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 funded
entirely by J. M. Cargal
Write to
jmcargal@sprintmail.com
or James M. Cargal, PO Box 210667, Montgomery AL 36121-0667.

This is the most recent
photograph of James M. Cargal (used with permission).
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 1.48 October 29, 2008: Additions include one
book in Real Analysis, one book on General Relativity, one book on Dynamical
Systems and three books on Combinatorics.
Edition 1.47 May 5, 2008: Additions include two books on real analysis,
two books on thermodynamics, one on linear algebra, one on logic (Gődel)
one on geometry. May 22: A book added on Maxwell's equations to
section on electromagnetism.
Edition 1.46 November 12, 2007: Additions to
Abstract Algebra, Linear Algebra, Topology, Vector Calculus, Real Analysis,
Geometry, Logic (Gӧdel), Evolution, Mechanics.
Edition 1.45 (May 28, 2007): Additions and changes to
Calculus. Additions to Combinatorial Mathematics, and Complex
Analysis.
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).
Additions 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 © 1998-2008
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
Pre-Calculus
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, non-linear, 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.
Never miss lecture.
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.
Back to Top.
Principles
of Learning Calculus
If you have not had
pre-calc for two years or more, retake pre-calc!
- 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 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.
- Back to Top
Calculus
Pedagogy
The battle between reform
calc and traditional calc is unimportant. The problem they are trying to
address 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.
- Back to Top
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.
- Back to Top
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
effort worthwhile.
- Back to Top
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
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.
- Back to Top
Pre-Calculus
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 Addison-Wesley.
- Back to Top
Trigonometry
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. Addison-Wesley. 02011-3332-6
- 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. S-V . 1998.
0387982892
- Back to Top
Calculus
First, see
Principle of Learning Calculus.
- The smart calculus
student will use a study guide. There are many competent study guides
for calculus. A venerable classic is:
- Thompson, Silvanus P.
Calculus
Made Easy. St. Martin's Press.
03121-8548-0.
- Another example that
should become a classic is most highly recommended!!!
-
Hass, Joel, Thompson and Adams. How
to Ace Calculus: the Streetwise Guide. W. H. Freeman. 1998.
07167-3160-6
- Note that there is a sequel
that covers the second and third semesters including multi-variable calculus.
- However, as of 2007 there are two great additions to this genre.
These two books are inexpensive and should cover all the needs of the
struggling student during the first two semesters..
- Banner, Adrian. The Calculus LifeSaver.
Princeton University Press. 2007. 978-0-691-13088-0
- This covers all of single variable calculus, i.e. first and
second semester calculus.
- Kelly, W. Michael. The Humongous Book of Calculus
Problems. Alpha. 2007. 978-1-59257-512-1
- Another book that works as a resource, particularly in the second
semester and seems to be aimed at engineering students is:
- Bear, H. S. Understanding Calculus, 2nd ed.
Wiley. 2003. 04714-3307-1
- Bear is one of the best writers on analysis and this book is quite good.
- Don't forget
Gullberg !!!
Regular Calculus Texts
- The modern calculus book (now the standard or traditional model) starts with the two
volume set written in the 20's by Richard Courant. (The final version
of this is Courant and John). Most modern calculus texts (the
standard model) are
remarkably alike with the shortest one in popular use being Varburg/Parcell (Prentice-Hall: 0-13-081137-8) (post 1980 volumes tend to be more
than 1000pp!). You can often find one on sale at large bookstores
(which are constantly selling off books obtained from college bookstores).
- If one standard
calculus text really stands out for quality of writing and
presentation it would be:
- Simmons, George F.
Calculus
with Analytic Geometry,
2nd ed. McGraw-Hill. 0070576424
- This is really a great text!
- Another book, that is standard in format and but may not be the best for
most students just beginning calculus, is the one by Spivak.
If you want to have one book to review elementary
calculus this might be it. It is an absolute favorite amongst
serious students of calculus and nerds everywhere.
- Spivak, Michael. Caculus, 3rd ed.
Publish or Perish. 0-914098-89-6
- Beginning students might find it as good as Simmons though.
- The reformed calculus
text movement is best typified by the work of the Harvard Calculus Consortium:
- Hughes-Hallett,
Deborah, William G. McCallum, Andrew M. Gleason, et al. Calculus:
Single and Multivariable. Wiley. 04714-7245-X
- However, I am not at all sold on this as a good start to
calculus. I suspect it might be useful for reviewing
calculus.
- There is another
unique treatment that does a great job of motivating the material and I
recommend it for students starting out.
This book is also particularly good for students who are restudying the
topic. It is an excellent resource for teachers (and is around 600 pages):
- Strang, Gilbert. Calculus.
Wellesley-Cambridge. 09614-0882-0
- Still another book that the beginning (serious) student might
appreciate, by one of the masters of math history is:
- Kline, Morris. Calculus: An Intuitive and Physical
Approach. Dover. 0-486-40453-6
Other Books on Calculus
- 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. 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 this book.
- Comenetz, Michael. Calculus: The Elements. World
Scientific. 2002. 9810249047
- See also
Bressoud .
-
Back to Top
Linear
Algebra
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.
07167-4174-1
- Another very friendly text is:
- Beatrous, Frank and Caspar Curjel. Multivariate Calculus:
A Geometric Approach. 2002. P-H.
0130304379
- 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 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
- Back to Top
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, 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. 070080194
- 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. 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. 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 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. 0-306-31060-0.
- 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.
1-85233-015-5
- The following is pedagogically exceptional. I like it a lot.
- Schiff, Joel L. The Laplace Transform. Springer.
1999. 233pp. 0-387-98698-7.
Partial Differential Equations
Difference
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. 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, 2nd ed.
AP . 2004.
978-0-12-349703-1
Three elementary books
follow. The second and third seem to be particularly suited as texts
at the sophomore-junior level. They emphasize linear algebra whereas
Acheson is more differential equations and physics.
- Scheinerman, Edward R.
Invitation to Dynamical Systems. PH
. 1996. 0131850008
- Sandefur, James T.
