of Two Variables. Chapman
and Hall. 1584881909
 Se also:
 Dineen, Seán. Multivariate Calculus and Geometry.
SV . 1998. 185233472X
 A quicker and more sophisticated approach but well written is:
 Craven, B.D. Functions of Several Variables. Chapman and
Hall. 0412233401
 An inexpensive Dover paperback that does a good job is:
 Edwards, C. H. Advanced Calculus of Several Variables.
Dover. 0486683362
 The following text is a true coffee table book with beautiful
diagrams. It uses a fair bit of linear algebra which is presented in
the text, but I suggest linear algebra as a prerequisite. Its
orientation is economics, so there is no Divergence Theorem or
Stokes Theorem.
 Binmore, Ken and Joan Davies. Calculus: Concepts and
Methods. 2001. Cambridge. 0521775418
 I think that following has real merit.
 Bachman, David. Advanced Calculus Demystified: A SelfTeaching Guide. 2007. McGraw Hill.
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Differential Equations
Like in some other areas, many books on differential equations
are clones. The standard text is often little more than a cookbook
containing a large variety of tools for solving d.e.'s. Most people
use only a few of these tools. Moreover, after the course, math
majors usually forget all the techniques. Engineering students on
the other hand can remember a great deal more since they often use
these techniques. A good example of the standard text is:
 Ross, Shepley L. Introduction to Ordinary Differential
Equations, 4^{th} ed. Wiley.1989. 0471098817
 Given the nature of the material one could much worse for a text
than to use the Schaum Outline Series book for a text, and like all
of the Schaum Outline Series it has many worked examples.
 Bronson, Richard. Theory and Problems of Differential
Equations, 2^{nd} ed. Schaum (McGrawHill). 1994.
070080194
 Still looking at the standard model, a particularly complete and
enthusiastic volume is:
 Braun, Martin. Differential Equations and Their
Applications, 3^{rd} ed. SV
. 1983. 0387908471
 An extremely well written volume is:
 Simmons, George F. Differential Equations with Applications
and Historical Notes, 2^{nd} ed. McGrawHill. 1991.
070575401
 The following book is the briefest around. It covers the main
topics very succinctly and is well written. Given its very modest
price and clarity I recommend it as a study aid to all students in
the basic d.e. course. Many others would appreciate it as well.
 Bear, H. S. Differential Equations: A Concise Course.
Dover. 1999. 0486406784
 Of the volumes just listed if I were choosing a text to teach
out of, I would consider the first two first. For a personal library
or reference I would prefer the Braun and Simmons.

An introductory volume that emphasizes ideas (and the graphical
underpinnings) of d.e. and that does a particularly good job of
handling linear systems as well as applications is:
 Kostelich, Eric J., Dieter Armbruster. Introductory
Differential Equations From Linearity to Chaos. AW
. 1997. 0201765497

Note that this volume sacrifices the usual compendium of
techniques found in most first texts.
 Another book that may be the best textbook here which is strong
on modeling is
 Borrelli and Coleman. Differential Equations: A Modeling
Perspective. Wiley. 1996. 0471433322

Of these last two books I prefer to use Borelli and Coleman in the
classroom, but I think Kostelich and Armbruster is a better read.
Both are quite good.
 The following book can be considered a supplementary text for
either the student or the teacher in d.e.
 Braun, Martin, Courtney S. Coleman, Donald A. Drew. ed's.
Differential Equation Models. SV
. 1978. 0387906959
 The following two volumes are exceptionally clear and well
written. Similar to the Kostelich and Armruster volume above these
emphasize geometry. These volumes rely on the geometrical view all
the way through. Note that the second volume can be read
independently of the first.
 Hubbard, J. H., B. H. West. Differential Equations: A
Dynamical Systems Approach. SV.
Part 1. 1990. 0387972862 (Part II)
HigherDimensional Systems. 1995. 0387943773
 The following text in my opinion is a fairly good d.e. text
along traditional lines. What it does exceptionally well is to use
complex arithmetic to simplify complex problems.
 Redheffer, Raymond M. Introduction to Differential
Equations. Jones and Bartlett. 1992. 0867202890
 The following rather small book is something of a reader.
Nonetheless, it is aimed at roughly the junior level.
 O'Malley, Robert E. Thinking About Ordinary Differential
Equations. Cambridge. 1997. 0521557429
 For boundary value problems see Powers .

An undergraduate text that emphasizes theory and moves along at a
fair clip is:
 Birkhoff, Garrett. GianCarlo Rota. Ordinary Differential
Equations. Wiley. 1978. 0471860034

Note that both authors are very distinguished mathematicians.
See Dynamical Systems and
Calculus.
The Laplace Transform
 I have three books to list on this topic.

Kuhfittig, Peter K. F. Introduction to the Laplace
Transform. Plenum. 1978. 205pp.
0306310600.
 The following text is a little more abstract and as the title
implies also covers Fourier series and PDE's.

Dyke, P. P. G. An Introduction to Laplace Transforms and
Fourier Series. Springer. 2001. 250pp.
1852330155
 The following is pedagogically exceptional. I like it a
lot.
Partial Differential Equations
Difference Equations
Dynamical Systems and Chaos
Two classics that precede the current era of hyperinterest in
this area are (both are linear algebra intensive)
 Luenberger, David G. Introduction to Dynamic Systems:
Theory, Models, & Applications. Wiley. 1979.
0471025941

I think this has been reprinted by someone.
 Hirsch, Morris W. and Stephen Smale. Differential Equations,
Dynamical Systems, and Linear Algebra.
AP . 1974. 0123495504

There is now a second edition of the Hirsch and Smale (Note
the change in title):
Hirsch, Morris W., Stephen Smale and Robert L.
Devaney. Differential Equations, Dynamical Systems &
An Introduction to Chaos, 2^{nd} ed. AP
. 2004. 9780123497031
 Four more books at the junior senior level that can double as
references on differential equations:
 Hale, J. and H. koçak. Dynamics and Bifurcations.
SV . 1991. 079231428X

Verhulst, Ferdinand. Nonlinear Differential Equations and
Dynamical Systems. SV . 1985.
3540609342

Strogatz, Steven H. Nonlinear Dynamics and Chaos with
Applications to Physics, Biology, Chemistry, and Engineering.
AW . 1994. 3540609342

Banks, John, Valentina Dragan and Arthur Jones. Chaos:
A Mathematical Introduction. Cambridge. 2003.
0521531047
 A book that I think should be of interest to most applied
mathematicians:

Schroeder, Manfred. Fractals, Chaos, Power Laws:
Minutes From an Infinite Paradise. Freeman. 1991.
0716721368
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Real Analysis
 There are two fantastic books that almost make a library by themselves. These are big and sumptious. The first is a solid course in undergraduate real analysis. The second is graduate level. To some extent they are available for download at their authors' web site.
 Thomson, Brian S., Judith B. Bruckner, Andrew M. Bruckner. Elementary Real Analysis, 2nd ed. 2008. www.classicalrealanalysis.com. 9781434843678.
 Bruckner, Andrew M., Judith B. Bruckner, Brian S. Thomson. Real Analysis, 2nd ed. 2008. www.classicalrealanalysis.com. 9781434844125.

For the student seeing analysis for the first time and who is
overwhelmed by analysis, there are a few books out there. A good
candidate is
 Bryant, Victor W. Yet Another Introduction to Analysis.
Cambridge. 1990. 052138835X
 A good text at the junior level is
 Reed, Michael. Fundamental Ideas of Analysis. Wiley. 1998.
0471159964

This book is unusual amongst its kind for its inclusion of
applications.
 There are two books for the serious student of real analysis by
Bressoud. These are books I recommend to grad students and
faculty; but one is at the undergraduate level. Very good on
history and motivation. Exceptional!!!!!

