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Jun 15
  1. Honors Calculus/Advanced Calculus/Elementary Real Analysis(Theory of Calculus on the Real Line and R )

  1. All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove. - G.H. Hardy

  2. One Variable Advanced Calculus Kenneth Kuttler BYU April 13, 2013 (PG)
  3. Multivariable Advanced Calculus Kenneth Kuttler Brigham Young University March 4,2014 (PG-13) These are the follow-up courses to Kuttler's calculus notes-and the resulting textbooks-we commented on earlier.  Those courses were rigorous and careful, as standard advanced calculus/ real analysis courses are. As a result, it's hard to see how an advanced calculus course that followed such courses would look like in 2014. Well, it's clear that these are general advanced calculus texts, not assuming more then a standard course in calculus as prerequisites. As a result, there's a considerable overlap between those earlier calculus texts and these advanced calculus textbooks, but there are striking differences as well. As expected, the more applied sections of the calculus texts are either omitted or downplayed and the theoretical sections are greatly expanded. A comparison of the table of contents of each corresponding volume in the calculus texts and advanced calculus texts will give specifics of how they differ. The first volume contains the usual theoretical treatments of naive set theory, limits of sequences and functions, continuity and differentiability, the Riemann integral and infinite series and sequences.  There is an explicit construction of the real numbers via Cauchy sequences in an appendix, which I think any serious undergraduate real analysis course shouldhave. The first book does have some unusual touches-particularly the inclusion of 2 quite good chapters on Fourier series and the Henstock-Kurtzwell integral. My one complaint is  the treatment of the last topic is somewhat cursory, so I don't really know what was gained by adding it. The differences become more profound in the second volume on multivariable calculus. Again, as in the case of the honors calculus text, there is a substantial treatment of vector spaces following a further development of set theory, in the first chapter. The linear algebra treatment in the advanced calculus version is longer, deeper and more substantial for several reasons.Firstly, to develop the topology of Rn via the normed space properties. As far as I know, there's only one advanced calculus textbook that makes this its central development of open and closed sets: Kenneth Hoffman's classic Analysis in Euclidean Space.  The main reason for Kuttler doing this is a simple one: so there is a unified language for advanced calculus, linear algebra, topology and geometry in Rand he doesn't have to develop a separate terminology for each. He develops many of the standard topics for such a course: convergence of vector valued sequences, limits and continuity of functions, the total derivative as a linear transformation, partial derivatives, the implicit and inverse function theorems and much more. But the other 2 main  differences with his honors calculus text comes with integration theory: Kuttler chooses to develop measure theory and the main theorems of Lebesgue integration on R.He does this so that he can develop completely general versions of  the three major integration theorems of vector analysis-Green's, Gauss' and Stokes' theorems-on all integrable spaces. Which brings us to the last major difference with the earlier books: From this point, he develops the material in terms of n-dimensional manifolds and differential forms in Rn. He also develops the Brower degree in Euclidean space in order to give a rigorous definition of orientation on a boundary that doesn't depend on smoothness. I know, this is all the rage for courses at this level now. I have mixed feelings about it as I've expressed elsewhere.  But  more profoundly,I don't know if relaxing smoothness conditions to this level is really useful in an advanced calculus where students' grasp of the basic theory of calculus is shaky at best. That caveat aside-the author has a very carefully constructed vision of what a 2 year advanced calculus course should look like and does an outstanding job of presenting the material both clearly and in depth in both volumes. There are lots of examples, good pictures and a diverse collection of exercises.I'm not sure I agree with everything in Kuttler's vision of what advanced calculus should look like, but he certainly has made a strong argument for it with these books. Very highly recommended for both students and teachers of undergraduate real analysis.