Discrete Dynamical Systems: Theory and Applications. Oxford. 1990.
0198533845
- Acheson, David. From
Calculus to Chaos: An Introduction to Dynamics. Oxford. 1997.
0198500777
- 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. 3540609342
- 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.
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
- Back to Top
Real
Analysis
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,
2nd ed.
MAA. 2006.
978-0883857472
- Bressoud, David. A Radical Approach to
Lebesgue's Theory of Integration.
MAA.
2008. 978-0-521-71183-8
- 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. 978-0691136264
- 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 pre-requisites.
- 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, 2nd ed.
S-V
. 1991. 0387941088
- A very large (and
historic) lovely and complete two volume set is
- 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 terminology) is
- Hardy, G. H. A
Course in Pure Mathematics. Cambridge. 0521092272
- A great read in
analysis and best seller is
- Boas, R. P. A
Primer of Real Functions 4th 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,
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. 0-7637-1708-8
- 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. 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, 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. 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 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.
2004. 0883857375
- A very interesting book:
- Dunham, William. The Calculus Gallery: Masterpieces from
Newton to Lebesque. Princeton. 2005. 0691095655
- See also Courant and John.
- Back to Top
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. Non-Standard
Analysis. North-Holland. 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
- Back to Top
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.
0-691-12798-0
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, 2nded. S-V
. 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 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. 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, 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 .
1114121770
- 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. 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, 2nd
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
- Back to Top
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 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. 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. 3-540-76180-2
- 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 .
0813340802
- The following is concise and offers an introduction to tensors, may be
the best intro:
- Matthews, P. C. Vector Calculus. Springer. 1998.
3-540-76180-2
- 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. 0127425101
- A book that does a
good job of introducing differential forms is:
-
Bressoud,
David M. Second Year Calculus. S-V
. 1991. 038797606X
- Back to Top
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 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,
2nd ed. Oxford. 1998. 0-8194-2616-4
- 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: 978-0471198260
- 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. 0-691-07078-4
- 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. 3-540-65058-X
- Vol II/1 3-540-66569-2
- Vol II/2 3-540-66570-6
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 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:
-
Stillwell,
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. 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
- Back to Top
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.
978-0-691-11880-2
- 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
- (First published in 1949.)
- Back to Top
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, 2nd 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 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. 2nd.
ed. A-W . 1994. 0201558025
- Back to Top
Combinatorics
(Including Graph Theory)
The serious student who
wants to specialize in combinatorics should not specialize too much.
In particular you should take courses in number theory and
probability. Abstract algebra, linear algebra, linear
programming-these and other areas can be useful.
The first two books are good introductions of reasonable length (that
is they can be covered in one semester). The Polya-Tarjan book which
is superb are the notes from a course. The book by Ross is more
elementary and is more along the lines of the finite mathematics that
every student should know whereas Polya-Tarjan is dedicated
combinatorics including of course a nice treatment of Polya's
Counting Theory.
- As nice an
introduction as you will ever see (junior-senior level) is this:
- Pólya, George.
Robert E. Tarjan. Donald R. Woods. Notes on Introductory Combinatorics.
Birkhäuser. 1983. 0817631704
- Here are four books at roughly the junior-senior
level. These books are all readable and are selective in their
topics. By this I mean they avoid the too common approach of
throwing in everything including the kitchen sink.
- Anderson, Ian. A First Course in
Discrete Mathematics. Springer. 2001.
1-85233-236-0
- This (Anderson) is a great introduction for
the undergraduate math major.
- Lovász, L., J. Pelikán and K. Vesztergombi.
Discrete Mathematics: Elementary and Beyond.
2003. Springer. 978-0-387-95585-8
-
Andreescu, Titu and Zuming Feng. A Path to Combinatorics
for Undergraduates. Birkhäuser. 2004. 0-8176-4288-9
-
Benjamin, Arthur T. and Jennifer J. Quinn. Proofs that
Really Count: The Art of Combinatorial Proof.
MAA.
2003. 0-88385-333-7
- In some ways a masterly book. Great for
self study. A lot on Fibonacci numbers.
- Another brief introduction at the sophomore level with some
emphasis on logic and Boolean algebra. Does not touch either
probability or number theory.
- Haggarty, Rod. Discrete Mathematics for Computing.
A-W.
2002.
- A pedagogically solid book at the senior-graduate level devoted
to counting is
- Martin, George E. Counting: The Art if Enumerative Combinatorics.
Springer. 2001.
- This is in Springer's Undergraduate Text series but the first
hundred pages (out of 250) cover generating functions and get well
into Polya's counting theory.
- Another book that is
quite formalistic and dry and reflects pre-computer science and yet I
come back to again and again and is simply a favorite is:
- Berge, Claude. Principles
of Combinatorics. AP . 1971.
0120897504
- It also has an excellent treatment of Polya's counting theory.
- A book that is quite
comprehensive and that is well written is:
- Cameron, Peter J. Combinatorics:
Topics, Techniques, Algorithms. Cambridge. 1994.
0521457610
- This is a great book!
Its level is roughly senior to graduate school. (It is divided into
undergraduate and graduate halves.)
- A classic text at the senior/graduate level that covers lattices,
generating functions, matroids, incidence functions and other stuff
- Aigner, Martin. Combinatorial Theory. Springer.
1997. 3-540-61787-6
- The majority of standard texts on Discrete mathematics can be
quite uninspiring. If I have to pick a single junior-senior text that
is fairly conprehensive and seems designed for the classroom (with
like most such texts enough material for at least two semesters) I
would choose:
- Biggs, Norman L. Discrete Mathemtics, 2nd ed.
Oxford. 1993. 0198507186
- The following two books are at an undergraduate level but of
interest to many professionals. They are both good reads and they
overlap a number of disciplines, but arguably belong most to
combinatorics. Note they do not belong in Foundations
like the book by Ebbinghaus, H.-D. Et al. The book by Bunch is
excellent for the serious freshman-sophomore. The second book is more
advanced and includes a nice treatment of Conway's own surreal numbers.
