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 associ ated. (Other views are incorrect.) This site does not take money from publishe rs, authors, or their agents. It is funded entirely by J. M. Cargal Write to jmcargal@sprintmail.com or James M. Cargal, PO Box 242548, Montgomery A L 36124.

This is the most recent photograph of James M. Caral (used with permission). Edition 1.52 April 1, 2012. Three books added on real analysis. One on advanced calculus. Two on combinatorics. One on group theory. Edition 1.5 October 14, 2011: An essay: Elements of Boolean Algebra (22 pages) Note that there is also a chapter on Boolean Algebra in the Lectures on algorit hms, number theory, probability and other stuff link below. Edition 1.49 January 26, 2009: One book on General Advanced Mathematics. One book on General Applied Mathematics. Three books added to Combinatorics ? two on Fibonacci numbers (the other is very strong on Fibonacci numbers as well). O ne book on evolution. Edition 1.4 (Jan 19, 2006): Due to the efforts of Bob Hofacker I have added ISB N numbers to most books here. However, these are here only as an aid. It is ea sy to switch them around or have the wrong edition. Also added here are two boo ks on Abstract Algebra and one on Logic. Edition 1.31 (June 7, 2003): Cargal's lecture on The EOQ Formula for manufactur ing (added to section on Inventory). Aitions in 1.3 (Jan 22, 2003) : Two books in Number Theory. Also a new section : Lectures on algorithms, number theory, probability and other stuff. Site Created December 1998. Copyright ? 1998-2012 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 prerequisi te. The belief that it will come back quickly has scuttled thousands of careers. Study every day C 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 le arn will only go into short term memory. Eventually, you will reach a level wher e 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 book s, 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 you r 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 aress is that most people come out of the calculus sequence with superficial knowledge of the subject. However, the students who survive wi th 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 d eeper 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 p lace. Delta-epsilon proofs in the initial sequence are generally a waste and are abusi ve. 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 seq uence 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 th e most elementary works. For example, even if you have a Ph.D. in physics, if yo u are trying to learn number theory but have no knowledge of the subject go ahea d and start with the most elementary texts available. You are likely to find tha t 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 interest ing). Often if one comes back to a discipline after a six-month layoff (from tha t discipline, not from math) it seems so different and much easier than it was b efore. 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 book s 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 diff erent viewpoints will reinforce each other in a manner that makes the effort wor thwhile. 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 st upid questions is a sign of intelligence. Back to Top

Two Books for Undergraduates in the Mathematical Sciences Jan Gullberg was a Swedish surgeon. When his son decided to major in engineerin g, Dr. Gullberg sat down and wrote a book containing all the elementary mathemat ics he felt every beginning engineer should know (or at least have at his dispos al). He then produced the book in camera-ready English. The result is almost a m asterpiece. It is the most readable reference around. Every freshman and sophomo re in the mathematical sciences should have this book. It covers most calculus a nd everything up to calculus, including basic algebra, and solutions of cubic an d quartic polynomials. It covers some linear algebra, quite a bit of geometry, t rigonometry, and some complex analysis and differential equations, and more. A g reat book: Gullberg, Jan. Mathematics From the Birth of Numbers. Norton. 1997. 1093pp. 039 304002X There are loads of books at many levels on mathematics for engineers and/or scie ntists. The following book is as friendly as any, and is well written. In many w ays it is a companion to Gullberg in that it starts primarily where Gullberg lea ves off. (There is some overlap, primarily basic calculus, but I for one don't t hink that is a bad thing.) It covers much of the mathematics an engineer might s ee 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 En gineering. Cambridge. 1997. 1008pp. 05218-9067-5 I might mention that Mathematical Methods for Physicists by Arfken and Weber ( A P ) has a very good reputation, but I can't vouch for it personally (since I hav e 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 alway s 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 m ove 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 Aison-Wesle y. Back to Top Trigonometry Trig like pre-calculus algebra and calculus itself tends to be remarkably simila r from one text to another.

