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.
  
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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.
  
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Principles of Teaching and Learning Mathematics

People like to go from simple models and examples to abstraction later. This is the normal way to learn.


• There is nothing wrong to learning the syntax of the area before the theory.
  
• Too much motivation can be as bad as too little.
  
• As you learn concepts, let them digest; play with them and study them some more before moving on to the next concept.
  
• When you get into a new area, there is something to be said for starting with the most elementary works. For example, even if you have a Ph.D. in physics, if you are trying to learn number theory but have no knowledge of the subject go ahead and start with the most elementary texts available. You are likely to find that you will penetrate the deeper works more ably than if you had started off with deeper works.
  
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Study it Twice!

A basic principle is this: most serious students of mathematics start to achieve depth in any given area the second time they study it. If it has been three or four years since you had the calculus sequence, go back and study your old text; you might be surprised by how different (and easier) it seems (and how interesting). Often if one comes back to a discipline after a 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.


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Two Books for Undergraduates in the Mathematical Sciences

Jan  Gullberg was a Swedish surgeon. When his son decided to major in engineering, Dr. Gullberg sat down and wrote a book containing all the elementary mathematics he felt every beginning engineer should know (or at least have at his disposal). He then produced the book in camera-ready English. The result is almost a masterpiece. It is the most readable reference around. Every freshman and sophomore in the mathematical sciences should have this book. It covers most calculus and everything up to calculus, including basic algebra, and solutions of cubic and quartic polynomials. It covers some linear algebra, quite a bit of geometry, trigonometry, and some complex analysis and differential equations, and more. A great book:

• Gullberg, Jan. Mathematics From the Birth of Numbers. Norton. 1997. 1093pp.  039304002X
• There are loads of books at many levels on mathematics for engineers and/or scientists. The following book is as friendly as any, and is well written. In many ways it is a companion to Gullberg in that it starts primarily where Gullberg leaves off. (There is some overlap, primarily basic calculus, but I for one don't think that is a bad thing.) It covers much of the mathematics an engineer might see in the last year as an undergraduate. Not only are there the usual topics but topics one usually doesn't see in such a book, such as group theory.
  • K. F.  Riley, Hobson, M. P., Bence, N. J. Mathematics Methods for Physics and Engineering. Cambridge. 1997. 1008pp. 05218-9067-5
    • I might mention that Mathematical Methods for Physicists by Arfken and Weber ( AP ) has a very good reputation, but I can't vouch for it personally (since I have never studied it). It is aimed at the senior level and above.
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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.
  
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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
    • Also in General Math .
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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, 2^nd 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, 2^nd 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, 3^rd 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 .
  
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Linear Algebra

There are a great many competent texts in this area. The best is

• Strang, Gilbert. Linear Algebra and Its Applications. 3^rd ed. HBJ .  0155510053
  
• This book is must have. It undoubtedly the most influential book in its area since Halmos's Finite Dimensional Vector Spaces.  S-V.  1124042660
• Strang has a second book on linear algebra. This is a more appropriate text for the classroom, especially at the sophomore level:
  • Strang, Gilbert. Introduction to Linear Algebra. 3^rd Ed. Wellesley-Cambridge.  2003.  0961408898
    • My thinking at this writing is that this is the best first text to use.
    • Also, I think that with the third edition the may supersede the HBJ text as the best single book on LA.
• The prototype of the abstract linear algebra text is Finite Dimensional Vector Spaces by Paul Halmos ( S-V ). A more recent book along similar lines is:
  • Curtis, Morton L. Abstract Linear Algebra. S-V .  03879-7263-3
• A slightly more elementary treatment of abstract linear algebra than either of these is:
  • Axler, Sheldon. Linear Algebra Done Right. 2^nd ed. S-V .   0387982590
    • I like this book a lot.
• An advanced applied text is:
  • Lax, Peter D. Linear Algebra. Wiley.  0471111112
• I am not alone in arguing that the most important perspective on linear algebra is its connection with geometry. A book emphasizing that is:
  • Banchoff, Thomas, and John Wermer. Linear Algebra Through Geometry. 2^nd ed. 1992. S-V .  0387975861
    • Still whether this is a good text for a first course is arguable. It is certainly an interesting text after the first course.
• I think that maybe the best text for the traditional course at the sophomore-junior level may well be:
  • Lay, David C. Linear Algebra and Its Applications, 2^nd ed. A-W . 1998.  0201824787
    • There are a lot of subtle points to his treatment. He does a nice job of introducing a surprising number of the key ideas in the first chapter. I think somehow that this has a great pedagogical payoff. Although it is very similar to many other texts, I like this particular text a great deal.
• If choosing a text for a sophomore level course, I myself would choose the book by Lay or the one by Strang (Wellesley-Cambridge Press).
• The following book has merit and might work well as an adjunct book in the basic linear algebra course.  It is the book for the student just learning mathematics who wants to get into computer graphics.
  • Farin, Gerald and Dianne Hansford.  The Geometry Toolbox:  For Graphics and Modeling.  A. K. Peters.  1998.  1568810741
• The following book is concise and very strong on applications:
  • Liebler, Robert A.  Basic Matrix Algebra with Algorithms and Applications.  Chapman and Hall.  2003.  1584883332
• The following book is a good introduction to some of the more abstract elements of linear algebra.  Also strong on applications.  An excellent choice for a second book:
  • Robert, Alain M.  Linear Algebra:  Examples and Applications.   World Scientific.  2005.  981-256-499-3
• The following is also a great text to read after the first course on LA.  It is well written and is abstract but will throw in a section for physicists.  I like this book quite a bit.
  • Jänich, Klaus.  Linear Algebra.   Springer-Verlag.  1994.  0-387-94128-2
• A good book explicitly designed as a second book is:
  • Blyth, T. S. and  E. F. Robertson.  Further Linear Algebra.  2002.  Springer.  1-85233-425-8
    