  4. Calculus Nai-Sher Yeh Fu-Jen Catholic University (Tiawan) 2010 (PG-13) Tiawan was the 10th highest scoring country for mathematics and science in the world according to a 2010 Organization for Economic Cooperation and Development report of student scores in 65 countries. Reading these amazing lecture notes by Yeh, I can completely believe it. This is the text for a first year course in calculus at Fu-Jen Catholic University and they actually dowhat so many  American and European courses claim to do: Combine thoroughly the theory and applications of calculus into a complete course. After a review of naïve set theory, Yeh gives a completely rigorous discussion of limits, continuity and derivatives. Up until that point, it looks like a real analysis course-until Yeh begins discussing related rates, maximum and minimum problems in physics and geometry. He then goes on to rigorously prove Rolle’s and theMean Value Theorum-and then he goes on to discuss applications of the MVT in economics. And the notes proceed entirely in this fashion, through Riemannsums, integration methods, applications such as work and the moment of inertia, infinite series and convergence tests and the course wraps with the calculus of vector functions in n-dimensional space and their applications. This is what every hard core mathematician dreams the course they could teach in calculus could be-and it shows what’s possible when the system cares enough to prepare students properly for it. There are even some proofs that usually aren’t given correctly-such as a full careful proof of L’Hopital’s rule. The one weird thing about the notes is that were apparently many figures in the original and theyeither didn’t make it into the final version or more likely, the practice was for the students to fill in the figures from the oral lectures. These are easily filled in by students or instructors using these wonderful notes. Very strongly recommended for serious mathematics students and their instructors.
  5. Advanced Calculus James S. Cook Liberty University  Fall 2011 (PG)This is kind of strange course on advanced calculus.It's not exactly an applied course-but it's not exactly a hardcore rigorous one either. On the one hand, it wants to cover calculus of several variables in a sufficiently sophisticated manner that the major results on differential manifolds embedded in Euclidean space can be covered in detail. However, the author is hesitant to cover more analytic aspects of the subject-such as the theory of integration on forms-because the aspects get "technical". Frankly, you get the impression reading Cook's website that a) his students are generally mediocre at theoretical mathematics and b) the author himself is more interested in applications and intuitive content then pure theory. That being said, the notes are quite interesting. Cook emphasizes the geometry and linear algebra of Euclidean space and it's embedded manifolds, an aspect that sometimes gets lost in more rigorous treatments. As a result, he produces a very visual and lucid set of notes with many examples and insights that would complement a more careful treatment very well. He takes great pains to not only precisely define everything he introduces, he often uses multiple viewpoints and definitions from the literature to give students a broad understanding of the many new concepts that are introduced. For example, when he defines manifolds, he gives several examples of different charts on the exact same subset of Euclidean space that results in radically different structures. He also gives one of the clearest and simplest presentations of multilinear algebra, the exterior product and differential forms I've ever seen, using just the elements of linear algebra in it's development. There's a terrific presentation of classical electromagnetics in the language of differential forms. For all it's shortcomings and flaws, this is a very informative presentation well worth studying, if only as a supplement to a more abstract treatment. Another fine set of lecture notes from Cook. Highly recommended.
  6. Advanced Calculus II Jie Wu Department of Mathematics National University of Singapore (PG) A relatively brief course on advanced calculus that focuses on series and sequences of numbers and functions and their theory. Very careful and concise, with lots of nice examples. It reminds me in some ways of the old book, A Textbook of Convergence by W. Ferrar. Within it's limited scope, these notes will be useful for it's target audience. Recommended.
  7. Honors  Calculus Part I Math 117 Lecture Notes James Muldowney University of Alberta 1999 (PG) HonorsCalculus Part II Math 118 Lecture Notes James Muldowney University of Alberta 1999  (PG)    This is a remarkable set of notes whose typed version was scanned in 1999 before LaTex-ing notes became standard practice.  A completely rigorous single variable calculus course that begins with an axiomatic development of the real numbers and proofs of basic inequalities and develops it from there through 2 semesters the standard topics: sequences and convergence, limits of functions, continuity, derivatives, Riemann sums and the integral, the gamma and hyperbolic functions, trigonometric functions, the techniques of integration (integration by parts, partial fractions,etc.), areas and volumes by shells and washers, infinite sequences and series. What's truly remarkable about these notes is that they are one of the few courses of this type I've seen that attempt to balance theory and application in equal measure. There are clear epsilon-delta arguments interwoven with many standard applications to geometry and physics, including surface area, volume, work, velocity and many more. There are many, many examples and pictures. Lastly, there are surprising, creative touches that illuminate rigorously several aspects of calculus that are virtually never explained in calculus at any level. For example, Muldowney explains the technique of partial fractions by explaining that the decomposition theorum is a consequence of the fact that the set of all nth degree polynomials forms a vector space and therefore, each polynomial can be decomposed into a linear combination of basis polynomials that span the space. I'd never even considered explaining it to beginning calculus students that way-I was brainwashed into thinking the theory behind the technique was Galois theory. But every field is a vector space over itself, so it makes perfect sense!!! I am so going to steal that explanation for my students who have linear algebra!  But that's only one example-these notes are truly a treasure trove for both students and teachers of calculus and analysis at any level and are a definite must read. The highest possible recommendation.