A good example of the genre is: Keedy, Mervin L., Marvin Bittinger. Trigonometry : Triangles and Functions. Aison-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 sophist icated 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 . Back to Top

Calculus First, see Principle of Learning Calculus. The smart calculus student will use a study guide. There are many competent stud y 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 aitions 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. 97 8-0-691-13088-0 This covers all of single variable calculus, i.e. first and second semester calc ulus. 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 s eems 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 t his is Courant and John). Most modern calculus texts (the standard model) are r emarkably alike with the shortest one in popular use being Varburg/Parcell (Pren tice-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 of f books obtained from college bookstores). If one standard calculus text really stands out for quality of writing and prese ntation it would be: Simmons, George F. Calculus with Analytic Geometry, 2nd ed. McGraw-Hill. 007057 6424 This is really a great text! Another book, that is standard in format and but may not be the best for most st udents just beginning calculus, is the one by Spivak. If you want to have one b ook 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 materi al 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. 0-486-40 453-6 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. T he 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 04 71000078 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 wor k 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 with out 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. 981024904 7 See also Bressoud . Back to Top

Linear Algebra There are a great many competent texts in this area. The best is Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. HBJ . 0155510053 This book is must have. It undoubtedly the most influential book in its area sin ce 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. 4th Ed. Wellesley-Cambridge. 2 009. 978-0-980232-71-4 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 S paces 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 t

hese is: Axler, Sheldon. Linear Algebra Done Right. 2nd 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. 2nd ed. 1992 . S-V . 0387975861 Still whether this is a good text for a first course is arguable. It is certainl y an interesting text after the first course. The following may be the most poplular text on Linear Algebra: Lay, David C. Linear Algebra and Its Applications, 2nd ed. A-W . 1998. 02018247 87 There are a lot of subtle points to his treatment. He does a nice job of introdu cing 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 ot her texts, I like this particular text a great deal. Personally though I prefer the introductory text by Strang 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 mathematic s who wants to get into computer graphics. Farin, Gerald and Dianne Hansford. The Geometry Toolbox: For Graphics and Mode ling. 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. Chap man 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 seco nd 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. J?nich, 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 Back to Top 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 s ell these sections as separate texts as an option. For example the Harvard Calcu lus 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: t he 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 partic

ularly good example is: Kaplan, Wilfred. Advanced Calculus, 3rd ed. A-W . 0201799375 A nice introductory book: Dineen, Sen. Functions of Two Variables. Chapman and Hall. 1584881909 Se also: Dineen, Sen. 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 linea r algebra as a prerequisite. Its orientation is economics, so there is no Diverg ence Theorem or Stokes Theorem. Binmore, Ken and Joan Davies. Calculus: Concepts and Methods. 2001. Cambridge. 0521775418 I think that following has real merit. Bachman, David. Advanced Calculus Demystified: A Self-Teaching Guide. 2007. McGr aw Hill. Back to Top Differential Equations Like in some other areas, many books on differential equations are clones. The s tandard 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, a fter the course, math majors usually forget all the techniques. Engineering stud ents 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 Serie s 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 vo lume 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 succin ctly 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 appr eciate 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 co nsider the first two first. For a personal library or reference I would prefer t he Braun and Simmons. An introductory volume that emphasizes ideas (and the graphical underpinnings) o f 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 mos t 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. 199 6. 0471433322 Of these last two books I prefer to use Borelli and Coleman in the classroom, bu t 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 t he Kostelich and Armruster volume above these emphasize geometry. These volumes rely on the geometrical view all the way through. Note that the second volume ca n 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 li nes. What it does exceptionally well is to use complex arithmetic to simplify co mplex 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 aim ed at roughly the junior level. O'Malley, Robert E. Thinking About Ordinary Differential Equations. Cambridge. 1 997. 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 cover s Fourier series and PDE's. Dyke, P. P. G. An Introduction to Laplace Transforms and Fourier Series. Sprin ger. 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 . 5th 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. 0471238 872 Another new book by one of the best writers alive on applied math, corresponds p recisely to a one-semester course: Logan, J. David. Applied Partial Differential Equations. Springer. 1998. 03872-

0953-0 Back to Top

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 consider ably 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 Back to Top Dynamical Systems and Chaos Two classics that precede the current era of hyper-interest in this area are (bo th are linear algebra intensive) Luenberger, David G. Introduction to Dynamic Systems: Theory, Models, & Applicat ions. 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. 978-0-12-34 9703-1 Three elementary books follow. The second and third seem to be particularly suit ed 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. 199 7. 0198500777 Four more books at the junior senior level that can double as references on diff erential equations: Hale, J. and H. ko?ak. 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, B iology, Chemistry, and Engineering. A-W . 1994. 3540609342 Banks, John, Valentina Dragan and Arthur Jones. Chaos: A Mathematical Introduc tion. 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 Para dise. Freeman. 1991. 0716721368 Back to Top Real Analysis

