Recommended Books in the
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recommendations here, are those of J. M. Cargal and do not reflect
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James M. Cargal
- This is the most recent photograph of James M. Caral (used
- 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
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 © 1998-2012
can copy, but with proper attribution.
of Teaching and Learning Mathematics
Study it Twice
Two Books for
Undergraduates in the Mathematical Sciences
(ODE's and PDE's)
Dynamical Systems and Chaos
Calculus (modern theory of infinitesimals)
Calculus, Tensors, Differential Forms
General Applied Math
General Advanced Mathematics
General Computer Science
Combinatorics (including Graph
Logic and Abstract Automata
(logic and set theory)
Operations Research (and
linear, non-linear, integer programming, and simulation)
Stochastic Processes (and
Inventory Theory and Scheduling
Thermodynamics and Statistical
Electricity and Electromagnetism
on algorithms, number theory, probability and other stuff
Related Sites for Mathematical
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
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
- 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.
If you have not had pre-calc for two years or more, retake
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 aid-a book of the type: "calculus for absolute
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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The battle between reform calc and traditional calc is
unimportant. The problem they are trying to aress is that
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.
Delta-epsilon 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 delta-epsilon
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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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
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 six-month 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
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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 camera-ready 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. 05218-9067-5
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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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
With the preceding in mind I prefer books in the workbook format.
An excellent textbook series is the series by Bittinger published by
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Trig like pre-calculus 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. Aison-Wesley.
There is an excellent treatment of trig in
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
- There are many short fascinating articles on trigonometry in:
- Apostol, Tom M., et al. Selected Papers on Precalculus.
- 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.
S-V . 1998. 0387982892
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Principle of Learning
- 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.
0471000051 and 0471000078
- A book that is about calculus but falls short of analysis is:
- Klambauer, Gabriel. Aspects of Calculus. S-V
. 1986. 03879-6274-3
- 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
non-standard analysis (although Comenetz is clearly familiar with
it). I think many engineers and physicists would love
- Comenetz, Michael. Calculus: The Elements.
World Scientific. 2002. 9810249047
- See also Bressoud .
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- 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. 978-0-521-14568-8
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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
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.
- Another very friendly text is:
- Beatrous, Frank and Caspar Curjel. Multivariate Calculus:
A Geometric Approach. 2002. P-H.
- Often texts in advanced calculus concentrate on multivariable
calculus. A particularly good example is:
- Kaplan, Wilfred. Advanced Calculus, 3rd ed.
A-W . 0201799375
- A nice introductory book:
- Dineen, Seán. Functions of Two Variables. Chapman
and Hall. 1584881909
- Se also:
- Dineen, Seán. Multivariate Calculus and Geometry.
S-V . 1998. 185233472X
- A quicker and more sophisticated approach but well written is:
- Craven, B.D. Functions of Several Variables. Chapman and
- An inexpensive Dover paperback that does a good job is:
- Edwards, C. H. Advanced Calculus of Several Variables.
- 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
- 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 Self-Teaching Guide. 2007. McGraw Hill.
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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, 4th ed. Wiley.1989. 04710-9881-7
- 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, 2nd ed. Schaum (McGraw-Hill). 1994.
- Still looking at the standard model, a particularly complete and
enthusiastic volume is:
- Braun, Martin. Differential Equations and Their
Applications, 3rd ed. S-V
. 1983. 0387908471
- An extremely well written volume is:
- Simmons, George F. Differential Equations with Applications
and Historical Notes, 2nd ed. McGraw-Hill. 1991.
- 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. A-W
. 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. S-V
. 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. S-V.
Part 1. 1990. 0-387-97286-2 (Part II)
Higher-Dimensional Systems. 1995. 0-387-94377-3
- 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. 08672-0289-0
- 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. Gian-Carlo Rota. Ordinary Differential
Equations. Wiley. 1978. 0471860034
Note that both authors are very distinguished mathematicians.
See Dynamical Systems and
The Laplace Transform
- I have three books to list on this topic.
Kuhfittig, Peter K. F. Introduction to the Laplace
Transform. Plenum. 1978. 205pp.
- 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.
- The following is pedagogically exceptional. I like it a
Partial Differential Equations
Dynamical Systems and Chaos
Two classics that precede the current era of hyper-interest in
this area are (both are linear algebra intensive)
- Luenberger, David G. Introduction to Dynamic Systems:
Theory, Models, & Applications. Wiley. 1979.
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, 2nd ed. AP
. 2004. 978-0-12-349703-1
- Four more books at the junior senior level that can double as
references on differential equations:
- Hale, J. and H. koçak. Dynamics and Bifurcations.
S-V . 1991. 079231428X
Verhulst, Ferdinand. Nonlinear Differential Equations and
Dynamical Systems. S-V . 1985.