Bressoud, David. A Radical Approach to Real Analysis,
2^{nd} ed. MAA.
2006. 9780883857472

Bressoud, David. A Radical Approach to Lebesgue's Theory
of Integration. MAA.
2008. 9780521711838
 Comparable to Bressoud's books there is another historical book
on analysis that I have found readable, informative and useful (for
example I think the short chapter on Lebesgue is a good introduction
to Lebesgue theory). I like it a lot.

Dunham, William. The Calculus Gallery: Masterpieces
from Newton to Lebesgue. Princeton. 2005.
9780691136264
 One of the most popular texts currently (2004) that does a nice
job for a first course is by Abbott. It does not do as much
hand holding as Bryant, which is arguably too much. It appears
to designed for a onesemester course, though you could probably
squeeze it into two semesters (with no difficulty at most
universities). Might be a nice resource for the student taking
the twosemester sequence out of another text. Minimal
prerequisites.
 Abbott, Stephen. Understanding Analysis.
Springer. 2001. 0387950605
 A remarkably similar book to Abbott is the one by Pedrick.
Is even briefer, but could probably fit into two semesters at most
schools.

Pedrick, George. A First course in Analysis.
Springer. 1994. 0387941088
 A more complete book at that level (more than two semesters in
my slow teaching) is
 Protter, M. H., and C. B. Morrey. A First Course in Real
Analysis, 2^{nd} ed. SV
. 1991. 0387941088
 A very large (and historic) lovely and complete two volume set
is
 Courant, Richard. Fritz John. Introduction to Calculus and
Analysis. SV . 354065058X
 A thorough treatment of undergraduate analysis is given in
 Bartle, Robert G. The Elements of Real Analysis, 2^{nd}
ed. Wiley. 0471054623
 A resource wonderful for its proofs and examples (and outdated
terminology) is
 Hardy, G. H. A Course in Pure Mathematics. Cambridge.
0521092272
 A fairly large book that is very good on undergraduate analysis and is applied is
 Estep, Donald. Practical Analysis in One Variable. 2002. Springer. 0387954848
 It is a good book for the numerical analyicist.
 A great read in analysis and best seller is
 Boas, R. P. A Primer of Real Functions 4^{th}
ed. MAA.
088385029X
 See also Simmons .

The following book is very well written it covers much of analysis
into Lebesgue measure. The chapter are short and break the material
into digestible chunks making the book a great reference, study
guide and first rate text. This may be the least appreciated book on
analysis.
 Bear, H. S. An Introduction to Mathematical Analysis.
AP. 1997. 0120839407
 The following texts I consider graduate level. These all cover
some abstract integration (almost always the Lebesque Integral).

The standard graduate text is
 Royden, H. L. Real Analysis, 3^{rd }ed. PH
. 1988. 0120839407
 If I had to recommend a single book, it might be:

Jones, Frank. Lebesgue Integration on Euclidean Space,
Revised ed. Jones and Bartlett. 2001.
0763717088

Don't be put off by the title, it is pedagogically very strong!!
 Books that are written to help the beleaguered student into
abstract analysis include:
 Burk, Frank. Lebesgue Measure and Integration: An
Introduction. Wiley. 1998. 0471179787

This may be the best of the lot.
 Bear, H. S. A Primer of Lebesgue Integration.
AP . 1995. 0471179787

Craven, Bruce D. Lebesgue Measure & Integral. Pitman.
1982. 0273017543
 The following excellent text may be the best introduction to the
Lebesque integral around. Very nice:
 Capinski, Marek and Ekkehard Kopp. Measure, Integral, and
Probability. Springer. 1999. 3540762604
 I like the following quite a bit:
 Chae, Soo Bong. Lebesgue Integration, 2^{nd} ed.
SV . 1995. 0387943579
 A classic book is
 Bartle, Robert G. The Elements of Integration and Lebesgue
Measure. Wiley. 1966 (new edition 1996). 0471042226
 A wonderful book that is strong on applications and should
probably belong to students of numerical analysis is:

Cooper, Jeffery. Working
Analysis. Elsevier. 2005. 0121876047

►►►Cooper is a must have for all serious
students of analysis. A great book!!!!
 Another classic which is fairly comprehensive is:
 Hewitt, Edwin, and Karl Stromberg. Real and Abstract
Analysis. SV . 1965.
0387901388
 Of the more advanced books that discuss the subject more deeply:
 Gordon, Russell A. The Integrals of Lebesgue, Denjoy,
Perron, and Henstock. American Mathematical Society. 1994.
0821838059
 A book influenced by Gordon's and also well written:

Burk, Frank. A Garden of Integrals.
MAA.
2007. 9 780883 853375
 Every graduate student of analysis should have:
 Carothers, N. L. Real Analysis. Cambridge. 2000.
0521497493
 Also recommended is the following senior level, very thorough
but friendly text (729pp):
 Strichartz, Robert S. The Way of Analysis. 2000. Jones
and Bartlett. 0763714976
 A superb book that treats the generalized Riemann integral
before going to the Lebesque is:
 Yee, Lee Peng. The Integral: An Easy Approach after Kurzweil
and Henstock. Cambridge. 2000. 0521779685
 The following magnum opus is the only one I've seen in this area
that can be useful to the nonspecialist.
 Schechter, Eric. Handbook of Analysis and Its Foundations.
AP. 1997. 0126227608
 Lastly any graduate student serious about analysis should also
have Korner .

The Mathematical Association of America publishes many works that
are intended as aids to teaching either calculus or analysis.
I do not know if these books are so useful to the teacher, but
they are great resources for the serious student. A recent
example is (that is particularly good):

Brabenec, Robert L. Resources for the
Study of Real Analysis. MAA.
2004. 0883857375
 A very interesting book:

Dunham, William. The Calculus Gallery: Masterpieces
from Newton to Lebesque. Princeton. 2005.
0691095655
 See also Courant and John.
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Infinitesimal Calculus (modern theory of infinitesimals)
This section is not for beginners! If you are just learning
calculus go to the section Calculus.
The
genesis, by the creator, is tough reading:
 Robinson, Abraham. NonStandard Analysis. NorthHolland.
1966. 0691044902
 The best introduction by far is:
 Henle and Kleinberg. Infinitesimal Calculus. MIT. 1979.
0486428869

This has been republished (2003) as inexpensive Dover
paperback.
 A book that is supposed to be easy but is very abstract is:
 Robert, Alain. Nonstandard Analysis. Wiley. 1985.
0486432793
 A quick, nice book with applications is:
 Bell, J. L. A Primer of Infinitesimal Analysis.
Cambridge. 1998. 0521624010
 A thorough, authoritative, and well written classic is
 Hurd, A. E. and P. A. Loeb. An Introduction to Nonstandard
Real Analysis. AP . 1985.
0123624401
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Complex Analysis
The following book is a primer on complex numbers that ends with a
short introduction to Complex Analysis. It is a perfect book
for the sophomore in math or engineering. Great book:
Nahin, Paul J. An Imaginary Tale: The Story
of √1. Princeton University. 1998.
0691127980
Perhaps the most remarkable book in this area; truly great book
is:
 Needham, Tristan. Visual Complex Analysis. Oxford.
1997. 0198534469