  8. Honors Advanced Calculus Part I 3rd edition  Lecture notes James S. Muldowney University of Alberta (PG )
  9. Honors Advanced Calculus Part II 3rd edition James Muldowney University of Alberta (PG)
  10. Honors Advanced Calculus: TeXed and Corrected version of the 1999 original Lecture Notes for Mathematics 217-317  James S. Muldowney The University of Alberta  January 10, 2010 (PG) These are the notes for Muldowney's year long advanced calculus course, a rigorous course in multivariable calculus. It also functions at the University of Alberta as the second year of the honors calculus sequence. There are 2 versions available-the original scanned 1999 version and a 2010 LaTeXed version by the author. They're identical except that, of course, the LaTeXed version takes up much less disk space. I'd download those for that reason. They begin with a rapid review of the axioms of the real numbers and all basic set theory. Linear algebra is assumed known. He proceeds to limits and continuity in several variables, differential calculus in several variables (the derivative and differential as linear
  11. transformations, partial derivatives and the Jacobian matrix, global and relative extrema in Rn , etc), the implicit and inverse function theorem, the Riemann integral in several variables, the change of variables theorem and line and surface integrals.The presentation, while very careful and rigorous, is entirely classical-manifolds and differential forms don't come into play. Which is fine with me. In any event these notes, while excellent and very clear, are a bit of a disappointment. They're a disappointment because since they are supposed to be a continuation of the previous wonderful honors calculus lectures in one variable by the same author, I was expecting the same balance of theory and applications in the second part and the wonderful vistas of applications that the extension of single variable calculus to geometry and physics. Unfortunately, that's not the case. Except for some applications to geometry, such as arc length and the surface area of specific surfaces, there are no applications, not even in the exercises. It's basically just a standard advanced calculus course in several variables. Not that that's bad, mind you-it's very well done. It's just that after the joy that the honors single variable notes were, going back to standard fare is a letdown. Fortunately, we do have some good choices for a follow-up to those notes that preserve the balance-the second half of Kuttler's honors notes would work well for that purpose, so would Hwang's notes. As for these, even with the disappointing ordinariness, this is a very solid advanced calculus course and well worth checking out. Highly recommended.
  12. APPLIED ADVANCED  CALCULUS LECTURE NOTES JAN VRBIK Brock University  (PG)Very nice applied course in advanced calculus, with many examples and  engineers need to see as undergraduates: first, second and third order ordinary differential equations, vector analysis in R3 and complex analysis. Hardly a proof in sight, which will horrify many pure mathematics students. But this is what many students demand in these courses, sadly. It shouldn't be an either/or situation in my opinion. Still, the notes are very good at it's intended goal and it can act as a very nice supplement to a purely theoretical treatment to these subjects.
  13. Advanced Calculus Paul C DuChateau Colorado State University CourseMaterials   (PG)  Relatively brief and concise one semester set of lecture notes on one variable advanced calculus from an axiomatic development of the real numbers through limits of sequences and functions to continuity and differentiation and finishing with the Riemann integral. While the development is well, written and very careful with many examples and solved problems, many major results are left as exercises and the proofs that are given are given quite tersely. It does contain an unusually rigorous and detailed treatment of both finite and infinite decimals, something usually omitted in a first rigorous treatment of calculus.  But otherwise, it's a pretty standard first course in analysis, albeit more concise then most. If you like that kind of do-it-yourself mathematics, you'll probably like these notes a lot. But it'll probably be most helpful as a study aid for students preparing for major exams, prelims or qualifying exams on real
  14. analysis or honors calculus. Highly recommended as a study aid in analysis.
  15. Advanced Calculus I Martin Bohner University of Missouri at Rolla Fall 2002   (PG) Skeletal set of notes for an advanced calculus course covering the standard topics: axioms of the reals, sequences, functions, limits, derivative,the Riemann integral and infinite series. There are no proofs and barely definitions. I'd pass except to use as study material.