Strogatz, Steven H. Nonlinear Dynamics and Chaos with
Applications to Physics, Biology, Chemistry, and Engineering.
A-W . 1994. 3540609342
Banks, John, Valentina Dragan and Arthur Jones. Chaos:
A Mathematical Introduction. Cambridge. 2003.
- A book that I think should be of interest to most applied
Schroeder, Manfred. Fractals, Chaos, Power Laws:
Minutes From an Infinite Paradise. Freeman. 1991.
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- 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. 978-1434843678.
- Bruckner, Andrew M., Judith B. Bruckner, Brian S. Thomson. Real Analysis, 2nd ed. 2008. www.classicalrealanalysis.com. 978-1434844125.
For the student seeing analysis for the first time and who is
overwhelmed by analysis, there are a few books out there. A good
- 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.
This book is unusual amongst its kind for its inclusion of
- 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,
2nd ed. MAA.
Bressoud, David. A Radical Approach to Lebesgue's Theory
of Integration. MAA.
- 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.
- 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 one-semester 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 two-semester sequence out of another text. Minimal
- 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
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, 2nd ed. S-V
. 1991. 0387941088
- A very large (and historic) lovely and complete two volume set
- Courant, Richard. Fritz John. Introduction to Calculus and
Analysis. S-V . 354065058X
- A thorough treatment of undergraduate analysis is given in
- Bartle, Robert G. The Elements of Real Analysis, 2nd
ed. Wiley. 0471054623
- A resource wonderful for its proofs and examples (and outdated
- Hardy, G. H. A Course in Pure Mathematics. Cambridge.
- A fairly large book that is very good on undergraduate analysis and is applied is
- Estep, Donald. Practical Analysis in One Variable. 2002. Springer. 0-387-95484-8
- 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 4th
- 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
- 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, 3rd 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.
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. 0-471-17978-7
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.
- 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, 2nd ed.
S-V . 1995. 03879-4357-9
- 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. S-V . 1965.
- 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.
- A book influenced by Gordon's and also well written:
Burk, Frank. A Garden of Integrals.
2007. 9 780883 853375
- Every graduate student of analysis should have:
- Carothers, N. L. Real Analysis. Cambridge. 2000.
- 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 non-specialist.
- 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.
- A very interesting book:
Dunham, William. The Calculus Gallery: Masterpieces
from Newton to Lebesque. Princeton. 2005.
- 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.
genesis, by the creator, is tough reading:
- Robinson, Abraham. Non-Standard Analysis. North-Holland.
- The best introduction by far is:
- Henle and Kleinberg. Infinitesimal Calculus. MIT. 1979.
This has been republished (2003) as inexpensive Dover
- A book that is supposed to be easy but is very abstract is:
- Robert, Alain. Nonstandard Analysis. Wiley. 1985.
- 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.
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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.
Perhaps the most remarkable book in this area; truly great book
- Needham, Tristan. Visual Complex Analysis. Oxford.
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,
2nded. S-V . 1997.
- 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.
- The workhouse introduction, particularly suited to engineers has
- Brown, James Ward and Ruel V. Churchill. Complex Variables
and Applications 6th 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,
2nd ed. A-W . 1994.
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, 3rd
ed. P-H. 2003. 0133321487
- Two more introductions worth mentioning are:
- Palka, Bruce P. An Introduction to Complex Function Theory.
S-V . 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. A-W .
- A book this is maybe more thorough than those above is
- Marsden, Jerrold E. and Michael J. Hoffman. Basic
Complex Analysis, 2nd ed. Freeman. 1987.
- 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
- A classic work (first published in 1932) that is thorough.
- Titmarsh, E. C. The Theory of Functions, 2nd
ed. Oxford. 1997. 0198533497
Essentially the third correction (1968) of the second edition
- 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.
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Calculus, Tensors, Differential Forms
See Multivariable Calulus
See Courant and John
A great pedagogical work most highly recommended especially to
- Schey, H. M. Div, Grad, Curl, and All
That: An Informal Text on Vector Calculus 3rded..
Norton. 1997. 0393093670
- A fairly comprehensive work I like a lot is:
- Marsden, Jerrold E., Anthony J. Tromba. Vector Calculus,
4rd ed. Freeman.
- This may be the best book to have. It is very good.
- 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.
- A user friendly texts on vector calculus:
- Colley, Susan Jane. Vector Calculus, 2nd ed.
P-H. 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, 2nd
ed. S-V . 1994. 038794088X
Danielson, D. A. Vectors and Tensors in Engineering and Physics,
2nd ed. A-W .
- The following is concise and offers an introduction to tensors,
may be the best intro:
Matthews, P. C. Vector Calculus. Springer.
- On differential forms I recommend
- Bachman, David. A Geometric Approach to Differential
Forms. Birkhäuser. 2006. 0-8176-4499-7
- Edwards, Harold M. Advanced Calculus: A Differential Forms
Approach. Birkhäuser. 1994. 0817637079
Weintraub, Steven H. Differential Forms: A Complement to Vector
Calculus. AP . 1997.