Although this is written as an introductory text, I recommend it
as a second book to be read after an introduction. Also, it is a
great reference during the first course.
 A wonderful book that is concise, elegant, clear: a must have:
 Bak, Joseph and Donald J. Newman. Complex Analysis,
2^{nd}ed. SV . 1997.
0387947566
 The nicest, most elementary introduction is:
 Stewart, Ian and David Tall. Complex Analysis.
Cambridge. 1983. 0521287634
 The most concise work (100 pages) may be:

Reade, John B. Calculus with Complex Numbers.
Taylor and Francis. 2003. 0415308461
 A thorough well written text I like is:
 Ablowitz, Mark J. and Athanassios S. Fokas. Complex
Variables: Introduction and Applications. 1997. Cambridge.
0521534291
 The workhouse introduction, particularly suited to engineers has
been:
 Brown, James Ward and Ruel V. Churchill. Complex Variables
and Applications 6^{th} ed. 1996. 0079121470
 Another book very much in the same vein as Brown and Churchill
is preferred by many people,
 Wunsch, A. David. Complex Variables with Applications,
2^{nd} ed. AW . 1994.
0201122995

This is my favorite book for a text in CA.
 Still another superb first text is formatted exactly as
elementary calculus texts usually are:

Saff, E. B. and A. D. Snider. Fundamental of Complex
Analysis with Applications to Engineering and Science, 3^{rd}
ed. PH. 2003. 0133321487
 Two more introductions worth mentioning are:
 Palka, Bruce P. An Introduction to Complex Function Theory.
SV . 1991. 038797427X
 Priestley, H. A. Introduction to Complex Analysis.
Oxford. 1990. 0198525621
 An introduction based upon series (the Weierstrass approach) is
 Cartan, Henri. Elementary Theory of Analytic Functions of
one or Several Variables. AW .
1114121770
 A book this is maybe more thorough than those above is
 Marsden, Jerrold E. and Michael J. Hoffman. Basic
Complex Analysis, 2^{nd} ed. Freeman. 1987.
0716721058
 A book that I regard as graduate level has been described as the
best textbook ever written on complex analysis:
 Boas, R. P. Invitation to Complex Analysis. Birkhauser
Boston. 0394350766
 A classic work (first published in 1932) that is thorough.
 Titmarsh, E. C. The Theory of Functions, 2^{nd}
ed. Oxford. 1997. 0198533497

Essentially the third correction (1968) of the second edition
(1939).
 A reference that I expect to sell very well to a wide audience:
 Krantz, Steven G. Handbook of Complex Analysis.
Birkhäuser. 1999. 0817640118
 The following is in one of Springer's undergraduate series but I
think is more suited for grad work. The author says it should get
you ready for Ph.D. qualifiers. Definitely a superior work.
 Gamelin, Theodore W. Complex Analysis. Springer. 2000.
0387950699
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Vector
Calculus, Tensors, Differential Forms
See Multivariable Calulus
See Courant and John
A great pedagogical work most highly recommended especially to
electrical engineers
 Schey, H. M. Div, Grad, Curl, and All
That: An Informal Text on Vector Calculus 3^{rd}ed..
Norton. 1997. 0393093670
 A fairly comprehensive work I like a lot is:
 Marsden, Jerrold E., Anthony J. Tromba. Vector Calculus,
4^{rd} ed. Freeman.
 This may be the best book to have. It is very good.
0716724324
 A short (and cheap) work that is concise and well written is
 Hay, G. E. Vector and Tensor Analysis. Dover. 1953
(original date with original publisher). 0486601099
 Another short and concise treatment that is well written is

Matthews, P. C. Vector Calculus. Springer. 1998.
3540761802
 A user friendly texts on vector calculus:
 Colley, Susan Jane. Vector Calculus, 2^{nd} ed.
PH. 2002. 0130415316
In general there are plenty of good books on vectors with the
two books above being outstanding. Books on differential forms and
tensors can often merely enhance the reputations of those areas for
being difficult. However, there are exceptions.
On tensors I like two books which complement each other well. The
book by Danielson is more application oriented. If you are serious
about this area get both books. Also, the Schaum outline series
volume on tensors has merit.
 Simmonds, James G. A Brief on Tensor Analysis, 2^{nd}
ed. SV . 1994. 038794088X

Danielson, D. A. Vectors and Tensors in Engineering and Physics,
2^{nd} ed. AW .
0813340802
 The following is concise and offers an introduction to tensors,
may be the best intro:

Matthews, P. C. Vector Calculus. Springer.
1998. 3540761802
 On differential forms I recommend
 Bachman, David. A Geometric Approach to Differential
Forms. Birkhäuser. 2006. 0817644997
 Edwards, Harold M. Advanced Calculus: A Differential Forms
Approach. Birkhäuser. 1994. 0817637079

Weintraub, Steven H. Differential Forms: A Complement to Vector
Calculus. AP . 1997.
0127425101
 A book that does a good job of introducing differential forms
is:
 Bressoud, David M. Second Year
Calculus. SV . 1991.
038797606X
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General Applied Math
There are roughly 37 zillion books on applied math (with titles
like Mathematics for LeftHanded Quantum Engineers)

Check out Gullberg , it was specifically
written for engineering students though it is appropriate for all
students of math

A great book which, appropriate for its
author, emphasizes linearity is:
 Strang, Gilbert. Computational Science and Engineering.
WellesleyCambridge Press. 2007. 9780961408817

A masterpiece and a must have for the library of every applied
mathematician.
 A recent book that is pedagogically very nice and goes though
junior level material with wide coverage extending to group theory
is Riley et al.

A great tool for applied mathematicians:

Andrews, Larry C. Special Functions of Mathematics for
Engineers, 2^{nd} ed. Oxford. 1998.
0819426164
 A two volume set that is more appropriate for seniors and
graduate students is
 Bamberg, Paul G., Shlomo Sternberg. A Course in Mathematics
for Students of Physics. Cambridge. 1991. 052125017X
 A superb book at roughly the junior level, a book that could
double as a text in advanced calculus, is
 Boas, Mary. Mathematical
Methods in the Physical Sciences, 3^{rd}
ed. Wiley. 2005. ISBN10: 0471198269; ISBN13:
9780471198260

This book is regarded very highly by many students and researchers
for its clarity of writing and presentation. (Also, this
demonstrates how completely impartial I am, since Professor Boas
detests me.)
 A tour de force at the graduate level; a book for the serious
student:
 Gershenfeld, Neil. The Nature of Mathematical Modeling.
Cambridge. 1999. 0521570956
 The following book could be put in Real Analysis or even
Numerical Analysis. It is compact and very appealing (and hard to
describe):
 Bryant, Victor. Metric Spaces: Iteration and Application.
Cambridge. 1985. 0521318971
 The following is very interesting, definitely requires calculus:
 Nahin, Paul J. When Least is Best. Princeton.
2004. 0691070784
 I think that a fantastic book for teaching modelling is the one that follows. It covers all sorts of modelling and is superb at the sophomore/junior level.
 Shiflet, Angela B. and George W. Shiflet. Introduction to Computational Science: Mdeling and Simulation for the Sciences. Princeton University Press. 2006. 9780691125657.
 Courant and John
A
great reference is the last edition of Courant's great classic work
on calculus. This is two volumes stretched to three with
Volume II now becoming Volume II/1 and Volume II/2.
Nonetheless they are relatively not expensive and they are great
references. Volume I is a superb work on analysis.
Volume II/1 and the first part of Volume II/2 are a full course on
multivariable calculus. Volume II/2 constitutes a great text
on applied math including differential equations, calculus of
variations, and complex analysis.
 Courant, Richard and Fritz John. Introduction to Calculus
and Analysis. Springer. 1989.