  16. ADVANCED CALCULUS Gilbert Weinstein Majmah University 2010  (PG) This is a brutal Moore method type course that's even more skeletal in some ways  then Bohner's notes above. Here's how the author describes the course in the preface: The course is run in the following way. In these notes, you will and Defi nitions, Theorems, and Examples. I will explain the defi nitions. At home, on your own, you will try to prove the theorems and the statements in the examples. You will use no books and no help from anyone. It will be just you, the pencil and the paper. Every statement you make must be justi ed. In your arguments, you may use any result which precedes in the notes the item you are proving. You may use these results even if they have not yet been proven. However, you may not use results from other sources. Then, in class, I will call for volunteers to present their solutions at the board. Every correct proof is worth one point. If more than one person volunteer for an item, the one with the fewest points is called to the board, ties to be broken by lot. Your grade will be determined by the number of points you have accumulated during the term. Now that's about as raw and naked a Moore method course as you're gonna find. I've voiced elsewhere my reservations about running such a course. I've not even going to deal with my doubts as to whether or not in the information age, such a course can be realistically run. I'll just say here the notes are dry and tasteless.There are almost no proofs by design, although in fairness there are very clear definitions and quite a few examples.  I wonder if any but the very strongest students can actually gain any real understanding of analysis from them. If I really wanted to teach such a course, I'd be much more inclined to use either Erdman's much more pleasant and meaty text below or the old classic, Introductory Problem Courses in Elementary Topology And Analysis by E.E. Moise Unless you're a masochist, I'd pass.
  17. Advanced Calculus I Jim Brown Clemson University 2012 (PG)  This is one of Brown's many uploaded scanned handwritten notes from the various courses he's either taken as a student or taught  as a post-doc/graduate student over the years.  The handwriting is legible if not exceptional. The course again covers all the usual topics of a first semester advanced calculus course: axioms of the reals, sequences, functions, limits, derivative,the Riemann integral and infinite series with a brief discussion of limits and differentiation of functions of several variables at the end. Full, almost pedantic proofs are given along with some nice examples that are worked out in detail. The level of detail makes this good supplementary study material. But to be honest, there's nothing original or striking here-it could have been copied from any standard text. I'd download them for additional study material, but I'd rather have one of the other sources listed here as my primary text. Good but not great.
  18. Advanced Calculus Richard Barraclugh University of Burmingham MSM2G2  (PG) Very brief and concise set of notes largely focusing on second order ordinary and partial differential equations. Laplace transforms and generalized function solution methods are discussed. Ok, but why the hell are these notes called "advanced calculus"? Beats me.
  19. Advanced Calculus: Patterns of Proof in Analysis An introduction to thinking mathematically  Draft by Kenneth F. Klopfenstein  for course taught by Yongcheng Zhou Colorado StateUniversity Spring  2012 
    (PG) I made a mistake initially in thinking that Zhou was the author of this strikingly original book in progress on advanced calculus. The author is actually Klopfenstien, who is clearly drafting and class testing them.  Zhou doesn't seem to have added any course materials in addition-and looking at these notes, it's clear why he didn't. He'd just be wasting his time when he had this jewel to work from. I don't know if or when the notes will ever be published as an actual book, but I strongly recommend students download this version and then update it regularly, as  it may eventually be and then it'll be lost as a free  resource. That would be a shame if so. While the content of the book is pretty much standard, it's organization and style most definitely is not. Sequences and convergence, the arithmetic of sequences, the order properties of the real line, completeness and convergence,Cauchy
  20. sequences,subsequences,limits of functions,continuity at a point,uniform continuity and continuity properties, differentiation and Riemann integration.There are many elementary analysis textbooks that serve the double purpose of helping students make the transition from plug and chug mathematics to rigorous proof courses by teaching the basics of proof, logic and set theory while teaching the all-important rigorous course in the theory of calculus. But for most such courses, this is a secondary-a sort of "by-the-way" byproduct of the far more important task of explaining why calculus works. Klopfenstein takes almost the converse approach here-while he does treat the theory of single variable calculus fully and carefully, the thread of a transition to rigorous proof is just as-if not more important then-the analysis material. I think this is brilliant idea because it mirrors historically the transition from the "intuitive"  mathematics that was practiced since the middle ages and characterized the calculus from the time of Newton and Leibniz to modern rigorous analysis at the end of the 19th century. Many of the major techniques of proof and the foundations of mathematics were developed precisely in order to make the calculus precise by modern standards. But this is only one reason the notes are so effective. Another is the very effective "drill structure" of the text-each section begins with a "study guide" that outlines the goal of the chapter and relates it to what has gone before. There are many explicit ε-δ computations, particularly with sequences and series,most carried out step by careful step.There are also many wonderful quotations from the history of mathematics, many amusing and others insightful. But best of all are the exercises. They are very diverse and highly unusual. Some are the usual proofs of main lemmas and theorems or inequality computations-but a number are "thinking questions" that one almost never sees in undergraduate mathematics courses and would be enormously helpful in making students think about the material they're learning and why it's important. There are lots of great examples here, but I'll just name 2: 1)the wonderful list of "what if" questions at the end of chapter 1 asking students to judge how mathematicians would respond to conjectures, such as Perleman's claimed proof of the Poincare conjecture, and comparing it to standards of "proof" in other sciences.  2) Problem 7 at the end of chapter 3. In short, this is a gem and a possible classic in the making here. Studying it will not  only teach students a lot of great mathematics, but how to think about it. The highest possible recommendation.