- A book that does a good job of introducing differential forms
- Bressoud, David M. Second Year
Calculus. S-V . 1991.
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General Applied Math
There are roughly 37 zillion books on applied math (with titles
like Mathematics for Left-Handed 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.
Wellesley-Cambridge Press. 2007. 978-0-961408-81-7
A masterpiece and a must have for the library of every applied
- 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, 2nd ed. Oxford. 1998.
- 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, 3rd
ed. Wiley. 2005. ISBN-10: 0471198269; ISBN-13:
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
- A tour de force at the graduate level; a book for the serious
- 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
- 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.
- 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. 978-0691125657.
- Courant and John
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. 3-540-65058-X
- Vol II/1 3-540-66569-2
Vol II/2 3-540-66570-6
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 mid-forties.
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:
John. Numbers and Geometry. S-V
. 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. S-V.
- The following is a modern classic
- Davis, Phillip J., Reuben Hersh, Elena Marchisotto. The
Mathematical Experience. Birkhäuser. 1995.
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.
- 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.
- 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.
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,
ed. A-W . 1992. 0201504014
- Another fine book-a great
tutorial-seems 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
- 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. 2nd.
ed. A-W . 1994. 0201558025
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(Including Graph Theory)
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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.
This is a great book for projects and
for reading. I would like to know however how it has done as a
- A book by a great applied
mathematician that is worth having is:
- Hamming, R. W. Numerical
Methods for Scientists and Engineers, 2nd 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, 2nd ed. Brooks/Cole. 1998.
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 pseudo-code for
algorithms. This is okay with me for the following reasons. Given
a textbook with good pseudo-code, 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
- A very good treatment for the
student of logic:
- Smith, Peter. An
Introduction to Gődel's Theorems.
Cambridge. 2007. 978-0-521-67453-9
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, 2nd 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
- Eves, Howard.
Foundations and Fundamental concepts of Mathematics, 3rd
ed. PWS-Kent. 1990. 048669609X
- A book that fits as well into
foundations as anywhere is:
- Ebbinghaus, H.-D. Et al. Numbers.
S-V . 1990.
- A book I like a lot (senior level
in my view) is
- Potter, Michael. Set
Theory and its Philosophy. Oxford. 2004.
- 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) 0-486-43520-2
- See also the previous section.
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The four volumes of D. E. Knuth,
The Art of Computing, Aison-Wesley 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.
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
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; McGraw-Hill 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
- Aho, Alfred V., John E. Hopcroft,
and Jeffrey D. Ullman. The
Design and Analysis of Computer Algorithms.
A-W . 1974. 0201000296
Aho, Alfred V., John E. Hopcroft, and
Jeffrey D. Ullman. Data
Structures and Algorithms.
A-W . 1983. 0201000237
- A rather theoretical tour of
algorithmic theory and select topics:
- Kozen, Dexter C. The
Design and Analysis of Algorithms.
S-V . 1992. 0387976876
- I have not seen the following book
but it had a very tantalizing review (as an introduction) in the AMM
- Haupt, Randy and Sue Ellen Haupt.
Practical Genetic Algorithms. Wiley. 1998. 0471455652
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and Information Theory
The second edition will include
recommendations on books on Digital Filters and Signal Analysis
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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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, 8th ed. Iowa State.
- A newer book in the spirit of
Snedecor et al but requiring calculus is:
- McPherson, Glen. Applying and
Interpreting Statistics: A Comprehensive Guide, 2nd
ed. Springer. 2001. 0387951105
- Like Snedecor, this book is
packed with real-life 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, 3rd
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
- 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.
- Another book on mathematical
statistics that merits attention is
- Mood, Alexander McFarlane.
Introduction to the Theory of Statistics. McGraw-Hill.
- 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. Harper-Collins. 1993.
- 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
- 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.
- 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. S-V
. 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 non-technical introduction (sort of Sampling for
Dummies) by Stuart. The book by Thompson is for the
- Stuart, Alan. Ideas of
Sampling, 3rded. 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 built-in 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
- 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, 3rd ed. Edward Arnold. 1990.
Bloomfield, Peter. Fourier Analysis
of Time Series: An Introduction. Wiley. 1976. 0471889482
- A book on nonparametric methods:
- Conover, W. J. Practical
Nonparametric Methods, 2nd ed. Wiley. 1980.
- 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, 2nd
ed. Wiley. 1993. 0471371246
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Research (and linear, non-linear, integer programming, and
edition will cover both decision theory and games of
the J H. Conway variety.
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.
- 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
- Owen, Guillermo. Game
. 1995. 0125311516
- A good brief work that is also
- 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.