Vol I. 354065058X
 Vol II/1 3540665692
Vol II/2 3540665706
General Mathematics
Check out Gullberg .

A classic (originally published more than fifty years ago):
 Hogben, Lancelot. Mathematics for the Millions: How to
Master the Magic of Numbers. Norton. 1993. 0393063615
 This is a great classic first published in the midforties.
Although ostensibly written for the layman, it is not a light work.
Its treatment of geometry is particularly good
 Courant, Richard, Herbert Robins. Revised by Ian Stewart. What
is Mathematics. Oxford. 1997. 0195105192
 A book that might be better considered general mathematics:
 Stillwell,
John. Numbers and Geometry. SV
. 1998. 0387982892

The level is roughly first or second semester calculus.
 A sweet book that is similar in spirit to Stillwell's and that
should be of interest to students of analysis is
 Pontrjagin, Lev S. Learning Higher Mathematics. SV.
1984. 0387123512
 The following is a modern classic
 Davis, Phillip J., Reuben Hersh, Elena Marchisotto. The
Mathematical Experience. Birkhäuser. 1995.
0395929687

I recommend other books by Davis and Hersh as well as books by
Davis and Hersh each alone.
 The late Morris Kline wrote several good books for the layman
(as well as for the professional). My personal favorite is strong on
history and art and I think deserves more attention than it has ever
had. I think it is more important now then when it was first
published (in the 1950's):
 Kline, Morris. Mathematics in Western Culture. Oxford.
1965. 0195006038
 A book that does a great job on foundations, fundamentals, and
history is Eves .

The following is a book I think every undergraduate math major (who
is at all serious) should have:

Hewson, Stephen Fletcher. A Mathematical Bridge: An
Intuitive Journey in Higher Mathematics. World
Scientific. 2003. 9812385541
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General Advanced Mathematics
 The following book is sensationally good. There does not
seem to be any other single volume that compares.

Gowers, Timothy (ed.) The Princeton Companion to
Mathematics. 2008. Princeton.
9780691118802
 This book is true to its title and is a must for the grad
student. Still anyone who goes into grad school knowing all of this
does not need my help.
 Garrity, Thomas A. All The Mathematics You Missed [But Need
to Know for Graduate School]. Cambridge. 2002. 0521797071
 The following is a very short book that every student of
abstract algebra should have:

Litlewood, D. E. The Skeleton Key of Mathematics: A
Simple Account of Complex Algebraic Theories. Dover.
2002. 0486425436
General Computer
Science
The books here tend cover algorithms and computability but don't
forget to go the sections Algorithms and
Logic and Computability .

A. K. Dewdney wrote a book of 66
chapters to briefly and succinctly cover the interesting topics of
computer science. The emphasis here is theory. This is a book every
computer science major should have, and probably every math major
and certainly anyone with a serious interest in computer science.
 Dewdney, A.
K. The New Turing Omnibus. Freeman. 1993. 0716782715
 A nice introduction that is good at
introducing the concepts and philosophy of computer algorithms is
 Harel, David. Algorithmics:
The Spirit of Computing,
2^{nd}
ed. AW . 1992. 0201504014
 Another fine booka great
tutorialseems to be out of print, but thankfully you can get it
online from the author at www.cis.upenn.edu/~wilf/AlgComp2.html
 Wilf, Herbert S. Algorithms and
Complexity. 1568811780
 A great book for the serious student of mathematics and computer
science is (senior level):
 Graham, Ronald, Oren Patashnik, Donald E. Knuth. Concrete
Mathematics: A Foundation for Computer Science. 2^{nd.}
ed. AW . 1994. 0201558025
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Combinatorics
(Including Graph Theory)
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Numerical
Analysis
Most books on numerical analysis are
written to turn off the reader and to encourage him or her to go
into a different, preferably unrelated, field. Secondly, almost all
of the books in the area are written by academics or researchers at
national labs, i.e. other academics. The kind of industry I use to
work in was a little different than that. The problem is partly
textbook evolution. I've seen books long out of print that would
work nicely in the classroom. However, textbook competition requires
that newer books contain more and more material until the book can
become rather unwieldy (in several senses) for the classroom. The
truth is that the average book has far too much material for a
course. Numerical analysis touches upon so many other topics this
makes it a more demanding course than others.

A marvelous exception to the above is
the book by G. W. Stewart. It avoids the problem just mentioned
because it is based upon notes from a course. It is concise and
superbly written. (It is the one I am now teaching out of.)
 Stewart, G. W. Afternotes on
Numerical Analysis. SIAM. 1996. 0898713625
 Volume II, despite the title, is
accessible to advanced undergraduates. If you liked the first text
you want this:
 Stewart, G. W. Afternotes
goes to Graduate school: Lectures on Advanced Numerical Analysis.
SIAM. 1998. 0898714044
 Two great books on the subject are
written by a mathematician with real industrial experience. The
first is absolutely superb. Both books are great to read, but I
don't like either as a text.
 Acton, Forman. Real Computing
Made Real: Preventing Errors in Scientific and Engineering
Calculations. Princeton. 1995. 0691036632

Acton, Forman. Numerical
Methods That Work.
MAA . 1990. 1124037799

This is a reprint with corrections of
an earlier work published by another publisher.
 An interesting book that seems in
the spirit of the first book by Acton (above) is:
 Breuer, Shlomo, Gideon Zwas.
Numerical Mathematics: A Laboratory Approach. Cambridge.
!993. 0521440408

This is a great book for projects and
for reading. I would like to know however how it has done as a
text.
 A book by a great applied
mathematician that is worth having is:
 Hamming, R. W. Numerical
Methods for Scientists and Engineers, 2^{nd} ed..
Dover. 1987. 0486652416
 The book I use in the classroom is
(although I intend to try G. W. Stewart).:
 Asaithambi, N. S. Numerical
Analysis: Theory and Practice. Saunders. 1995. 0030309832
 A textbook that looks very
attractive to me is:
 Fairs, J. Douglas, Richard Burden.
Numerical Methods, 2^{nd} ed. Brooks/Cole. 1998.
0534392008

This is about as elementary as I can
find. This is the problem with teaching the course. On the flip
side of course, it covers less material (e.g. fixed point
iteration is not covered). Also, it does not give pseudocode for
algorithms. This is okay with me for the following reasons. Given
a textbook with good pseudocode, no matter how much I lecture the
students on its points and various alternatives, they usually copy
the pseudocode as if it the word of God (rather than regarding my
word as the word of God). It is useful to make them take the
central idea of the algorithm and work out the details their
selves. This text also has an associated instructors guide and
student guides. It refers also to math packages with an emphasis
on MAPLE and a disk comes with the package, which I have ignored.
See the book by Cooper.
 A quicker treatment
than even that is in the first three pages of Smullyan's book on
Gödel above. This is the
book to have.
 This is definitely a useful
book.
 A very good treatment for the
student of logic:
 Smith, Peter. An
Introduction to Gődel's Theorems.
Cambridge. 2007. 9780521674539
By foundations I do not
mean fundamentals. Of the books listed here the only one of
serious interest to the specialist in logic is the one by Wilder.