There are two fantastic books that almost make a library by themselves. These ar e big and sumptious. The first is a solid course in undergraduate real analysis. The second is graduate level. To some extent they are available for download at their authors' web site. Thomson, Brian S., Judith B. Bruckner, Andrew M. Bruckner. Elementary Real Analy sis, 2nd ed. 2008. www.classicalrealanalysis.com. 978-1434843678. Bruckner, Andrew M., Judith B. Bruckner, Brian S. Thomson. Real Analysis, 2nd ed . 2008. www.classicalrealanalysis.com. 978-1434844125. For the student seeing analysis for the first time and who is overwhelmed by ana lysis, there are a few books out there. A good candidate is Bryant, Victor W. Yet Another Introduction to Analysis. Cambridge. 1990. 052138 835X 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 undergrad uate level. Very good on history and motivation. Exceptional!!!!! Bressoud, David. A Radical Approach to Real Analysis, 2nd ed. MAA. 2006. 9780883857472 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 ch apter 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 ar guably too much. It appears to designed for a one-semester course, though you c ould probably squeeze it into two semesters (with no difficulty at most universi ties). Might be a nice resource for the student taking the two-semester sequenc e 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 . 3540 65058X 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 fairly large book that is very good on undergraduate analysis and is applied i s Estep, Donald. Practical Analysis in One Variable. 2002. Springer. 0-387-95484-8 It is a good book for the numerical analyicist. A great read in analysis and best seller is Boas, R. P. A Primer of Real Functions 4th 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 ma king the book a great reference, study guide and first rate text. This may be th e 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 int egration (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 B artlett. 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 in clude: 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 integr al 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 st udents 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. 0387 901388 Of the more advanced books that discuss the subject more deeply: Gordon, Russell A. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Amer ican 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 L ebesque is: Yee, Lee Peng. The Integral: An Easy Approach after Kurzweil and Henstock. Cambr idge. 2000. 0521779685 The following magnum opus is the only one I've seen in this area that can be use ful 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 a s aids to teaching either calculus or analysis. I do not know if these books ar e so useful to the teacher, but they are great resources for the serious studen t. 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 s ection 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 . 1 985. 0123624401 Back to Top Complex Analysis The following book is a primer on complex numbers that ends with a short introdu ction to Complex Analysis. It is a perfect book for the sophomore in math or en gineering. 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 boo k to be read after an introduction. Also, it is a great reference during the fir st course. A wonderful book that is concise, elegant, clear: a must have: Bak, Joseph and Donald J. Newman. Complex Analysis, 2nded. S-V . 1997. 03879475 66 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. 0415 308461 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 6th ed. 1996. 0079121470 Another book very much in the same vein as Brown and Churchill is preferred by m any people, Wunsch, A. David. Complex Variables with Applications, 2nd ed. A-W . 1994. 0201 122995 This is my favorite book for a text in CA. Still another superb first text is formatted exactly as elementary calculus text s 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. 03879742 7X 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 Variabl es. 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. Fre eman. 1987. 0716721058 A book that I regard as graduate level has been described as the best textbook e ver 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. Birkh?user. 1999. 0817640118 The following is in one of Springer's undergraduate series but I think is more s uited for grad work. The author says it should get you ready for Ph.D. qualifier s. 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 engine ers 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 be ing outstanding. Books on differential forms and tensors can often merely enhanc e the reputations of those areas for being difficult. However, there are excepti ons. On tensors I like two books which complement each other well. The book by Daniel son is more application oriented. If you are serious about this area get both bo oks. 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. Birkh?user. 2006. 0-8176-4499-7 Edwards, Harold M. Advanced Calculus: A Differential Forms Approach. Birkh?user. 1994. 0817637079 Weintraub, Steven H. Differential Forms: A Complement to Vector Calculus. AP . 1 997. 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 Pres s. 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 mater ial 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. Oxfo rd. 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 Phys ics. 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 clari ty of writing and presentation. (Also, this demonstrates how completely imparti al 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. 052157 0956 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. 0521 318971 The following is very interesting, definitely requires calculus: Nahin, Paul J. When Least is Best. Princeton. 2004. 0-691-07078-4 I think that a fantastic book for teaching modelling is the one that follows. It covers all sorts of modelling and is superb at the sophomore/junior level. Shiflet, Angela B. and George W. Shiflet. Introduction to Computational Science: Mdeling and Simulation for the Sciences. Princeton University Press. 2006. 9780691125657. Courant and John A great reference is the last edition of Courant's great classic work on calculu s. 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 f irst 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, ca lculus of variations, and complex analysis.