- Bunch, Bryan. The Kindom of Infinite Number: A Field Guide.
Freeman. 2000.
- Conway, J. H. and R. K. Guy. The Book of Numbers.
Copernicus (S-V). 1996. 038797993X
- A superb first book on
graph theory is:
- Hartsfield, Nora,
Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introduction,
Revised ed. AP . 1994.
- In truth it is not
comprehensive. Secondly, although it covers algorithms it is not
computer oriented. Algorithms are very much secondary.
- For finite geometries go to Batten.
- Graph theory has
become important precisely because of algorithms. Let me mention two
excellent books in order of my preference.
-
Gibbons,
Alan. Algorithmic Graph Theory. Cambridge. 1985. 0521288819
- Even, Shimon. Graph Algorithms.
Computer Science Press. 1979. 0914894218
- Again, thinking of
computer science, let me mention another book:
- Stanton, Dennis,
Dennis White. Constructive Combinatorics. S-V
. 1986. 0387963472
- A very nice book at the senior-graduate level strictly devoted to
generating functions:
- Wilf, Herbert S. Generatingfunctionology, 2nd.
ed. AP . 1994. 0127519556
- For a more complete listing of works on graph theory go to
http://www.math.fau.edu/locke/graphstx.htm
- There are many books on Fibonacci numbers (and the
golden ratio). The following two are exceptionally clear and well
written. See also the book above by Benjamin and Quinn.
- Vorobiev, Nicolai N. Fibonacci Numbers.
Birkhäuser. 2002. 3-7643-6135-2
- Posamentier, Alfred S.
and Ingmar Lehman. The (Fabulous) Fibonacci Numbers.
Prometheus. 2007.
Back to Top
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, 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. 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
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.
-
Back to Top
Fourier
Analysis
The best book on Fourier
analysis is the one by Korner. However, it is roughly at a first year
graduate level and is academic rather than say engineering oriented.
Any graduate student in analysis should have this book.
- Korner,
T. W. Fourier Analysis. Cambridge. 1990. 0521389917
- My favorite work on
Fourier analysis (other than Korner) is by a first rate electrical engineer:
- Bracewell, Ronald. The
Fourier Transform and Its Applications, 2nd ed.
McGraw-Hill. 1986.
- Another book in a
similar vein has been reprinted recently (I think):
- Papoulis, Athanasios.
The
Fourier Integral and Its Applications. McGraw-Hill. 1962.
- A book with many
applications to engineering is
- Folland, Gerald B. Fourier
Analysis and its Applications. Wadsworth and Brooks/Cole. 1992.
0534170943
- The best first book
for an undergraduate who is not familiar with the material is very likely:
- Morrison, Norman. Introduction
to Fourier Analysis. Wiley. 1994. 047101737X
- This book is very user friendly!
- A fairly short book
(120pp) that is worthwhile is:
- Solymar, L. Lectures
on Fourier Series. Oxford. 1988. 0198561997
- A concise work (189pp), well written, senior level, which assumes
some knowledge of analysis, very nice:
- Pinkus, Allan, and Samy Zafrany. Fourier Series and Integral Transforms.
Cambridge. 1997. 0521597714
- A truly great short introduction:
- James, J. F. A Student's Guide to Fourier Transforms with Applications
in physics and Engineering. Cambridge. 1995. 052180826X
- It is now out in a second edition.
- Another short concise work:
- Bhatia, Rajendra. Fourier Series.
MAA. 2005.
- Back to Top
Number
Theory
Number theory is one of
the oldest and most loved mathematical disciplines and as a result
there have been many great books on it. The serious student will also
need to study abstract algebra and in particular group theory.
- Let me list four
superb introductions. These should be accessible to just about
anyone. The book by Davenport appears to be out of print, but not
long ago it was being published by two publishers. It might return
soon. The second book by Ore gives history without it getting in the
way of learning the subject.
- Ore, Oystein. Invitation
to Number Theory. MAA
. 1969. 1114251879
- Davenport, Harold. The
Higher Arithmetic: an Introduction to the Theory of Numbers.
0090306112
- Ore, Oystein. Number
Theory and its History. Dover. 0486656209
- Friedberg, Richard.
An
Adventurer's Guide to Number Theory. Dover. 1994. 0486281337
- There have been many
great texts on NT, but most of them are out of print. Here are five
excellent elementary texts that (last I knew) are still in print.
- Silverman, Joseph H. A Friendly Introduction to Number Theory,
3rd Ed. PH.. 2006.
0131861379
- Excellent text (Silverman) for undergraduate course!
- Dudley, Underwood. Elementary
Number Theory, 2nd Ed. Freeman. 1978. 071670076X
- Rosen, Kenneth R. Elementary
Number Theory and its Applications, 5th ed.
A-W
. 2005. 0201870738
- This text (Rosen) has evolved
considerably over the years into a lush readable text, strong on
applications, and basically a great text. Maybe the text to
have.
- Burton, David M. Elementary
Number Theory, 4th Ed. McGraw-Hill. 1998. 0072325690
- Burton is not the most elementary. He gets into arithmetic
functions before he does Euler's generalization of Fermat's Little Theorem.
However, many of the proofs are very nice. I like this one quite bit.
Like Rosen, the later editions are indeed better.
- An
Introductory Text that has a lot going for it is the one by Stillwell.
It has great material but is too fast for most beginners. Should
require a course in abstract algebra. Maybe the best second book
around on number theory.
- Stillwell, John. Elements of Number Theory. Springer.
2003. 0387955879
- A standard text that
is quite a bit more comprehensive than the four just given is:
- Niven, Ivan, Herbert
S. Zuckerman, Hugh L. Montgomery. An Introduction to the Theory of Numbers,
5thed. Wiley. 0471625469
- A remarkably concise
text (94pp) that covers more than some of the books listed above is:
- Baker, Alan. A
Concise Introduction to the Theory of Numbers. Cambridge. 1990.