The best book is, I think,
 Wilder, Raymond L Introduction
to the Foundations of Mathematics, 2^{nd} ed. Krieger.
 One of the most underrated books I
know is this book by Eves. It does a very credible job of covering
foundations, fundamentals and history. It is quite a little gem (344
pp).
 Eves, Howard.
Foundations and Fundamental concepts of Mathematics, 3^{rd}
ed. PWSKent. 1990. 048669609X
 A book that fits as well into
foundations as anywhere is:
 Ebbinghaus, H.D. Et al. Numbers.
SV . 1990.
 A book I like a lot (senior level
in my view) is
 Potter, Michael. Set
Theory and its Philosophy. Oxford. 2004.
0199270414
 This book is indeed very good.
I strongly recommend it.
 A slightly more elementary text is:
 Tiles, Mary. The
Philosophy of Set Theory: An Historical Introduction to
Cantor's Paradise. Dover. 2004. Reprint of
1989 edition) 0486435202
 See also the previous section.
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Algorithms
The four volumes of D. E. Knuth,
The Art of Computing, AisonWesley are more or less
a bible. They are comprehensive, authoritative, brilliant. They are
mathematically sophisticated and are considered by most people to be
references more than texts.

See General
Computer Science .

For graph algorithms specifically see
the books by Gibbons and Even .

For algorithms on optimization and
linear programming and integer programming go to the appropriate
sections.

The best single book on the subject is
the one by Cormen, Leiseron, and Rivest. It covers a great deal of
ground; it is well organized; it is well written; it reviews
mathematical topics well; it has good references; the algorithms are
stated unusually clearly.
 Cormen, Thomas H., Charles E.
Leiserson, and Ronald L. Rivest. Introduction to
Algorithms. MIT for individual copies; McGrawHill for
large quantities. 1990. 1028 pp. 0262531968
 Aho, Hopcroft, and Ullman wrote two
texts on algorithms. The second one is slightly more elementary and
is better written. If I were to choose one I would choose this one
(1983).
 Aho, Alfred V., John E. Hopcroft,
and Jeffrey D. Ullman. The
Design and Analysis of Computer Algorithms.
AW . 1974. 0201000296

Aho, Alfred V., John E. Hopcroft, and
Jeffrey D. Ullman. Data
Structures and Algorithms.
AW . 1983. 0201000237
 A rather theoretical tour of
algorithmic theory and select topics:
 Kozen, Dexter C. The
Design and Analysis of Algorithms.
SV . 1992. 0387976876
 I have not seen the following book
but it had a very tantalizing review (as an introduction) in the AMM
telegraph reviews:
 Haupt, Randy and Sue Ellen Haupt.
Practical Genetic Algorithms. Wiley. 1998. 0471455652
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Coding
and Information Theory
The second edition will include
recommendations on books on Digital Filters and Signal Analysis
Probability
Fuzzy
Stuff (logic and set theory)
Some books in this area are better
than others. By in large though, it is a lot of bull about ad hoc,
not particularly robust, algorithms. Claims of anything new and
profound are general pompous bullstuff. Fuzzy methods are trivial if
you have knowledge of probability and logic. In
my view the aspiring applied mathematician can not do better than to
study probability .
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Statistics
A book of practical statistics
as opposed to mathematical or theoretical statistics is the one by
Snedecor and Cochran. It is rigorous but does not use calculus. It
uses real life biological data for examples but is fascinating. It
is a wonderfully well written and clear book. A real masterpiece.
Anyone who actually does statistics should have this book. But
remember, though it does not require calculus it does require
mathematical maturity. My feeling is that if you want to use this
book but do not know calculus you should go back and take calculus.
 Snedecor, George W. and William G.
Cochran.Statistical Methods, 8^{th} ed. Iowa State.
1989. 0813815614
 A newer book in the spirit of
Snedecor et al but requiring calculus is:
 McPherson, Glen. Applying and
Interpreting Statistics: A Comprehensive Guide, 2^{nd}
ed. Springer. 2001. 0387951105
 Like Snedecor, this book is
packed with reallife examples. A great book.
 The best books about statistics for
the layman are very likely:
 Tanur, Judith
M. et al. Statistics: A Guide to the Unknown, 3^{rd}
ed. Wadsworth. 1989. 0534094929
 Again, students almost invariably
get through the basic course on statistics without knowing what
statistics (the field) is and how statisitics are actually used.
This is a great book. See also Bennett.
 Salsburg, David. The Lady
Tasting Tea. Freeman. 2001. 0805071342
 This is a history of statistics
that is a very quick read. Without using a single formula it does
a much better job of telling the layman what statistics is about
than does the usual introductory text. It is also of interest to
the professional.
 A classic applied book that is
readable and thorough and good to own is:
 Neter, John, Michael K. Kutner,
Christopher J. Nachtsheim, William Wasserman. Applied Linear
Statistical Models,4th ed. Irwin. 1996. 0256117365
 1407 pages on linear
regression and ANOVA.
 My favorite text on mathematical
statistics is definitely the following. It is a large text with
enough material for a senior level sequence in mathematical
statistics, or a more advanced graduate sequence in mathematical
statistics. It is very well done.
 Dudewicz, Edward J. and Satya N.
Mishra. Modern Mathematical Statistics. Wiley. 1988.
0471814725
 Another book on mathematical
statistics that merits attention is
 Mood, Alexander McFarlane.
Introduction to the Theory of Statistics. McGrawHill.
1974. 0070428646
 For the student who needs help in
the sophomore statistics course in business or the social sciences,
let me say first, that this site is far people with more advanced
problems. Still, I can heartily recommend the following:
 Gonick, Larry and Woolcot Smith.
The Cartoon Guide to Statistics. HarperCollins. 1993.
0062731025
 If this book only had exercises I
would suggest its use as a textbook.
 An elementary book that does a nice
job on statistical tests and which might be of interest to the
practitioner is:
 Langley, Russell. Practical
Statistics Simply Explained. Dover. 1971. 0486227294
 In the area of design of
experiments and analysis of variance, the book by Hicks is a good
standard reference. The book by Box, Hunter and Hunter is wonderful
at exploring the concepts and underlying theory. The book by Saville
and Wood is worth considering by the serious student. Although its
mathematics is simple and not calculus based this is the way theory
was developed (and this is also touched upon in the book by Box,
Hunter, and Hunter.
 Hicks, Charles R. Fundamental
Concepts in the Design of Experiments. Oxford. 1993.
0195122739
 Box, George E. P., J. Stuart
Hunter, and William Gordon Hunter. Statistics for Experimenters:
An Introduction to Design, Data Analysis, and Model Building.
Wiley. 1978. 0471093157
 This is a wonderful book!
 Saville, David J. And Graham R.
Wood. Statistical Methods: A
Geometric Primer. SV
. 1996. 0387975179
 Note that these authors have an
earlier slightly more advanced book covering the same topic.
 My favorite book on regression is
the one by Draper and Smith. The book by Ryan is particularly
elementary and thorough.
 Draper, Norman R. and Harry Smith.
Applied Regression Analysis. Wiley. 1998. 0471029955

Ryan, Thomas P. Modern Regression
Methods. Wiley. 1997. 0471529125
 For sampling theory there is
actually a nontechnical introduction (sort of Sampling for
Dummies) by Stuart. The book by Thompson is for the
practitioner.
 Stuart, Alan. Ideas of
Sampling, 3^{rd}ed. Oxford. 1987. 0028530608

Thompson, Steven K. Sampling.
Wiley. 1992. 0471558710
 I personally think that time series
analysis for forecasting is usually worthless. If forced to use time
series analysis for purposes of forecasting I almost always will use
double exponential smoothing possibly embellished with seasonal
attributes and builtin parameter adjusting. The bible of times
series analysis is Box and Jenkins. The book by Kendall and Ord is
fairly complete in its survey of methods. I like the book by
Bloomfield.
 Box, George E. P., Gwilym M.
Jenkins, Gregory C. Reinsel. Times Series Analysis: Forecasting
and Control. Wiley. 1994. 0130607746
 Kendall, Sir Maurice and J. Keith
Ord. Time Series, 3^{rd} ed. Edward Arnold. 1990.
0195205707