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Multivariable Calculus

See Vector Calculus, Tensors, and Differential Forms.  Also see Courant and John..

Most standard calculus texts have a section on multivariable calculus and many sell these sections as separate texts as an option. For example the Harvard Calculus Consortium mentioned in Calculus sell their multivariable volume separately.


• The most informal treatment is the second half of a series. This is a great book for the student in third semester calculus to have on the side.
  • Adams, Colin, Abigail Thompson and Joel Hass. How to Ace the Rest of Calculus: the Streetwise Guide. Freeman. 2001.  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, 3^rd 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
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Differential Equations

Like in some other areas, many books on differential equations are clones. The standard text is often little more than a cookbook containing a large variety of tools for solving d.e.'s. Most people use only a few of these tools. Moreover, after the course, math majors usually forget all the techniques. Engineering students on the other hand can remember a great deal more since they often use these techniques. A good example of the standard text is:

• Ross, Shepley L. Introduction to Ordinary Differential Equations, 4^th ed. Wiley.1989.  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, 2^nd 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, 3^rd ed. S-V . 1983.  0387908471
• An extremely well written volume is:
  • Simmons, George F. Differential Equations with Applications and Historical Notes, 2^nd 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

The standard text in this area has been:

• Ward, James Brown. Ruel V. Churchill. Fourier Series and Boundary Value Problems. 5^th ed. McGraw-Hill. 1993.  070082022
• I like the following:
  • Farlow, Stanley J. Partial Differential Equations for Scientists and Engineers. Dover. 1993.   048667620X
    • Very nice formatting. Lots of pictures.
• A new book that is also very attractive:
  • O'Neil, Peter V. Beginning Partial Differential Equations. Wiley. 1999.  0471238872
• Another new book by one of the best writers alive on applied math, corresponds precisely to a one-semester course:
  • Logan, J. David. Applied Partial Differential Equations. Springer. 1998.  03872-0953-0
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Difference Equations

A Classic introduction. Elementary and a quick read.

• Goldberg, Samuel. Introduction to Difference Equations. Dover. $9.  11240-4587-2
• There are two fairly recent texts that I think are attractive. Both are considerably more in depth than Goldberg's. (Read his first.)
  • Elaydi, Saber, N. An Introduction to Difference Equations, 2nd ed. S-V . 1999.  0387230599
    
  • Kelley, Walter G. and Allan C. Peterson.  Difference Equations: An Introduction with Applications. Wiley. 1991.  012403330X
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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
• 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
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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, 2^nd ed.  MAA. 2006.  978-0883857472
  • Bressoud, David.  A Radical Approach to Lebesgue's Theory of Integration.  MAA.  2008.  978-0-521-71183-8
• 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, 2^nd 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, 2^nd ed. Wiley.  0471054623
• A resource wonderful for its proofs and examples (and outdated terminology) is
  • Hardy, G. H. A Course in Pure Mathematics. Cambridge.    0521092272
• A great read in analysis and best seller is
  • Boas, R. P. A Primer of Real Functions 4^th ed. MAA.   088385029X
• See also Simmons .
  
• The following book is very well written it covers much of analysis into Lebesgue measure. The chapter are short and break the material into digestible chunks making the book a great reference, study guide and first rate text. This may be the least appreciated book on analysis.
  • Bear, H. S. An Introduction to Mathematical Analysis. AP. 1997.  0120839407
• The following texts I consider graduate level. These all cover some abstract integration (almost always the Lebesque Integral).
  