  21. Sequences and Series: A Sourcebook Pete L. Clark University of Georgia 2012   (PG)The cosmic irony of Pete Clark's name appearing several times on this list can't be understated. As I said in the introduction-which probably seems like eons ago for those of you who've been actually scrolling through the website and reading all the reviews and links one at a time- one of the main inspirations for this Herculean undertaking was The Chicago Undergraduate Mathematics Bibliography. Clark was one of the Bibliography's original authors-then an extraordinarily talented graduate student at The University of Chicago. (The fact they considered Micheal Spivak's A Comprehensive Introduction To Differential Geometry an undergraduate textbook says all you need to know about the caliber of mathematics student that exists at that esteemed university! ) Pete's now an assistant professor at the University of Georgia who's area of research is the emerging field of arithmetic geometry. Arithmetic geometry, for those of you scratching your heads, is yet another branch of algebraic geometry where the geometric results of varieties and sheaves are used to determine number theoretic results and vice versa. In any event, Pete ( who, despite never having met me,very nicely told me it was okay to call him by his first name when we were internet comrades at the MathStackExchange) in-between developing this nascent field, not only teaches many different mathematics courses at UGA, he works very hard to develop his own notes for those courses. When I found out he did this, my respect for him tripled. First up is his lengthy notes on sequences and series for the honors calculus/ advanced calculus/ elementary real analysis course that follows a nonrigorous calculus course.The idea of basing a first exposure to rigorous analysis by focusing on sequences and series isn't a new idea, of course-there's Konrad Knopp's Theory And Application of Infinite Series and T.Ferrar's A Textbook of Convergence. And of course, most first analysis courses in the UK are structured that way. It does make a pedagogical kind of sense-think about how we usually explain limits carefully to students. Isn't the intuitive picture really that of a converging sequence? Contents: The real numbers as an ordered field, binary operations, open and closed intervals and convergence to a point, Archimedian property, least upper bounds,monotone sequences,  the extended real numbers and the Bolzano-Weierstrass Theorem, Cauchy sequences, infinite series with positive or negative terms and their sums, power series, sequences and series of functions and complex sequences. These notes are very careful with Pete's usual literate lucidity, with quite a few examples, developing the basics of convergence and limits of both real and complex valued sequences and series, both finite and infinite, as well as lots of good problems of varying difficulty. He also has many very deep observations that he states in an amusing and/or entertaining manner. For example, he calls the fact every infinite sequence has a monotone subsequence The Rising Sun Theorem (Get it?) . He also includes a number of topics you don't normally see in an introductory analysis course, such as an example of a  sequentially complete non-Archimedian ordered field,  partial limits and equidistribution in Fourier series. In short, it's a wonderful resource for learning analysis via sequences and series, especially for strong students. Very highly recommended. Interestingly, Pete's colleague at UGA, Malcom Adams, has written and posted a similarly themed set of lecture notes and it's hard not to compare the two.My full commentary on the Adams notes can be found here.
  22. ANALYSIS J.K. Langley University of Nottingham G11AN1 1993 (PG) Fairly standard first course in theoretical analysis in the concise "Cambridge" style. Sequences, Functions, Limits and continuity. Differentiability. Power series. Representing functions by power series. Indeterminate forms. Integration. Improper Integrals. Some good examples and good exposition. Solid, but really nothing special. And the exercises are missing, sadly.