0521286549
- Let me list a few more
very worthy books:
- Andrews, George E. Number
Theory. Dover. 1971. 0486682528
- Stark, Harold M. An
Introduction to Number Theory. 1991. MIT. 0262690608
- Rademacher, Hans. Lectures
on Elementary Number Theory. Krieger. 1984. 1114123064
- Hardy, G. H. and E. M. Wright.
The Theory of Numbers. 5th ed. Oxford. 354064332X
- This is
classic text but is somewhat advanced.
- Schroeder,
M. R. Number Theory in Science and Communication, 3rd
ed. S-V . 1997. 0387158006
- Also, see
Childs .
- A book I like a lot is the one by Anderson and Bell. Although
they give the proper definitions (groups on p. 129), I recommend it
to someone who already has had a course in abstract algebra. It has
applications and a lot of information. Well laid out. Out a very good
book to have.
- Anderson, James A. and James M. Bell. Number Theory with Applications. P-H
. 1997. 0131901907
- The first graduate level book to have on number theory might be
- Ireland, Kenneth and Michael Rosen. A Classical Introduction
to Modern Number Theory. 2nd ed. S-V.
1990. 038797329X
- Be careful on this book. The first edition was a different title
and publisher but, of course, the same authors.
- A very short work (115 pages) at the first year graduate
level covers a good variety of topics:
- Tenenbaum, G. and M. M. France. The Prime Numbers and Their Distribution.
American Mathematical Society. 2000. 821816470
- One book that I assume must be great is the following. I base
this on the references to it. However, I have never seen it and at
$180, the last I checked, I can't afford it.
- Sierpinski, Waclaw. Elementary Theory of Numbers. 2nd.ed.
North-Holland. 1987.
- A reissued classic that is well written requires, I think, a
decent knowledge of abstract algebra.
- Weyl, Hermann. Algebraic Theory of Numbers. Princeton.
1998. (First around 1941.) 0691059179
- The following text makes for a second course
in number theory. It requires a first course in abstract algebra (it
often refers to proofs in Stewart's Galois Theory which is listed in
the next section (Abstract Algebra)).
- Analytic Number Theory is a tough area and it is an area where I am
not the person to ask. However, in the early 2000's there appeared
three popular books on the Riemann Hypothesis. All three received
good reviews. The first one
(Derbyshire) does the best job in explaining the mathematics (in my
opinion). Although the subject is tough these books are
essentially accessible to anyone.
- Derbyshire, John. Prime Obsession. Joseph
Henry Press. 2003. 0309085497
- This is an offshoot of the National Academy of Sciences.
- Sabbagh, Karl. The Riemann Hypothesis: The
Greatest Unsolved Problem in Mathematics. Farrar, Straus,
Giroux. 2002. 1843541009
- Sautoy, Marcus du. The Music of the Primes:
Searching to Solve the Greatest Mystery in Mathematics.
Perennial. 2003. 0060935588
- A recent book that is a solid accessible introduction to analytic
number theory and highly recommended is
- Stopple, Jeffrey. A Primer of Analytic Number Theory:
From Pythagoras to Riemann. Cambridge. 2003.
0-521-01253-8
- Back to Top
Abstract
Algebra
Note, that at this time
the only book I have listed here that could be considered really
elementary is the one by Landin.
- Landin, Joseph. An
Introduction to Algebraic Structures. Dover. 1989. 0486659402
- A standard text is:
- Fraleigh, John B. A
First Course in Abstract Algebra, 5th ed.
A-W
. 1994.
0201763907
- It is a text for a
tough two semester course through Galois Theory.
- Herstein was one of
the best writers on algebra. Some would consider his book as more
difficult than Fraleigh, though it doesn't go all the way through
Galois Theory (but gets most of the way there). He is particularly
good (I think) on group theory.
- Herstein, I. N. Abstract
Algebra, 3rded. PH
. 1990. 0471368792
- Hernstein has a great book on abstract algebra at the graduate
level. It is thorough, fairly consise and beautifully written. He is
very strong on motivation and explanations. This is a four-star book
(out of four stars). It is one of the best books around on group
theory. His treatment there I think should be read by anyone
interested in group theory.
- Herstein, I. N. Topics in Algebra,
2nd. ed. Wiley. 1975. 1199263311
- The book by
Childs covers quite a bit of number theory as well as a whole
chapters on applications. It is certainly viable as a text, and I
definitely recommend it for the library.
- Childs, Lindsay N. A
Concrete Introduction to Higher Algebra, 2nd ed.
S-V
. 1995. 0387989994
- The following text may
be the best two-semester graduate text around. Starting with matrix
theory it covers quite a bit of ground and is beautifully done. I
like it a great deal. Note that some people consider this book
undergraduate in level.
- Artin, Michael. Algebra.
1991. PH. 0130047635
- A nice book for a
single semester course at the undergraduate level is:
- Maxfield, John E.
Margaret W. Maxfield. Abstract Algebra and Solution by Radicals.
Dover. 1971. 0486671216
- This book is a nice introduction to Galois Theory.
- The following is a fairly complete text which is strong on group
theory besides other topics.
- Hungerford, Thomas W. Abstract Algebra: An Introduction,
2nd ed.. Saunders. 1997. 0030105595
- The following, though, is the same author's graduate text which is
something of a standard.
- Hungerford, Thomas W.
Algebra. Springer. 1974. 978-0-387-90518-1
- A book I like at the graduate level is:
- Dummit, David S., Richard M. Foote. Abstract algebra, 2nd
ed. Wiley. 1990. 0471433349
- A Carus Monograph that spends time on field extensions and covers
some basic Number Theory over Gaussian Integers:
- Pollard, Harry and Harold Diamond. The Theory of Algebraic Numbers,
2nd ed. MAA.
1975. 0486404544
- Another book that I
like and which is a credit to one's library is:
- Dobbs, David E. and
Robert Hanks. A Modern Course on the Theory of Equations.
Polygonal Press. 1980. 0936428147
- Despite the title, the following is a book I think most students of
abstract algebra should check it out.
- Alaca, Şaban, and Kenneth S Williams.