Courant, Richard and Fritz John. Introduction to Calculus and Analysis. Spring er. 1989. Vol I. 3-540-65058-X Vol II/1 3-540-66569-2 Vol II/2 3-540-66570-6

Back to Top 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 Numb ers. 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 par ticularly good Courant, Richard, Herbert Robins. Revised by Ian Stewart. What is Mathematics. O xford. 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 inte rest 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 . Birkh?user. 1995. 0395929687 I recommend other books by Davis and Hersh as well as books by Davis and Hersh e ach alone. The late Morris Kline wrote several good books for the layman (as well as for th e professional). My personal favorite is strong on history and art and I think d eserves more attention than it has ever had. I think it is more important now th en 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 se rious) should have: Hewson, Stephen Fletcher. A Mathematical Bridge: An Intuitive Journey in Highe r 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 s ingle volume that compares. Gowers, Timothy (ed.) The Princeton Companion to Mathematics. 2008. Princet on. 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 th e sections Algorithms and Logic and Computability . A. K. Dewdney wrote a book of 66 chapters to briefly and succinctly cover the in teresting topics of computer science. The emphasis here is theory. This is a boo k every computer science major should have, and probably every math major and ce rtainly 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 c omputer algorithms is Harel, David. Algorithmics: The Spirit of Computing, 2nd ed. A-W . 1992. 020150 4014 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 (sen ior level): Graham, Ronald, Oren Patashnik, Donald E. Knuth. Concrete Mathematics: A Foundat ion 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 speciali ze too much. In particular you should take courses in number theory and probabil ity. Abstract algebra, linear algebra, linear programming-these and other areas can be useful. There are two books that are extremely good one-volume introductions at the unde rgraduate level. Tehy are very well written. I said in printed review that book by Mazur is the best book ever published on combinatorics, or something like tha t. The second book compares quite favorably. They are both junior-senior level. Mazur, Barry. Combinatorics: A Guided Tour. 2010. MAA. 978-0-88385-762-5 Allenby, R.B.J.T., Alan Slomson. How To Count: An Introduction to Combinatorics, 2nd ed. 2011. CRC Press. 978-1-4200-8260-9 The Polya-Tarjan book is superb. It is based on the notes from a course. As nice an introduction as you will ever see (junior-senior level) is this: Plya, George. Robert E. Tarjan. Donald R. Woods. Notes on Introductory Combinatori cs. Birkh?user. 1983. 0817631704 Here are four books at roughly the junior-senior level. These books are all rea dable and are selective in their topics. By this I mean they avoid the too comm on approach of throwing in everything including the kitchen sink. Anderson, Ian. A First Course in Discrete Mathematics. Springer. 2001. 1-852 33-236-0 This (Anderson) is a great introduction for the undergraduate math major. Lovsz, L., J. Pelikn and K. Vesztergombi. Discrete Mathematics: Elementary and Bey ond. 2003. Springer. 978-0-387-95585-8 Andreescu, Titu and Zuming Feng. A Path to Combinatorics for Undergraduates. B irkh?user. 2004. 0-8176-4288-9 Benjamin, Arthur T. and Jennifer J. Quinn. Proofs that Really Count: The Art o f 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 an

d 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. 19 94. 0521457610 This is a great book! Its level is roughly senior to graduate school. (It is div ided into undergraduate and graduate halves.) A classic text at the senior/graduate level that covers lattices, generating fun ctions, 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 pr ofessionals. They are both good reads and they overlap a number of disciplines, but arguably belong most to combinatorics. Note they do not belong in Foundation s like the book by Ebbinghaus, H.-D. Et al. The book by Bunch is excellent for t he serious freshman-sophomore. The second book is more advanced and includes a n ice 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. 03879 7993X A superb first book on graph theory is: Hartsfield, Nora, Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introd uction, 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 mentio n 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. 03879634 72 A very nice book at the senior-graduate level strictly devoted to generating fun ctions: 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.e du/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 Benja min and Quinn. Vorobiev, Nicolai N. Fibonacci Numbers. Birkh?user. 2002. 3-7643-6135-2 Posamentier, Alfred S. and Ingmar Lehman. The (Fabulous) Fibonacci Numbers. Pr ometheus. 2007. Back to Top