Bloomfield, Peter. Fourier Analysis
of Time Series: An Introduction. Wiley. 1976. 0471889482
 A book on nonparametric methods:
 Conover, W. J. Practical
Nonparametric Methods, 2^{nd} ed. Wiley. 1980.
0471160687
 Any statistical practitioner should
have the following:
 Noreen, Eric W. Computer
Intensive Methods for Testing Hypotheses: An Introduction.
Wiley. 1989. 0471611360
 A Simple book that simply contains
information on distributions:
 Evans, Merran, Nicholas Hastings,
and Brian Peacock. Statistical Distributions, 2^{nd}
ed. Wiley. 1993. 0471371246
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Operations
Research (and linear, nonlinear, integer programming, and
simulation)
A future
edition will cover both decision theory and games of
the J H. Conway variety.
Game Theory
An early classic of extremely
elementary nature is the one by Williams. It precedes the widespread
use of linear programming.
 Williams, J. D. The Complete
Strategyst: Being a Primer on the Theory of Games. Dover 1986.
1131977025
 This is the listing I have, but I
suspect the spelling in the title is still as was: Compleat.
 See Thie .

A fine elementary book is:
 Straffin, Philip D. Game
Theory and Strategy.
MAA . 1993. 0883856379
 A standard reference that is fairly
technical:
 Owen, Guillermo. Game
Theory, 3^{rd
}ed. AP
. 1995. 0125311516
 A good brief work that is also
fairly technical:
 Aumann, Robert J. Lectures on
Game Theory. Westview. 1989.
 A well written text at the senior
level emphasizing economics is:
 Romp, Graham. Game Theory:
Introduction and Applications. Oxford. 1997. 0198775016
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Stochastic (Markov) Decision
Processeswill be covered in a future edition.
Stochastic
Processes (and Queueing)
Inventory
Theory and Scheduling
I am not to smitten with the books
in this area. For the second edition I will try to do better. Until
then, there is one excellent book in print. There is almost
certainly an excellent book to appear. The book by French is
excellent and is out of print and shouldn't be. The books by Conway
et al and Hadley et al were published in the sixties and are out of
print and despite that are first rate if you can get your hands on
them.

The book to have these days:
 Silver, Edward A., David F. Pyke,
and Rein Peterson. Inventory Management and Production Planning
and Scheduling, 3^{rd} ed. Wiley. 1998.
0471119474
 The following book is written by
top authorities who can write. So I would bet this will be a must
have book for its area:
 Lawler, E. L., J. K. Lenstra, and
A. H. G. Rinooy Kan. Theory of Sequences and Scheduling.
Wiley. Scheduled for 2000.
 A book that never should have gone
out of print:
 French, Simon. Sequencing and
Scheduling: An Introduction to the Mathematics of the JobShop.
Ellis Horwood. 1982. 0470272295
 Two outofprint classics:
 Conway, Richard W., William L.
Maxwell, and Louis Miller. Theory
of Scheduling. AW
. 1967. 1114499161

Hadley, G. and Whitin, T. M. Analysis
of Inventory Systems. PH
. 1963. 0130329533
 Another wellthought of book that
is out of print:
 Baker, Kenneth R. Introduction
to Sequencing and Scheduling. Wiley. 1974. 0471045551
 See also Cargal's lecture on The
EOQ Formula
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Investment Theory
This is a new area for me. There are a lot of books giving
contradictory advice or useless advice. Investment theory is
inherently mathematical, but there is a mathematical offshoot known
as "technical analysis." I have dealt with it for
more than twenty years myself, and I consider it generally nonsense.
Some of it is as bad as astrology. The better (technical
analysis) stuff is basically a dead end, or perhaps I should say
deadly end. The book by Malkiel aresses it well.
 One of the most readable books that seems to cover the topics
very well is:
 Paulos, John Allen. A Mathematician Plays the Stock
Market. Basic Books. 2003. 0465054811
 This book serves, to me, much like a glossary. It gives
descriptions and discussions of basic terminology.
 Fontanills, George A. and Tom Gentile. The Stock
Market Course. Wiley. 2001. 0471393150
 This book serves the same purpose is briefer and more readable
in my view. It covers wider ground than the first which seems
dedicated primarily to stocks.
 Caruso, David and Robert Powell. Decoding Wall
Street. McGraw Hill. 2002. 0071379533
 David Luenberger and Sheldon Ross are great writers on
operations research and applied mathematics, and are brilliant.
Luenberger is at Stanford and Ross is at Berkeley. Their books
on investment are for anyone who has a good knowledge of
undergraduate applied math. These books could easily be the
best two books on the subject. I would say Ross is the more
elementary. Get both.
 Luenberger, David. Investment Science.
Oxford. 1998. 0195108094; 0195125177
 Ross, Sheldon. An Elementary Introduction to
Mathematical Finance, 2^{nd} ed. Cambridge.
2003. 0521814294
 Don't let the title fool you. The book requires a
knowledge of calculus and some mathematical maturity.
 The following opus was a classic from its first edition in
1973. The second edition is thoroughly brought up to date.
 Malkiel, Burton G. A Random Walk Down Wall Street:
The Time Tested Strategy for Successful Investing 2^{nd}
ed. Norton. 2003. 0393325350
 I do not claim that the next book is useful for investing.
Perhaps it should be elsewhere. It is purely philosophical and
could be viewed as the Zen meditation guide that accompanies Random
Walk (the preceding book). It is however an interesting book.
 Taleb, Nassim Nicholas.
Fooled by Randomness: The
Hien Role of Chance in the Markets and in Life, 2^{nd}
ed. Texere. 2004. 0812975219
 This last work appears to present a contrary view to Random Walk
(Malkiel) but is not nearly as contrary as its title suggests.
A very interesting book. Perhaps I should have included it
with the first four.
 Stein, Ben and Phil DeMuth. Yes, You Can Time the
Markets. Wiley. 2003. 0471430161
 Two books about crashes (kind of). The book by Mandelbrot
is a good read. He has some major points. He can be
vague on mathematical details.
 Mandelbrot, Benoit and Richard L. Hudson. The
(mis)Behavior of Markets: A Fractal View of Risk, Ruin, and
Reward. Basic. 2004. 0465043550
 Sornette, Didier. Why Stock Markets Crash:
Critical Events in Complex Financial Systems. Princeton.
2003. 0691118507
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General
Physics
I really haven't gotten around to
this area yet. Secondly, I prefer to learn most physics from
specialized sources (for example to study mechanics, how about using
a book just on mechanics?). One series you are sure to hear about is
the great series by Feynman. Be aware, that it is probably more
useful to people who already have a knowledge of the subjects. Also,
it is a great reference. It deserves its reputation as a work of
genius, but in gneral I would not recommend it to someone just
beginning to learn physics.
 Feynman, Richard, Robert Leighton
and Matthew Sands. The
Feynman Lectures on Physics.
Three volumes. AW. 1964.
0201500647
 There are many fine one volume
summaries of physics aimed at an audience with some knowledge of
mathematics. The following, my favorite du jour, requires a
good knowledge of basic calculus through vector calculus.
 Longair, Malcolm.
Theoretical Concepts in Physics: An Alternative View,
2^{nd} ed. Cambridge. 2003. 052152878X
 The following book is good
exposition and is strong on mechanics and a good introduction to
tensors.
 Menzel, Donald H.
Mathematical Physics. Dover. 1961.
0486600564
 The following book is quite remarkable. It is a brief summary of physics. It seems to require some undergraduate mathematics. It is perfect for the mathematical scientist who did not study physics but wants an overview. It is an amazing book.
 Griffiths, David J. Revolutions in Twentieth Century Physics. Cambridge. 2013. 9781107602175