Introductory Algebraic Number Theory. Cambridge.
2004. 0-521-54011-9
- Let me mention several
books on Galois Theory.
- As a rule even if some of these books do not presume a prior knowledge of
group theory, you should learn some group theory before hand.
- The first of these
books has a lot of other information and is certainly one of the best:
- Hadlock, Charles
Robert. Field Theory and Its Classical Problems.
MAA
. 1978. 0883850206
- Another nice
introduction is:
- Stewart, Ian. Galois
Theory, 3rd ed. Chapman and Hall. 2004.
1584883936
- This third edition is a significant update to the second edition.
May be the best introduction.
- My favorite is the
book by Stillwell. I don't think much of it as text, but it is a
great book to read. Despite the title, it is very much a book on
Galois Theory.
- Stillwell, John. Elements
of Algebra: Geometry, Numbers, Equations. S-V
. 1994. 0387942904
- Another book that is unusually clear and well written:
- Howie, John M. Fields and Galois Theory. Springer.
2006. 1-85233-986-1
- A succinct book and a classic is:
- Garling, D. J. H. A
Course in Galois Theory. Cambridge. 1986. 0521312493
- The most succinct book is
- Artin, Emil. Galois Theory.
Notre Dame. 1944. 0486623424
- It is beautifully
written but is not for the beginning student.
- Another succinct book similar to Artin's in every way is
- Postnikov, M. M. Foundations of Galois Theory. Dover.
2004. 0-486-43518-0
- Another book, that is very concise, is great for the reader who
already is fairly comfortable with group theory and ring theory. (It
is nota book for a first course in abstract algebra.)
- Rotman, Joseph. Galois Theory, 2nded. S-V
. 1998. 0387985417
- A book that is quite concrete on Galois Theory:
- Cox, David. Galois Theory. Wiley. 2004.
0-471-43419-1
- A unique book that
deserves mention here is:
- Fine, Benjamin, and
Gerhard Rosenberger. The Fundamental Theorem of Algebra.
S-V
. 1997. 0387946578
- This book ties
together algebra and analysis at the undergraduate level. Great
special study.
- If you are looking for
applications of abstract algebra, you should look first to Childs
. An elementary undergraduate small collection of applications is
given in:
- Mackiw, George. Applications
of Abstract Algebra. Wiley. 1985. 0471810789
- The following applied book strikes me as more of a resource than
a text.
- Hardy, Darel W. and Carol L. Walker. Applied Algebra: Codes,
Ciphers, and Discrete Algorithms. P-H.
2003. 0130674648
- A more advanced and
far more ambitious undertaking is:
-
Lidl,
Rudolf, and Günter Pilz. Applied Abstract Algebra.
S-V
. 1984. 0387982906
- The previous book overlaps another book also coauthored by Lidl:
- Lidl, Rudolp and Harald Niederreiter. Introduction to Finite
Fields and Their Applications, Revised Edition. Cambridge. 1994.
0521460948
- See also (for applications) Schroeder .
-
A senior level work on ring theory.
- Cohn, P. M. An Introduction to Ring Theory. Springer. 2000.
- See also the book on Fermat's last theorem by
Stewart and Tall in the
Number Theory section.
- The following book intends to shed light on Wiles's proof of Fermat's
Last Theorem. Supposedly it is aimed at an audience with minimal
mathematics, but it should be enlightening to students who have had a course
in Abstract Algebra who might find it fascinating.
- Ash, Avner, and Robert Gross. Fearless Symmetry: Exposing
the Hidden Patterns of Numbers. Princeton. 2006.
0-691-12492-2
- Group Theory
- Virtually all books on abstract algebra and some on number theory
and some on geometry get into group theory. I have indicated which of
these does an exceptional job (in my opinion). Here we will look at
books devoted to group theory alone.
- One of the most elementary and nicest introductions is:
- Grossman, Israel and Wilhelm Magnus. Groups and Their Graphs.
MAA. 1964.
088385614X
- This is my favorite introductory treatment. However, if you are
comfortable with groups, but are not acquainted with graphs of groups
(Cayley diagrams) get this book. Graphs give a great window to the subject.
- The next book is an introduction that goes somewhat further than
the Grossman book. It is quite good. I think it needs a second
edition. The first few sections strike me as a little kludgy (I know,
there should be a better word-but how much am I charging you for
this?) and might give a little trouble to a true beginner.
- Armstrong, M. A. Groups and Symmetry. S-V.
1988. 0387966757
- The following two books may be the best undergraduate texts on group
theory.
- Smith, Geoff and Olga Tabachnikova. Topics in Group theory.
S-V.
2000.
0852332352
- I like this a lot. I think this is the best on undergraduate
group theory. Would be a good text (does anyone have an undergraduate
course in group theory?)
- Humphreys, John F. A Course in Group Theory. Oxford.
1996. 0198534590
- This appears to be a standard reference in much of the elementary
literature.
- A rather obscure book that deserves some attention; despite the
title, this book is more groups than geometry (there are books on
groups and geometry in the geometry section). Also, it has some
material on rings and the material on geometry is non-trivial. It is
very good on group theory. Excellent at the undergraduate level for
someone who has already had exposure to groups.
- Sullivan, John B. Groups and Geometry. William C. Brown. 1994.
0697205851
- Perhaps the best (first) graduate books on group theory are
- Cameron, Peter J. Permutation Groups. Cambridge. 1999.
0521388368
- Cameron is one of the best writers in mathematics. See
combinatorics.
- Rotman, Joseph J. An Introduction to the Theory of Groups. 4th
ed. S-V. 1995. 0387942858
- I like this book a great deal.
- Another book that goes into graduate level that is worth a look
and quite inexpensive is
- Rose, John S. A Course on Group Theory. Dover. 1978.
0486681947
- A very good for group theory is the book
Topics
in Algebra by Herstein. Note both books by Herstein do a
good job, but the second is the one to have.
- See also in the section on
Abstract Algebra the books by Hungerford and
by Dummit and Foote.