Numerical Analysis Most books on numerical analysis are written to turn off the reader and to encou rage him or her to go into a different, preferably unrelated, field. Secondly, a lmost all of the books in the area are written by academics or researchers at na tional labs, i.e. other academics. The kind of industry I use to work in was a l ittle different than that. The problem is partly textbook evolution. I've seen b ooks long out of print that would work nicely in the classroom. However, textboo k competition requires that newer books contain more and more material until the book can become rather unwieldy (in several senses) for the classroom. The trut h is that the average book has far too much material for a course. Numerical ana lysis touches upon so many other topics this makes it a more demanding course th an others. A marvelous exception to the above is the book by G. W. Stewart. It avoids the p roblem just mentioned because it is based upon notes from a course. It is concis e 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 l iked the first text you want this: Stewart, G. W. Afternotes goes to Graduate school: Lectures on Advanced Numeric al Analysis. SIAM. 1998. 0898714044 Two great books on the subject are written by a mathematician with real industri al 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 Eng ineering 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 publi sher. 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. Cambr idge. !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. 1 987. 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. 0030 309832 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 i teration is not covered). Also, it does not give pseudo-code for algorithms. Thi s is okay with me for the following reasons. Given a textbook with good pseudo-c ode, no matter how much I lecture the students on its points and various alterna tives, they usually copy the pseudocode as if it the word of God (rather than re garding 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 packa ges with an emphasis on MAPLE and a disk comes with the package, which I have ig nored. 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 a t a first year graduate level and is academic rather than say engineering orient ed. 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 elec trical engineer: Bracewell, Ronald. The Fourier Transform and Its Applications, 2nd ed. McGraw-Hi ll. 1986. Another book in a similar vein has been reprinted recently (I think): Papoulis, Athanasios. The Fourier Integral and Its Applications. McGraw-Hill. 19 62. A book with many applications to engineering is Folland, Gerald B. Fourier Analysis and its Applications. Wadsworth and Brooks/C ole. 1992. 0534170943 The best first book for an undergraduate who is not familiar with the material i s 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. Cambrid ge. 1997. 0521597714 A truly great short introduction: James, J. F. A Student's Guide to Fourier Transforms with Applications in physic s 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 a s 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 wa s being published by two publishers. It might return soon. The second book by Or e 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 Numbe rs. 0090306112 Ore, Oystein. Number Theory and its History. Dover. 0486656209 Friedberg, Richard. An Adventurer's Guide to Number Theory. Dover. 1994. 048628 1337 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 . 2 005. 0201870738 This text (Rosen) has evolved considerably over the years into a lush readable t ext, strong on applications, and basically a great text. Maybe the text to have . Burton, David M. Elementary Number Theory, 4th Ed. McGraw-Hill. 1998. 007232569 0 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 pr oofs 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 ha s 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 Th eory 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. 11141230 64 Hardy, G. H. and E. M. Wright. The Theory of Numbers. 5th ed. Oxford. 354064332 X 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 prop er 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. Wel l laid out. Out a very good book to have. Anderson, James A. and James M. Bell. Number Theory with Applications. P-H . 199 7. 0131901907 The first graduate level book to have on number theory might be Ireland, Kenneth and Michael Rosen. A Classical Introduction to Modern Number Th eory. 2nd ed. S-V. 1990. 038797329X Be careful on this book. The first edition was a different title and publisher b ut, of course, the same authors. A very short work (115 pages) at the first year graduate level covers a good va riety of topics: Tenenbaum, G. and M. M. France. The Prime Numbers and Their Distribution. Americ an 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 refere nces 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 fi rst course in abstract algebra (it often refers to proofs in Stewart's Galois Th