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Mechanics
There is a great classic, very
readable, by a major thinker, full of history, that goes back to
1893:
 Mach, Ernst. The Science of
Mechanics, 6^{th} (English) ed. Open Court. 1960.
0875482023
 Perhasp the best introduction for
the engineering or physics undergraduate is the following:
 Taylor, John R. Classical
Mechanics.
University Science Books. 2005. 189138922X
 A solid large exposition, fairly
slow:
 French, A. P. Newtonian
Mechanics. Norton. 1971. 0177710748
 French is one of the best
expositors of basic physics at the university level.
 A couple of concise well written
first books for the student who has been through the calculus
sequence:
 Smith, P, and R. C. Smith.
Mechanics, 2^{nd} ed. Wiley. 1990. 0471927376

Lunn, Mary. A First Course in
Mechanics. Oxford. 1991. 0198534337
 Books, still elementary, suitable
for a second look at mechanics:
 Kibble, T. W. B. and F. H.
Berkshire. Classical Mechanics, 4^{th}ed. Longman.
1996.
 Knudsen, J. M., and P. G. Hjorth.
Elements of Newtonian
Mechanics. SV
. 1995. 3540608419

Barger, Vernon, Martin Olsson.
Classical Mechanics: A Modern Introduction, 2^{nd}
ed. McGrawHill. 1995. 0070037345
 A more advanced book that
introduces Langrangian and Hamiltonian dynamics.
 Woodhouse, N. M. J. Introduction
to Analytical Dynamics. Oxford. 1987.
 A couple of thorough books:
 Greenwood, Donald T. Principles
of Dynamics, 2^{nd}
ed. PH . 1988. 0137099819
 Chorlton, F. Textbook of
Dynamics, 2^{nd} ed. Wiley (actually it is not clear
who published this). 1983. 0792353293
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Fluid
Mechanics
Thermodynamics
and Statistical Mechanics
There are several books for laymen
on the second law of thermodynamics. The first by Atkins is well
illustratedbasically it is a coffee table book. It is very good.
Atkins is one of the best science writers alive. The book by the
Goldsteins does a thorough job of discussing the history and
concepts of thermodynamics. It is also very good.
 Atkins, P. W. The Second Law.
Freeman. 1994. 071675004X

Atkins, Peter. Four
Laws that Drive the Universe.
Oxford. 2007. 9780199232369
 Goldstein, Martin, Inge F.
Goldstein. The Refrigerator and the Universe: Understanding the
Laws of Energy. Harvard. 1993. 0674753240

BenNaim, Arieh. Entropy
Demystified: The Second Law Reduced to Common Sense.
World Scientific. 2007. 9789812700520
 This book assumes no knowledge of
probability. It is probably of less interest to nerds.
 An unusual book in format that is
aimed at the serious student, but is definitely worth having:
 Perrot, Pierre. A to Z of
Thermodynamics. Oxford. 1998. 0198565569
 Three books that are as elementary
as can be at the calculus level are:
 Ruhla, Charles. The Physics of
Chance. Oxford. 1989. 0198539606

Whalley, P. B. Basic Engineering
Thermodynamics. Oxford. 1992. 0198562543

Van Ness, H. C. Understanding
Thermodynamics. Dover. 1969. 103pp. 0486632776
 Some more advanced texts that are
still at the undergraduate level. The book by Lawden is fairly
brief.
 Lawden, D. F. Principles of
Thermodynamics. Wiley. 1987. 0486446476

Lavenda, Bernard H. Statistical
Physics: A probabilistic Approach. Wiley. 1991.
0471546070
 A great undergraduate survey:
 Carter, Ashley H. Classical
and Statistical Thermodynamics.
PH. 2001. 0137792085
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Electricity
and Electromagnetism
Quantum
Mechanics
There are books that try to explain
quantum physics to the layman, i.e. without mathematics. For the
most part it is like trying to explain Rembrandt to a person who has
never possessed sight.

To start off with I'll mention one of
the nonmathematical (coffeetable) works:
 Hey, Tony and Patrick Walters. The
Quantum Universe. Cambridge. 1987. 0521564573
 Let me mention two that have a
minimal amount of mathematics (for books on QM).
 Ponomarev, L. I. The Quantum
Dice. Institute of Physics. 1993. 0750302518

Albert, David Z. Quantum Mechanics
and Experience. Harvard. 1992. 0674741129
 The book by Albert goes better
with some knowledge of linear algebra.
 Two rather unusual references:
 Brandt, Siegmund and Hans Dieter
Dahmen. The Picture Book of
Quantum Mechanics,
2^{nd}
ed. SV . 1995. 0387943803

Atkins, P. W. Quanta: A Handbook of
Concepts, 2^{nd} ed. Oxford. 1991. 0198555733
 Very nice technical introductions:
 Chester, Marvin. Primer
of Quantum Mechanics. Dover. 1987. 0486428788
 Phillips, A. C. Introduction
to Quantum Physics. Wiley. 2004. 0470853247
 Haken, H. Wolf, H. C. The
Physics of Atoms and Quanta: Introduction to Experiments and
Theory, 4^{th}ed.
SV . 1994. 0387583637
 French, A. P. and Edwin F. Taylor.
An Introduction to Quantum Physics. Norton. 1978.
0393091066
 Baggott, Jim. The Meaning of
Quantum Theory. Oxford. 1992. 019855575X

LévyLeblond, JeanMarc, and
François Balibar. Quantics: Rudiments of Quantum Physics.
North Holland. 1990.
 A much more comprehensive treatment
that can be a little hairy but nonetheless is as readable as this
stuff gets:
 Zee, A. Quantum Field
Theory in a Nutshell. Princeton. 2003.
0691010196
 A book for the individual with
comfort in QM.
 Bell, J. S. Speakable and
Unspeakable in Quantum Mechanics. Cambridge. 1993.
0521818621
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Relativity
The reason that there are so many
expositions of relativity with little more than algebra is
that special relativity can be covered with little more
than algebra. It is however rather subtle and deserves a lot of
attention. (A literature professor would explain that the special
relativity is a nuanced paradigm reflecting in essence
Einstein's misogyny.) As to general relativity it can not be
understood with little more than algebra. Rather, it can be
described technically as a real motherlover.