• The standard graduate text is
  • Royden, H. L. Real Analysis, 3^rd ed. PH . 1988.  0120839407
• If I had to recommend a single book, it might be:
  • Jones, Frank.  Lebesgue Integration on Euclidean Space, Revised ed.  Jones and Bartlett.  2001.  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, 2^nd 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
    • Strongly recommended!
• 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.
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Infinitesimal Calculus (modern theory of infinitesimals)

This section is not for beginners! If you are just learning calculus go to the section Calculus.
The genesis, by the creator, is tough reading:

• Robinson, Abraham. 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
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Complex Analysis

The following book is a primer on complex numbers that ends with a short introduction to Complex Analysis.  It is a perfect book for the sophomore in math or engineering.  Great book:

• Nahin, Paul J.  An Imaginary Tale:  The Story of √-1.  Princeton University.  1998.  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, 2^nded. 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
    • Has good examples.
• A thorough well written text I like is:
  • Ablowitz, Mark J. and Athanassios S. Fokas. Complex Variables: Introduction and Applications. 1997. Cambridge.  0521534291
• The workhouse introduction, particularly suited to engineers has been:
  • Brown, James Ward and Ruel V. Churchill. Complex Variables and Applications 6^th ed. 1996.  0079121470
• Another book very much in the same vein as Brown and Churchill is preferred by many people,
  • Wunsch, A. David. Complex Variables with Applications, 2^nd ed. 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, 3^rd 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, 2^nd ed. Freeman. 1987.  0716721058
• A book that I regard as graduate level has been described as the best textbook ever written on complex analysis:
  • Boas, R. P. Invitation to Complex Analysis. Birkhauser Boston.  0394350766
• A classic work (first published in 1932) that is thorough.
  • Titmarsh, E. C. The Theory of Functions, 2^nd ed. Oxford. 1997.  0198533497
    • Essentially the third correction (1968) of the second edition (1939).
• A reference that I expect to sell very well to a wide audience:
  • Krantz, Steven G. Handbook of Complex Analysis. Birkhäuser. 1999.  0817640118
• The following is in one of Springer's undergraduate series but I think is more suited for grad work. The author says it should get you ready for Ph.D. qualifiers. Definitely a superior work.
  • Gamelin, Theodore W. Complex Analysis. Springer. 2000.  0387950699
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Vector Calculus, Tensors, Differential Forms

See Multivariable Calulus

See Courant and John

A great pedagogical work most highly recommended especially to electrical engineers

• Schey, H. M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus 3^rded.. Norton. 1997.  0393093670
• A fairly comprehensive work I like a lot is:
  • Marsden, Jerrold E., Anthony J. Tromba. Vector Calculus, 4^rd ed. Freeman.
    • This may be the best book to have. It is very good.  0716724324
• A short (and cheap) work that is concise and well written is
  • Hay, G. E. Vector and Tensor Analysis. Dover. 1953 (original date with original publisher).  0486601099
• Another short and concise treatment that is well written is
  • Matthews, P. C.  Vector Calculus.  Springer.  1998.  3-540-76180-2
• A user friendly texts on vector calculus:
  • Colley, Susan Jane. Vector Calculus, 2^nd 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, 2^nd ed. S-V . 1994.  038794088X
    
  • Danielson, D. A. Vectors and Tensors in Engineering and Physics, 2^nd 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
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General Applied Math

There are roughly 37 zillion books on applied math (with titles like Mathematics for Left-Handed Quantum Engineers)


• Check out Gullberg , it was specifically written for engineering students though it is appropriate for all students of math
  
• A very nice book which, appropriate for its author, emphasizes linearity is:
  • Strang, Gilbert. Introduction to Applied Mathematics. Wellesley Press.  0961408804
• A recent book that is pedagogically very nice and goes though junior level material with wide coverage extending to group theory is Riley et al.
• A great tool for applied mathematicians:
  • Andrews, Larry C.  Special Functions of Mathematics for Engineers, 2^nd ed.  Oxford.  1998.  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, 3^rd 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

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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
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General Advanced Mathematics

• 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.)
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General Computer Science

The books here tend cover algorithms and computability but don't forget to go the sections Algorithms and Logic and Computability .