- Back to Top
Geometry
- If I were to recommend just one book on geometry to an undergraduate
it would probably be
- Stillwell, John. The Four Pillars of
Geometry. Springer. 2005. 0-387-25530-3
- An even more recent book by Stillwell that can be classified as
geometry is the following. It recapitulates parts of several of
his earlier works and is a great pleasure to read (even if you have read
the others). It might make sense to read this first and then Four Pillars (immediately above).
- Stillwell, John. Yearning for the
Impossible: The Surprising Truths of Mathematics. A. K.
Peters. 2006. 1-56881-254-X
- For a general introduction
to much of geometry from a master:
- Coxeter, H. S. M. Introduction
to Geometry, 2nd ed. Wiley. 1969. 0471504580
- Another rather
extensive book by an authority second only to Coxeter is:
- Pedoe, Dan. Geometry:
A Comprehensive Course. Dover. 1970. 0486658120
- The title is correct;
this book makes for a comprehensive course, and in my view does it
better than does the book by Coxeter.
- A less ambitious but
readable work is:
- Roe, John.
Elementary Geometry. Oxford. 1994. 0198534566
- It covers affine and
projective geometries (only a little on projective), traditional
analytic geometry a little beyond a thorough treatment of the conics.
The last two chapters cover volume and quadric respectively. This is
a very viable text for an undergraduate course.
- The following two
books are intended as undergraduate texts. Both volumes are slim and
do a short course on Euclidean geometry and the development of
non-Euclidean geometry followed by affine and projective geometries.
The book by Sibley touches on a few other topics and may be a little
easier to read. I believe it was influenced heavily by Cederberg's
text. The design is very similar. She is better on projective
geometry though; I suspect he will touch that up for a second
edition. Also, when he does iterated fractal systems in 2 or 3
pages-I am not sure that that is worth the effort; do it thoroughly
or leave it.
- Cederberg, Judith N.
A
Course in Modern Geometries, 2nd ed.
S-V
. 1989. 0387989722
- Sibley, Thomas Q. The
Geometric Viewpoint: A Survey of Geometries. A-W
. 1998. 0201874504
- A book that is great for library and that is particularly strong
on affine and projective geometries is:
- Polster, Buckard. A Geometrical Picture Book.
S-V
. 1998. 0387984372
- Let me list four
excellent texts for the course on traditional Euclidean geometry and
the development of non-Euclidean geometry (principally hyperbolic geometry).
- Greenberg, Marvin Jay.
Euclidean
and Non-Euclidean Geometries: Development and History, 3rd
ed. Freeman. 1993. 0716724464
- Gans, David. An
Introduction to Non-Euclidean Geometry. AP
. 1973.
- For a quick introduction to hyperbolic geometry, I would suggest Gans. (Also
covers elliptic geometry.) 0122748506
- Martin, George E. The
Foundations of Geometry and the Non-Euclidean Plane.
S-V
. 1975. 0387906940
- A thorough treatment, perhaps compares to Hartshone (below).
- Trudeau, Richard J.
The
Non-Euclidean Revolution. Birkhäuser. 1987. 0817633111
- The four books listed above are all excellent! but there is a new
book on the same topic, by a great geometer, that I think is a
masterpiece. If this topic (traditional Euclidean geometry and the
development of non-Euclidean geometry) interests you, then you want
the damn book.
- Hartshone, Robin. Geometry: Euclid and Beyond. Springer. 2000.
0387986502
- A book devoted to the (complex) half-plane model of hyperbolic
geometry:
- Anderson, James W. Hyperbolic Geometry, 2nd ed.
Springer. 2005. 1-85233-934-9
- Two books devoted only
to groups and geometry:
- Nikulin, V. V. and I.
R. Shafarevich. Geometries and Groups. S-V
. 1987. 0387152814
- Lyndon, Roger C. Groups
and Geometry. Cambridge. 1985. 0521316944
- Many of the books
listed here spend much time on projective geometry. However, let me
list two books just on projective geometry, the more elementary book first:
- Coxeter, H. S. M. Projective
Geometry, 2nd ed. S-V
. 1987. 0387406239
- Coxeter, H. S. M. The
Real Projective Plane, 3rd ed. S-V
. 1993.
- The second book, in
particular, does stray from projective geometry a little.
- The following books
emphasize an analytic approach. Note, this is the mathematics that
lies under computer graphics. I like the book by Henle a great
deal. Note also that the analytic approach is treated nicely in the
books by Sibley, Cederberg, and Bennett.
- Henle, Michael. Modern
Geometries: The Analytic Approach. PH
. 1997. 013193418X
- I think that this is a
great book to have. I love it.
- Brannan, David A., Matthew F. Esplen and Jeremy J. Grey.
Geometry. Cambridge. 1999. 0521591937
- This book is a worthy competitor to Henle. Absolutely great:
- Ryan, Patrick J. Euclidean
and Non-Euclidean Geometry: An Analytic Approach. Cambridge. 1986.
0521276357
- Hausner, Melvin. A
Vector Approach to Geometry. Dover. 1998.
0486404528
- The following book emphasizes the connections between affine and
projective geometries with algebra. I think that the reader should
have some experience with these geometries and with abstract algebra.
- Blumenthal, Leonard M. A Modern View of Geometry.
Dover. 1980 (originally 1961).
- A concise well written summary of modern geometries which
(realistically) requires a course in linear algebra:
- Galarza, Ana Irene Ramirez and José Seade.
Introduction to Classical Geometries. Birkhauser.
2007. 978-3-7643-7517-1
- Other books of note.
- Bennett, M. K.
Affine and projective Geometry. Wiley. 1995. 0471113158
- Stillwell, John. Geometry
of Surfaces. S-V . 1992.
0387977430
- Sved, Marta. Journey
into Geometries. MAA
. 1991. 0883855003
- Coxeter, H. S. M. Non-Euclidean
Geometry. MAA
. 1998. 0883855224
- This is a
republication of a much older classic.
- Batten, Lynn Margaret. Combinatorics of
Finite Geometries, 2nd ed. Cambridge. 1997.