On the subject of general relativity
and covering special relativity as well, there is a magnum opus,
perhaps even a 44 magnum opus. This book is the book for any serious
student. I would imagine that graduate students in physics all get
it. It is 1279 pages long and it takes great pains to be
pedagogically sweet. Tensors and everything are explained ex vacua
(that is supposed to be Latin for out of nothing it probably
means death to the lefthanded). I have trouble seeing this
all covered in two semesters at the graduate level. It is formidable
but it is also magnificent.
 Misner, Charles W., John Archibald
Wheeler, Kip Thorne. Gravitation. Freeman. 1973.
0716703440
 Similarly, if I have to pick one
book on special relativity it would the following. The only caveat
here is that there are many fine books on special relativity and
some of them are less technical. Nonetheless the book avoids
calculus.
 Taylor, Edwin F. and John
Archibald Wheeler. Spacetime Physics: Introduction to Special
Relativity, 2nd ed. Freeman. 1992. 0716723271
 They now have a wonderful sequel on
general relativity. Although it can be read independently, I
strongly recommend reading Spacetime Physics first.
 Taylor, Edwin F. and John
Archibald Wheeler. Exploring
Black Holes: Introduction to General Relativity.
AWL. 2000. 020138423X
 Of the next four books on special
relativity, the first is less technical than the others.
 Epstein, Lewis Carroll. Relativity
Visualized. Insight Press. 1991. 0935218033

French, A. P. Special Relativity.
Norton. 1968. 1122425813

Born, Max. Einstein's Theory of
Relativity. Dover. 1965. 111452400X
 This has a last short chapter on
general relativity. (Born was a Nobel laureate.)
 Rindler, Wolfgang. Introduction
to Special Relativity, 2^{nd} ed. Oxford. 1991.
0198539525
 Two great introductions to general
relativity are:
 Callahan, J. J. The
Geometry of Spacetime: An Introduction to Special and General
Relativity. SV.
2000. 0387986413

Hartle, James B. Gravity:
An Introduction to Einstein's General Relativity.
AWL. 2003. 0805386629
 Here are five excellent books that
get into general relativity. The last two (Harpaz and Hakim) are
very mathematical and in my judgement Harpaz is the more elementary
of the two. The book by Bergman is wonderfully concise and clear.
 Gibilisco, Stan. Understanding
Einstein's Theories of Relativity: Man's New Perspective on the
Cosmos. Dover. 1983. 0486266591

Bergmann, Peter G. The Rile of
Gravitation. Dover. 1987. 1199965642
 Geroch, Robert. General
Relativity from A to B. University of Chicago. 1978.
0226288633

Harpaz, Amos. Relativity Theory:
Concepts and Basic Principles. A. K. Peters. 1993.
1568810261

Hakim, Rémi. An Introduction
to Relataivistic Gravitation. Cambridge. 1999. 0521459303
 Lastly, there is a reprint of a
1945 classic on special and general relativity by Lillian Lieber
with illustrations by her husband Hugh. This is an amazing
book; sort of Dr. Seuss with tensors.
 Lieber, Lillian. The
Einstein Theory of Relativity: A Trip to the Fourth
Dimension. Paul Dry
Books. 2008. 9781589880443
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Waves
Evolution
Anyone that argues that evolution is
improbable either does not understand natural selection or
probability and usually both.
 A similar statement can be made
about the 2^{nd} law of thermodynamics argument

Likewise the logictautology argument.
 The disagreement between the
Dawkin's (The Selfish Gene and all that) crowd and Gould and
Eldredge (see below) is to me, a nonargument. Dawkin's view is
perfectly logical. It is a hardcore Darwinistic viewpoint. Arguments
that it is missing something seem to me to miss the point. In the
end some genes survive and spread and others do not. Explanations of
why are basically post hoc rationalizations. That is not at all to
say that these rationalizations are without merit, but they in no
way mitigate against Dawkin's view.

Go to
http://www.nap.edu/readingroom/books/evolution98/
 A book by the 20^{th }century
master
 Mayr, Ernst. What Evolution Is.
Basic Books. 2001. 0465044263
 A great summary. Very readable. A
must for the library.
 If you want to point to a single
book that shows how natural selection accounts for evolution either
of the following two books do the job.

Carroll, Sean B. The
Making of the Fittest.
Norton. 2006. 9780393330519

Coyne, Jerry A. Why
Evolution is True.
Viking. 2009. 9780670020539

Whereas both of these books are
readable, the one by Coyne might be better for a general
audience.
 The most interesting book I've seen
recently is fascinating because of its refutation of creationist
arguments on one hand and ts arguement on the other hand that
natural selection is compatible with a loving God. The author's
scholarship is impressive.
 Miller, Kenneth. Finding
Darwin's God. Harper Collins. 1999. 0060930497
 A book that is good read but is
also a work of brilliance is
 Ruse, Michael. Can a Darwinian
be a Christian? Cambridge. 2001. 0521637163
 A best seller in 1999 that pretty
well demolishes the latest inanity from the creationists is:
 Pennock, Robert T. Tower of
Babel: The Evidence against the New Creationism. MIT. 1999.
0262661659
 Darwin's The Origin of the
Species, 1859, is still a great and marvelous book to read. I
suggest a reprint of the first edition.

A fascinating scholarly work about the
academic and intellectual framework under which Darwin worked is a
great companion to The Origin of the Species.
 Ruse, Michael. The Darwinian
Revolution: Science Red in Tooth and Claw, 2^{nd} ed.
The Uiversity of Chicago. 1999. 0226731650
 See also, Darwin's The Descent
of Man and Selection in Relation to Sex. (Princeton has
them in a single volume, 1981.

See BenAri.

The writing of E. O. Wilson is
generally recommended.

Dawkin's works The Blind Watchmaker,
Climbing Mount Improbable, and The Selfish Gene are
all recommended.

A recent book that I like a lot that I
think might appeal to math oriented readers is:
 Eldredge, Niles. The Pattern of
Evolution. Freeman. 1998. 219pp. 0716730464
 A very readable book about modern
genetic research is
 Sykes, Bryan. The Seven
Daughters of Eve. Norton. 2001. 0965026264
 Population Genetics.

The books I know on population
genetics–some classics and some out of print–tend to be
tomes. The following, at 174 pages, is more concise. It is readable
by someone with a basic course in probability and the elementary
sequence in calculus.
 Gillespie, John H. Population
Genetics: A Concise Guide. John Hopkins. 1998.
0801880084
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Philosophy
See
also Foundations (where two of the books
have the word Philosophy
in their titles).
 Feynman reportedly referred to
philosophy as "bullshit." I tend to agree although
philosophy of mathematics is important. There are good works
on it and there is serious bullshit. The following book is
delightful:

Casti, John L. The One True
Platonic Heaven. Joseph Henry Press (an imprint of the
National Academy of Sciences). 2003. 0309085470
 Feynman himself has a great book on
the nature of science. Far too clear and readable for
professional philsophers.

Feynman, Richard. The
Character of Physical Law. MIT. 1965.
 Another fine book on the nature of
science that is very readable and aresses recent controversies.

BenAri,
Moti. Just a Theory: Exploring the Nature of
Science. Prometheus Books. 2005.
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Science
Studies
 Science Studies is a new
discipline that began in Edinborough Scotland in the 1960's. It
claims to be interested in understanding the sociological workings
of science. However, practitioners explicitly assume that science
controversies are always resolved by politics and not by one theory
being actually better than another. They believe further that there
is no scientific method and the belief in such is naive. To them the
scientific method is a myth that is used by scientists as they
actually proceed through other means to achieve any consensus. Their
works invariably show that scientific results were the result of
politics and personalities and not based upon higher fundaments.
However, it is no great trick to prove a proposition when that
proposition happens to be your primary assumption!! The following
book is a brilliant scholarly work that touches upon science studies
and is the book that inspired Alan Sokal to perform his celebrated
hoax.
 Gross, Paul R. and Norman Levitt.
Higher Superstition: The Academic Left and Its Quarrels with
Science. John Hopkins. 1994. 0801847664
 See also articles on the Sokal
affair:

The Sokal Hoax: The Sham that
Shook the Academy. Bison Books. 2000.
0803279957
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Lectures on algorithms, number
theory, probability and other stuff
Another
Site
Abbreviations
MAA:
Mathematical Association of America
SV:
SpringerVerlag
AW:
AddisonWesley
AWL:
Addison Wesley Longman
HBJ:
Harcourt Brace Jovanovich.
AP:
Academic Press
PH:
Prentice Hall.
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