  23. Advanced Calculus I-II ZIQI SUN Wichita State University Spring 2013 (PG)This is an interesting if concise and relatively brief set of notes on advanced calculus which-as many of these notes and online books for a first rigorous analysis course do-also doubles as an introduction to methods of proof in rigorous mathematics. Sun packs a lot into these notes and he does so without the notes becoming either skeletal or impenetrably dense. These notes cover the theory of calculus on both R and Rn. The notes begin with a substantial first chapter on elementary logic and basic set theory and then proceed to the usual advanced calculus topics: limits and convergence of sequences, basic topology of Rn , limits of  functions, continuity, differentiation on both R and Rn , the Riemann integral ,improper integrals and integrals on unbounded regions,infinite series and sequences  Despite the conciseness, the notes contain a surprisingly large number of examples and exercises. Unfortunately,. he also throws a lot of deep results out there-like the Open Mapping Theorem-and either omits the proof without comment or doesn't encourage it's proof as an exercise. Also, there are no pictures, which could be very helpful with the multidimensional material in particular. Clearly, these notes are still being drafted and improvements can be made. Still, it's a very good set of notes and will be useful for students and teachers. Highly recommended.
  24. Advanced Calculus of One Variable J.W. Thomas Professor of Mathematics Colorado State University June 2007 (PG) Exactly what the title says it is. A substantial and extremely detailed-if unoriginal-development-of a first course in rigorous analysis on the real line. The standard topics are covered: axioms of the real number system, basic topology of R , the limits of sequences and functions, continuity, differentiation, the Darboux and Riemann integrals and infinite series and sequences.  The material is very well organized; definitions and theorems are stated very clearly and with full proofs. There are a very large number of examples and exercises-the examples in particular are unusual, very detailed and informative. But if you're looking for anything original here, forget it. Still,it's a very good treatment, detailed and well written, one that will be particularly useful for self study due to the level of detail. Worth a look for both students and teachers. Highly recommended.
  25. Advanced Calculus and Analysis MA1002 Ian Craw University of Aberdeen November 6, 2000, Version 1.5  (PG) Relatively brief but comprehensive, literate and insightful set of notes on the theory of calculus of one and several variables. Craw covers the standard material for such a course, but he's much broader in his coverage to include functions of several variables: axioms of the reals, sequences, functions, limits, derivative,the Riemann integral and infinite series,Euclidean spaces, the total derivative, partial derivatives, multiple integrals, line and surface integrals. Manifolds and forms are avoided, but the treatment is nevertheless very rigorous and careful, being based on linear algebra throughout. The strength of these notes is the very focused and thoughtful manner in which Craw asks why a rigorous approach to calculus is needed and shows the necessity as well as the advantages of it. The notes were originally written as a supplement to Spivak's Calculus and the focus on explaining the mechanics of rigorous limits and the use of inequalities in their proofs certainly shows it. There are many solved examples and exercises, geared to illustrate the points of the discussion. There are also a number of pictures which help clarify many points, particularly in functions of several variables in R3 . A very strong, beautifully written and informative set of notes for either an honors calculus, advanced calculus or elementary real analysis course. Very highly recommended.
  26. Advanced Calculus Ng Tze Beng National University of Singapore  (PG) Excellent set of notes on advanced calculus that focuses on conceptual understanding of why things work in calculus the way they do. The author is a dedicated teacher who has been working on his own version of Calculus reform for many years and has authored his own book on calculus. What's unique about this approach to the theory of calculus is that it emphasizes the role of convergent series and sequences: limits of functions and the usual suspects of derivative and integral aren't discussed until halfway through the book and when they finally are, these concepts are developed and proven through power series expansions! There are an enormous number of examples and solved computations with epsilon-delta arguments and lot of good exercises. Contents: Axioms of the real and complex numbers, sequences and convergence, monotone sequences, Cauchy sequences and the Bolzano-Weierstrass Theorem on the real line,power series and uniform convergence in complete spaces, limits and continuity of functions, derivatives and the Riemann integral. The handwriting is clear and legible in the scans (yes,if you're not going to type the notes, that matters a lot!) -they concentrate on the theory of calculus, which really is appropriate in a course at this level. Metric spaces are introduced in the next to last chapter in order to discuss sequences of functions and uniform convergence, but otherwise, most of the material is set on the real line. But it's really the emphasis on sequences and series that makes these notes stand out. An excellent and striking original treatment of the theory of calculus. I also recommend most strongly reading the author's website and articles, which are really deepening discussions of standard results like the Mean value theorem and the extreme value theorem. All in all, the associated website is a treasure trove for serious students of calculus and both students and teachers of honors and advanced calculus should make themselves very familiar with Professor Beng's site. Most highly recommended for anyone with a serious interest in the theory of calculus.