0521599938
- A very elementary book
of 80 pages (a good book for the talented high school student):
- Krause, Eugene F. Taxicab
Geometry: An Adventure in Non-Euclidean Geometry. Dover. 1986.
0201039346
- The book by Ogilvy is short and precious. It requires careful
study but is quite a gem. It covers inversion, conic sections,
and projective geometry and several other topics.
- Ogilvy, C. Stanley. Excursions in Geometry. Oxford.
1969. 0486265307
- Note that Ogilvy has been republished as a Dover paperback.
- Algebraic Geometry
- An elementary book in
algebraic geometry is:
- Bix, Robert. Conics
and Cubics: A Concrete Introduction to Algebraic Curves.
S-V
. 1998. 0387984011
- It is not as elementary as one might expect. It would be better
if it assumed knowledge of elementary linear algebra. I doubt that
individuals without this knowledge will read it.
- Another book that also is intended to be elementary is
- Gibson, C. G. Elementary Geometry of Algebraic Curves: An
Undergraduate Introduction. Cambridge. 1998. 0521646413
- Like most books with elementary intentions, it may require more
than it claims. Yes it provides the basic definitions of abstract
algebra, but I would recommend a course in abstract algebra before
reading this book.
- A more thorough and
advanced first book is
- Cox, David, John
Little, Donal O'Shea. Ideals, Varieties, and Algorithms: An
Introduction to Computational Algebraic Geometry and Commutative Algebra,
2nd ed. S-V . 1997.
0387946802
- Another much briefer
text is:
- Reid, Miles. Undergraduate
Algebraic Geometry. London Mathematical Society. 1988.
0521356628
- Differential Geometry
- A new book that is strong pedagogically and divides the material
into nice chunks (definitely senior level) is:
- Pressley, Andrew. Elementary Differential Geometry.
Springer. 2001. 1852331526
- A leisurely journey in a finely crafted book is:
- Stoker, J. J. Differential Geometry. Wiley. 1969.
0471828254
- This book has been reissued (2001?).
- Some elementary books
in ascending order of difficulty are
- Casey, James. Exploring
Curvature. Vieweg. 1996. 3528064757
- McCleary, John. Geometry
From a Differentiable Viewpoint. Cambridge. 1994. 0521424801
- Bruce, J. W., P.J.
Giblin. Curves and Singularities, 2nd ed.
Cambridge. 1992. 0521249457
- A great text that is
quite inexpensive is:
- Struik, Dirk J. Lectures
on Classical Differential Geometry, 2nd ed. Dover. 1961.
0486656098
- Other texts:
- Porteous, Ian R. Geometric
Differentiation for the Intelligence of Curves and Surfaces.
Cambridge. 1994. 0521002648
- Barrett O'Neill. Elementary
Differential Geometry, 2nd ed. AP
. 1998. 0125267452
- Do Carmo, Manfredo P.
Differential
Geometry of Curves and Surface. PH
. 1976. 0132125897
- Bloch, Ethan D. A
First Course in Geometric Topology and Differential Geometry.
Birkhäuser. 1997. 0817638407
- Gamkrelidze, R. V.
Editor. Geometry I. S-V . 1991.
0387519998
- Back to Top
Topology
I have yet to meet a book
that is on just point set topology that I adore. The following book
(which is not just on point set topology) is very good:
-
Simmons,
George F. Introduction to Topology and Modern Analysis.
Krieger. 1983. 0898745519
- The following is a very nice introduction that is as elementary a
treatment you will see of a great mix of topics:
- Crossley, Martin D. Essential Topology.
Springer. 2005. 1-85233-782-6
- Another book that is well written and inexpensive is:
- Mendelson, Bert. Introduction to Topology, 3rd
ed. Dover. 1990. 0486663523
- Another book with
quite a bit of point set topology is:
- Steen, Lynn Arthur, J.
Arthur Seebach, Jr.Counterexamples in Topology. Dover. 1978.
048668735X
- A fairly compact
covering of several topics (I am not sure if it really belongs in the
series "Undergraduate Texts in Mathematics"):
- Singer, I. M., J. A. Thorpe.
Lecture Notes on Elementary Topology and Geometry.
S-V
. 1967. 0387902023
- A very nice
algebraically oriented text (as well as combinatorial):
- Blackett, Donald W.
Elementary
Topology: A Combinatorial and Algebraic Approach.
AP
. 1982. 112405121X
- A superb text by one
of the best expository writers in mathematics:
- Stillwell, John. Classical
Topology and Combinatorial Group Theory, 2nd ed.
S-V
. 1993. 0387979700
- Three more texts in
algebraic topology:
- McCarty, George.
Topology, An Introduction with Application to Topological Groups.
Dover. 1967. 1124055053
- Croom, Fred H. Basic
Concepts of Algebraic Topology. S-V
. 1978. 0387902880
- Wall, C. T. C. A
Geometric Introduction to Topology. Dover. 1972. 0486678504
- Back to Top
Set
Theory
By set theory, I do not
mean the set theory that is the first chapter of so many texts, but
rather the specialty related to logic. See the section on
Foundations as there are books there with a
significant amount of set theory.
- A particularly fine
first book, if still in print, is
- Henle, James M. An
Outline of Set Theory. S-V . 1986.
0387963685
- Two superb texts are:
- Devlin, Keith. The
Joy of Sets: Fundamentals of Contemporary Set Theory.
S-V
. 1993. 0387940944
- Moschovakis, Yiannis
N. Notes on Set Theory. S-V . 1994.
0387941800
- A classic that should
be of interest to the serious student (specialist) is (it is also out
of print):
- Cohen, Paul J. Set
Theory and the Continuum Hypothesis. 0805323279
- Back to Top
Logic
and Abstract Automata (and computability and languages)
- For the specialist student
in logic, I think the Oxford publications of Raymond
Smullyan should be de rigueur.
- If you are going to have one book on logic, I recommend:
- Wolf, Robert S. A Tour Through Mathematical Logic.
MAA. 2005.