eory which is listed in the next section (Abstract Algebra)). Stewart, Ian and David Tall. Algebraic Number Theory and Fermat's Last Theorem, 3rd ed. A. K. Peters. 2002. 1568811195 Analytic Number Theory is a tough area and it is an area where I am not the pers on to ask. However, in the early 2000's there appeared three popular books on t he Riemann Hypothesis. All three received good reviews. The first one (Derbys hire) 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 Mathem atics. Farrar, Straus, Giroux. 2002. 1843541009 Sautoy, Marcus du. The Music of the Primes: Searching to Solve the Greatest My stery 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 Riema nn. 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 considere d really elementary is the one by Landin. Landin, Joseph. An Introduction to Algebraic Structures. Dover. 1989. 048665940 2 A standard text is: Fraleigh, John B. A First Course in Abstract Algebra, 5th ed. A-W . 1994. 020176 3907 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 T heory (but gets most of the way there). He is particularly good (I think) on gro up theory. Herstein, I. N. Abstract Algebra, 3rded. PH . 1990. 0471368792 Hernstein has a great book on abstract algebra at the graduate level. It is thor ough, fairly consise and beautifully written. He is very strong on motivation an d explanations. This is a four-star book (out of four stars). It is one of the b est 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 chapt ers on applications. It is certainly viable as a text, and I definitely recommen d 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 w ith matrix theory it covers quite a bit of ground and is beautifully done. I lik e it a great deal. Note that some people consider this book undergraduate in lev el. 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 Radical s. 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. 04 71433349 A Carus Monograph that spends time on field extensions and covers some basic Num ber 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. Po lygonal Press. 1980. 0936428147 Despite the title, the following is a book I think most students of abstract alg ebra should check it out. Alaca, ?aban, and Kenneth S Williams. Introductory Algebraic Number Theory. C ambridge. 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. 0 883850206 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 be st introduction. My favorite is the book by Stillwell. I don't think much of it as text, but it i s a great book to read. Despite the title, it is very much a book on Galois Theo ry. 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 fairl y comfortable with group theory and ring theory. (It is nota book for a first co urse 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 s pecial study. If you are looking for applications of abstract algebra, you should look first t o 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 Discre te Algorithms. P-H. 2003. 0130674648 A more advanced and far more ambitious undertaking is: Lidl, Rudolf, and Gnter 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 Ap plications, 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 The ory section. The following book intends to shed light on Wiles's proof of Fermat's Last Theor em. Supposedly it is aimed at an audience with minimal mathematics, but it shou ld be enlightening to students who have had a course in Abstract Algebra who mig ht find it fascinating. Ash, Avner, and Robert Gross. Fearless Symmetry: Exposing the Hien 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 ge ometry get into group theory. I have indicated which of these does an exceptiona l 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. 08838 5614X 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 MAA published a lavish book that seems to be designed to supplant Grossman a nd Magnus (just above this). I prefer Grossman and Magnus for their conciseness for the elementary material. Howeever, the newer book is dazzling. It spends a l ong time motivating the group concept emphasizing the graphical and other visual approaches. The second part goes much deeper than Grossman and Magnus and in pa rticular gives maybe the best treatment of the Sylow theorems that I have seen. Carter, Nathan. Visual Group Theory. MAA. 2009. 978-0-88385-757-1 The next book is an introduction that goes somewhat further than the Grossman bo ok. 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. 085233235 2 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 geo metry 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. 0 387942858 I like this book a great deal. Another book that goes into graduate level that is worth a look and quite inexpe nsive 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 bot h 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 Dummi t and Foote. Back to Top Geometry If I were to recommend just one book on geometry to an undergraduate it would pr obably 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 grea t pleasure to read (even if you have read the others). It might make sense to r ead this first and then Four Pillars (immediately above). Stillwell, John. Yearning for the Impossible: The Surprising Truths of Mathema tics. 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), tradit ional 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 t ext for an undergraduate course. The following two books are intended as undergraduate texts. Both volumes are sl im and do a short course on Euclidean geometry and the development of non-Euclid ean geometry followed by affine and projective geometries. The book by Sibley to uches 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 bett er on projective geometry though; I suspect he will touch that up for a second e dition. Also, when he does iterated fractal systems in 2 or 3 pages-I am not sur e 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. 0387989 722 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 p rojective geometries is: Polster, Buckard. A Geometrical Picture Book. S-V . 1998. 