Processes (and Queueing)
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
The book to have these days:
- Silver, Edward A., David F. Pyke,
and Rein Peterson. Inventory Management and Production Planning
and Scheduling, 3rd ed. Wiley. 1998.
- 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 Job-Shop.
Ellis Horwood. 1982. 0470272295
- Two out-of-print classics:
- Conway, Richard W., William L.
Maxwell, and Louis Miller. Theory
of Scheduling. A-W
. 1967. 1114499161
Hadley, G. and Whitin, T. M. Analysis
of Inventory Systems. PH
. 1963. 0130329533
- Another well-thought 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
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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, 2nd ed. Cambridge.
- 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 2nd
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, 2nd
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.
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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. A-W. 1964.
- 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,
2nd ed. Cambridge. 2003. 052152878X
- The following book is good
exposition and is strong on mechanics and a good introduction to
- Menzel, Donald H.
Mathematical Physics. Dover. 1961.
- 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. 978-1-107-60217-5
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There is a great classic, very
readable, by a major thinker, full of history, that goes back to
- Mach, Ernst. The Science of
Mechanics, 6th (English) ed. Open Court. 1960.
- Perhasp the best introduction for
the engineering or physics undergraduate is the following:
- Taylor, John R. Classical
University Science Books. 2005. 1-891389-22-X
- A solid large exposition, fairly
- 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
- Smith, P, and R. C. Smith.
Mechanics, 2nd 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, 4thed. Longman.
- Knudsen, J. M., and P. G. Hjorth.
Elements of Newtonian
. 1995. 3540608419
Barger, Vernon, Martin Olsson.
Classical Mechanics: A Modern Introduction, 2nd
ed. McGraw-Hill. 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, 2nd
ed. PH . 1988. 0137099819
- Chorlton, F. Textbook of
Dynamics, 2nd ed. Wiley (actually it is not clear
who published this). 1983. 0792353293
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and Statistical Mechanics
There are several books for laymen
on the second law of thermodynamics. The first by Atkins is well
illustrated--basically 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. 978-0-19-923236-9
- Goldstein, Martin, Inge F.
Goldstein. The Refrigerator and the Universe: Understanding the
Laws of Energy. Harvard. 1993. 0674753240
Ben-Naim, Arieh. Entropy
Demystified: The Second Law Reduced to Common Sense.
World Scientific. 2007. 978-981-270-052-0
- 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
- Lawden, D. F. Principles of
Thermodynamics. Wiley. 1987. 0486446476
Lavenda, Bernard H. Statistical
Physics: A probabilistic Approach. Wiley. 1991.
- A great undergraduate survey:
- Carter, Ashley H. Classical
and Statistical Thermodynamics.
P-H. 2001. 0137792085
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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 non-mathematical (coffee-table) 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
ed. S-V . 1995. 0387943803
Atkins, P. W. Quanta: A Handbook of
Concepts, 2nd 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
S-V . 1994. 0387583637
- French, A. P. and Edwin F. Taylor.
An Introduction to Quantum Physics. Norton. 1978.
- Baggott, Jim. The Meaning of
Quantum Theory. Oxford. 1992. 019855575X
Lévy-Leblond, Jean-Marc, 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
- Zee, A. Quantum Field
Theory in a Nutshell. Princeton. 2003.
- A book for the individual with
comfort in QM.
- Bell, J. S. Speakable and
Unspeakable in Quantum Mechanics. Cambridge. 1993.
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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 mother-lover.
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 left-handed). 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.
- 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
- 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, 2nd ed. Oxford. 1991.
- Two great introductions to general
- Callahan, J. J. The
Geometry of Spacetime: An Introduction to Special and General
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.
Harpaz, Amos. Relativity Theory:
Concepts and Basic Principles. A. K. Peters. 1993.
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. 978-1-58988-044-3
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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 2nd law of thermodynamics argument
Likewise the logic-tautology argument.
- The disagreement between the
Dawkin's (The Selfish Gene and all that) crowd and Gould and
Eldredge (see below) is to me, a non-argument. 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.
- A book by the 20th century
- 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. 978-0-393-33051-9
Coyne, Jerry A. Why
Evolution is True.
Viking. 2009. 978-0-670-02053-9
Whereas both of these books are
readable, the one by Coyne might be better for a general
- 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.
- 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, 2nd 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.
The writing of E. O. Wilson is
Dawkin's works The Blind Watchmaker,
Climbing Mount Improbable, and The Selfish Gene are
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.
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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
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
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.
Moti. Just a Theory: Exploring the Nature of
Science. Prometheus Books. 2005.
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- 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
- 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
The Sokal Hoax: The Sham that
Shook the Academy. Bison Books. 2000.
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Lectures on algorithms, number
theory, probability and other stuff
Mathematical Association of America
Addison Wesley Longman
Harcourt Brace Jovanovich.
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