• A. K. Dewdney wrote a book of 66 chapters to briefly and succinctly cover the interesting topics of computer science. The emphasis here is theory. This is a book every computer science major should have, and probably every math major and certainly anyone with a serious interest in computer science.
  • Dewdney, A. K. The New Turing Omnibus. Freeman. 1993.  0716782715
• A nice introduction that is good at introducing the concepts and philosophy of computer algorithms is
  • Harel, David. Algorithmics: The Spirit of Computing, 2^nd ed. 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. 2^nd. ed. A-W . 1994.  0201558025
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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
• 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 much more comprehensive text has a lot of room for the instructor to build a course of his (or her) choosing:
  • Maurer, Stephen B., Anthony Ralston. Discrete Algorithmic Mathematics, 2^nd ed. A. K. Peters. 1998.  1568810911
• 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, 2^nd 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, 2^nd. ed. AP . 1994.  0127519556
• For a more complete listing of works on graph theory go to http://www.math.fau.edu/locke/graphstx.htm
  
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Numerical Analysis

Most books on numerical analysis are written to turn off the reader and to encourage him or her to go into a different, preferably unrelated, field. Secondly, almost all of the books in the area are written by academics or researchers at national labs, i.e. other academics. The kind of industry I use to work in was a little different than that. The problem is partly textbook evolution. I've seen books long out of print that would work nicely in the classroom. However, textbook competition requires that newer books contain more and more material until the book can become rather unwieldy (in several senses) for the classroom. The truth is that the average book has far too much material for a course. Numerical analysis touches upon so many other topics this makes it a more demanding course than others.


• A marvelous exception to the above is the book by G. W. Stewart. It avoids the problem just mentioned because it is based upon notes from a course. It is concise and superbly written. (It is the one I am now teaching out of.)
  • Stewart, G. W. Afternotes on Numerical Analysis. SIAM. 1996.  0898713625
• Volume II, despite the title, is accessible to advanced undergraduates. If you liked the first text you want this:
  •  Stewart, G. W. Afternotes goes to Graduate school: Lectures on Advanced Numerical Analysis. SIAM. 1998.  0898714044
• Two great books on the subject are written by a mathematician with real industrial experience. The first is absolutely superb. Both books are great to read, but I don't like either as a text.
  • Acton, Forman. Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations. Princeton. 1995.  0691036632
    
  • Acton, Forman. Numerical Methods That Work. MAA . 1990.   1124037799
    • This is a reprint with corrections of an earlier work published by another publisher.
• An interesting book that seems in the spirit of the first book by Acton (above) is:
  • Breuer, Shlomo, Gideon Zwas. Numerical Mathematics: A Laboratory Approach. Cambridge. !993.  0521440408
    • This is a great book for projects and for reading. I would like to know however how it has done as a text.
• A book by a great applied mathematician that is worth having is:
  • Hamming, R. W. Numerical Methods for Scientists and Engineers, 2^nd ed.. Dover. 1987.  0486652416
• The book I use in the classroom is (although I intend to try G. W. Stewart).:
  • Asaithambi, N. S. Numerical Analysis: Theory and Practice. Saunders. 1995.  0030309832
• A textbook that looks very attractive to me is:
  • Fairs, J. Douglas, Richard Burden. Numerical Methods, 2^nd ed. Brooks/Cole. 1998.   0534392008
    • This is about as elementary as I can find. This is the problem with teaching the course. On the flip side of course, it covers less material (e.g. fixed point iteration is not covered). Also, it does not give 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 wih an emphasis on MAPLE and a disk comes with the package, which I have ignored.
• See the book by Cooper.
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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, 2^nd 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.
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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,  3^rd Ed.  PH..  2006.  0131861379
    • Excellent text (Silverman) for undergraduate course!
  • Dudley, Underwood. Elementary Number Theory, 2^nd Ed. Freeman. 1978.  071670076X
    
  • Rosen, Kenneth R. Elementary Number Theory and its Applications, 5^th 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, 4^th 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, 5^thed. 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. 5^th ed. Oxford.  354064332X
    •  This is classic text but is somewhat advanced.
• Schroeder, M. R. Number Theory in Science and Communication, 3^rd 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. 2^nd 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
    • I like this book a lot.
• 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. 2^nd.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)).
  • Stewart, Ian and David Tall.  Algebraic Number Theory and Fermat's Last Theorem, 3^rd ed.  A. K. Peters.  2002.  1568811195

• 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
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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, 5^th 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, 3^rded. 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, 2^nd 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, 2^nd 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, 2^nd 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, 2^nd 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, 3^rd 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, 2^nded. 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. 4^th 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.
  
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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, 2^nd 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, 2^nd 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, 3^rd 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, 2^nd 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, 2^nd ed. S-V . 1987.  0387406239
    
  • Coxeter, H. S. M. The Real Projective Plane, 3^rd 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
    • Compare this book with Banchoff and Wermer.  Also compare with Farin and Hansford.
• 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, 2^nd 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, 2^nd 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, 2^nd ed. Cambridge. 1992.  0521249457
• A great text that is quite inexpensive is:
  • Struik, Dirk J. Lectures on Classical Differential Geometry, 2^nd 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, 2^nd 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
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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, 3^rd 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, 2^nd ed. S-V . 1993.  0387979700
• Three more texts in algebraic topology:
  • M^cCarty, 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
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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
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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.