0387984372 Let me list four excellent texts for the course on traditional Euclidean geometr y and the development of non-Euclidean geometry (principally hyperbolic geometry ). Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries: Development and H istory, 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. Birkh?user. 1987. 0817633111 The four books listed above are all excellent! but there is a new book on the sa me topic, by a great geometer, that I think is a masterpiece. If this topic (tra ditional Euclidean geometry and the development of non-Euclidean geometry) inter ests 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. 1-85233-93 4-9 Two books devoted only to groups and geometry: Nikulin, V. V. and I. R. Shafarevich. Geometries and Groups. S-V . 1987. 038715 2814 Lyndon, Roger C. Groups and Geometry. Cambridge. 1985. 0521316944 Many of the books listed here spend much time on projective geometry. However, l et 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 mathematic s 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, Ceder berg, 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. Hausner, Melvin. A Vector Approach to Geometry. Dover. 1998. 0486404528 Compare this book with Banchoff and Wermer. Also compare with Farin and Hansfor d. The following book emphasizes the connections between affine and projective geom etries with algebra. I think that the reader should have some experience with t hese geometries and with abstract algebra. Blumenthal, Leonard M. A Modern View of Geometry. Dover. 1980 (originally 196 1). A concise well written summary of modern geometries which (realistically) requir es a course in linear algebra: Galarza, Ana Irene Ramirez and Jos 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. 19 97. 0521599938 A very elementary book of 80 pages (a good book for the talented high school stu dent): Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Dove r. 1986. 0201039346 The book by Ogilvy is short and precious. It requires careful study but is quit e a gem. It covers inversion, conic sections, and projective geometry and seve ral 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 kn owledge of elementary linear algebra. I doubt that individuals without this know ledge will read it. Another book that also is intended to be elementary is Gibson, C. G. Elementary Geometry of Algebraic Curves: An Undergraduate Introduc tion. 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 Int roduction 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 chunk s (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. 0521 424801 Bruce, J. W., P.J. Giblin. Curves and Singularities, 2nd ed. Cambridge. 1992. 0 521249457 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 Su rfaces. Cambridge. 1994. 0521002648 Barrett O'Neill. Elementary Differential Geometry, 2nd ed. AP . 1998. 012526745 2 Do Carmo, Manfredo P. Differential Geometry of Curves and Surface. PH . 1976. 0 132125897 Bloch, Ethan D. A First Course in Geometric Topology and Differential Geometry. Birkh?user. 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 f ollowing 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. 19 78. 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 Group s. 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 Foundatio ns 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 . 1 993. 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); Now reprinted by Dover!! 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. Cambri dge. 1989. 0521007585 The following are also good introductions: Hamilton, A. G. Logic for Mathematicians, revised ed. Cambridge. 1988. 05213686 50 Lyndon, Roger C. Notes on Logic. Van Nostrand. 1966. Enderton, Herbert B. A Mathematical Introduction to Logic. AP . 1972. 012238452 0 Cutland, N. J. Computability: An Introduction to Recursive Function Theory. Camb ridge. 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 Com plexity. 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. 19 97. 0763714224 This is the pedagogical work. It covers less than Hopcroft and Ullman and is aim ed at a slightly lower level, but is in many ways the best written book and is t he 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, Langua ges, 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 Compu ter 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 introduct ory and reasonably paced): Ebbinghaus, H.-D., J. Flum and W. Thomas.Mathematical Logic. S-V . 1984. 038794 2580 Smullyan, Raymond M. First-Order Logic. Dover. 1995. 0486683702 Smullyan, Raymond M. G?del's Incompleteness Theorems. Oxford. 1992. 0195046722 Smullyan, Raymond M. Recursion Theory for Metamathematics. Oxford. 1993. 019508 232X Matiyasevich, Yuri V. Hilbert's Tenth Problem. MIT. 1993. 0262132958 G?del There is a celebrated treatment for all readers of G?del's Incompleteness Theore m. This book received a Pulitzer and was a significant event. (More concisely, t he book received a lot of hype and derserved it.) Hofstadter, Douglas R. G?del, Escher, Bach: An Eternal Golden Braid. Basic Books . 1979. 0465026850 A nice very short treatment of G?del's incompleteness theorem it the article: Hehner, Eric C. R. Beautifying G?del pp. 163-172, in Beauty is Our Business: A B irthday Salute to Edgar W. Dijkstra. S-V. 1990. 3540972994 A quicker treatment than even that is in the first three pages of Smullyan's boo k on G?del above. This is the book to have. The following is a good introduction to Godel's incompleteness theorem as well a s 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 G?del'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. Kriege r. One of the most underrated books I know is this book by Eves. It does a very cre dible job of covering foundations, fundamentals and history. It is quite a littl e gem (344 pp). Eves, Howard. Foundations and Fundamental concepts of Mathematics, 3rd ed. PWS-K ent. 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 Canto r'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, Aison-Wesley are more or less a bible. They are comprehensive, au