0883850362
- See Dewdney .
- The following books
are very nice overview/introductions:
- Rosenberg, Grzegorz,
and Arto Saloma. Cornerstones of Undecidability.
PH
. 1994.
- Epstein, Richard L.
and Walter A. Carnielli. Computability: Computable Functions, Logic,
and the Foundations of Mathematics. Wadsworth and Brooks/Cole. 1989.
- Bridges, Douglas S.
Computability:
A Mathematical Sketchbook. S-V . 1994.
- Wang, Hao. Popular
Lectures on Mathematical Logic. Dover. 1981.
- Boolos, George S. and
Richard C. Jeffrey.Computability and Logic, 3rd ed.
Cambridge. 1989. 0521007585
- The following are also
good introductions:
- Hamilton, A. G. Logic
for Mathematicians, revised ed. Cambridge. 1988. 0521368650
- Lyndon, Roger C. Notes
on Logic. Van Nostrand. 1966.
- Enderton, Herbert B.
A
Mathematical Introduction to Logic. AP
. 1972. 0122384520
- Cutland, N. J. Computability:
An Introduction to Recursive Function Theory. Cambridge. 1980.
0521294657
- This is a great introduction on computability.
- Good books on just
automata and languages:
- Brookshear, J. Glenn.
Theory
of Computation: Formal Languages, Automata, and Complexity.
Benjamin/Cummings. 1989. 0805301437
- This is a more elementary or pedagogical work than Hopcroft and Ullman.
- Linz, Peter. An
Introduction to Formal Languages and Automata, 2nd ed. Heath. 1997.
0763714224
- This is the
pedagogical work. It covers less than Hopcroft and Ullman and is
aimed at a slightly lower level, but is in many ways the best written
book and is the
book to teach from.
- Kozen, Dexter C. Automata
and Computability. S-V . 1997.
0387949070
- See comment on the
next book
- Hopcroft, John E. And
Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. A-W
. 1979. 020102988X
- This is the standard, but is perhaps threatened by the more
recent Kozen.
- Loeckx, J. Computability
and Decidability: An Introduction for Students of Computer Science. S-V
. 1970. 0387058699
- This last book is
quite concise: 76pp.Its entire approach is via Turing machines.
- The following are a
little more advanced books on logic (but are still introductory and
reasonably paced):
- Ebbinghaus, H.-D., J.
Flum and W. Thomas.Mathematical Logic. S-V
. 1984. 0387942580
-
Smullyan,
Raymond M. First-Order Logic. Dover. 1995. 0486683702
- Smullyan, Raymond M.
Gödel's
Incompleteness Theorems. Oxford. 1992. 0195046722
- Smullyan, Raymond M.
Recursion
Theory for Metamathematics. Oxford. 1993. 019508232X
- Matiyasevich, Yuri V.
Hilbert's
Tenth Problem. MIT. 1993. 0262132958
Gödel
- There is a celebrated treatment for all readers of Gödel's
Incompleteness Theorem. This book received a Pulitzer and was a
significant event. (More concisely, the book received a lot of hype
and derserved it.)
- Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal
Golden Braid. Basic Books. 1979. 0465026850
- A nice very short treatment of Gödel's
incompleteness theorem it the article:
- Hehner, Eric C. R. Beautifying Gödel
pp.
163-172, in Beauty is Our Business: A Birthday Salute to
Edgar W. Dijkstra. S-V. 1990.
3540972994
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.
The following is a good introduction to Godel's
incompleteness theorem as well as providing a very useful discussion of
its abuses:
- Franzen, Torkel.
Godel's Theorem: An Incomplete Guide to its Use and Abuse A. K. Peters. 2005. 1-566881-238-8
- 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. 978-0-521-67453-9
Back to
Top
Foundations
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 pp).
- 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. 0-19-927041-4
- 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.
- Back to Top
Algorithms
The four volumes of D.
E. Knuth, The Art of Computing,
Addison-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.
- 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; 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 (1983).
- 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 telegraph reviews:
- Haupt, Randy and Sue Ellen Haupt. Practical Genetic Algorithms.
Wiley. 1998. 0471455652
- Back to Top
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 .
- Back to Top
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, 8th 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, 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 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.
McGraw-Hill. 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.
Harper-Collins. 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.
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 practitioner.
- 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 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, 3rd 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, 2nd 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,
2nd ed. Wiley. 1993. 0471371246
- Back to Top
Operations
Research (and linear, non-linear, 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, 3rd 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
- Back to Top
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, 3rd 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 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 EOQ
Formula
- Back to Top
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 addresses
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.
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 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
Hidden 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. 2003.
0691118507
- Back to Top.
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.
-
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 tensors.
- Menzel, Donald H. Mathematical Physics. Dover.
1961. 0-486-60056-4
- Back to Top
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, 6th (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. 1-891389-22-X
- 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, 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. 1996.
- Knudsen, J. M., and P.
G. Hjorth. Elements of Newtonian Mechanics.
S-V
. 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
- Back to Top
Fluid
Mechanics
Thermodynamics
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 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.
P-H. 2001. 0137792085
- Back to Top
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 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 Quantum Mechanics, 2nd 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 Theory,
4thed. S-V . 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
- 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 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
- Back to Top
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 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.
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, 2nd 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. S-V.
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
Riddle 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. 978-1-58988-044-3
- Back to Top
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 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.
- Go to
http://www.nap.edu/readingroom/books/evolution98/
A book by the 20th 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. 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 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, 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.See
Ben-Ari.
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
Back
to Top
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
addresses recent controversies.
- Ben-Ari, Moti. Just a Theory: Exploring the Nature of Science. Prometheus Books.
2005.
- Back to Top
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
- Back to Top
Lectures on algorithms, number theory, probability and other stuff
Another Site
Abbreviations
MAA:
Mathematical Association of America
S-V: Springer-Verlag
A-W: Addison-Wesley
AWL: Addison Wesley Longman
HBJ:
Harcourt Brace Jovanovich.
AP:
Academic Press
PH:
Prentice Hall.
Back to Top