  27. Calculus with Theory I MIT 18.014 course notes by James Munkres (PG)
  28. Calculus with Theory II Spring 2003 Anna Lachowska amd James Munkres MIT 18.024 Lecture Notes   (PG)These are the supplementary lecture notes of a remarkable year-long course given to honors freshman at the Massachucetts Institute of Technology, some of the strongest entering classes of mathematics majors in the world.  These notes were written by James Munkres for the course in 2003 and have been in use at  MIT ever since.  Looking at them, it’s not hard to see why. The same legendary teaching skill of Munkres’ that made his textbooks, such as Topology and Analysis on Manifolds,  absolute classics is present on each and every page of this wonderful source. However, it’s important to keep in mind these are supplementary lecture notes and they’re not really intended to serve as a stand-alone textbook-as a result, they’re very incomplete. But their coverage is outstanding for the topics they do cover-and Munkres’ choice of topics is excellent, as he focuses on points of real analysis that are usually not covered in the standard honors calculus  textbooks. The book for the course, for the record, was Tom Apostol’s classic 2 volume Calculus ,which sadly the average student’s parents have to take out a second mortgage to own now. Except for Spivak’s Calculus .this has been the book of choice for honors calculus classes at top universities. Using this as his template, Munkres begins with some development of some properties of the number systems that are useful in analysis-a full construction of the real numbers is not undertaken either here or in Apostol-particularly those related to the Archimedian property of the reals. He then moves on to the Riemann criteria of integrability for step functions and the essential properties of the Riemann integral via step functions. (Apostol is notorious for taking this “historical” development of calculus, doing integration first via step functions. I
  29. have mixed feelings about this approach because I think the careful development of Riemann sums is very helpful as a foundation for later studies, particularly in the theory of integration and the use of nets in point set topology. But it does simplify a rigorous presentation quite a bit.)  Other topics considered in the first semester’s notes include applications of the intermediate value theorem, a full proof of the extreme value theorem, integration techniques, the theory of power series and a brief introduction to Fourier series. The second semester’s focuses on the theory of functions of several variables and draws heavily from the author’s Analysis on Manifolds despite being pitched at a considerably lower level then that book. After a development of basic linear algebra, including matrix analysis and parametrization of curves, Munkres develops the essentials of vector calculus in the classical spirit of Apostol sans manifolds and differential forms. This includes differential calculus and completely rigorous-if classical-proofs of
  30. the fundamental theorems of n-dimensional calculus, Green’s amd Stokes’ theorems.   (Interestingly, he develops Riemann sums in the n-dimensional context in the course of these proofs.)  Like all Munkres’ expositions, the notes are detailed, careful and wonderfully insightful. They cannot be used on their own as a textbook as I’ve said, but they are an absolute must-read for students studying rigorous calculus for the first time. Combined with a more intuitive set of notes or text, of which there are many choices here, they would make for a wonderful course for honors students. One of my favs. Very strongly recommended.
  31.  Honors Analysis lecture notes John Labute Math 255 McGill University  (PG)These are notes for the second semester of the honors undergraduate  analysis sequence at McGill University-there are several alternate versions reviewed on this website of the lecture notes for each of this course's sequence.The Riemann-Stieltjes integral,sequences and series of functions,infinite series and convergence tests, elementary functions and metric spaces. There are some original touches, like the presentation of Riemann-Stieljes integral from multiple perspectives, such as the Darboux approach and step functions Very standard and nothing new overall, but impressively detailed and careful, with many examples,exercises and proofs. An excellent set of  notes for the second half of a single variable real analysis course. Well worth a look by those either taking or teaching such a course.Highly recommended.
  32. CONSTRUCTION OF NUMBER SYSTEMS N. MOHAN KUMAR University of Washington St Louis  (PG) I always get grief for saying this, but experience has taught me this the hard way and people will never convince me otherwise: students need to build all the number systems from scratch at least once as students. (Well,not literally from scratch-literally from scratch would mean to begin with a set-theoretic axiom system or something logically equivalent and build the foundations until you create Peano systems and then go from there, but you get my meaning..........) . Seriously, I believe most students struggle in their first rigorous analysis course because they don't really understand the real numbers.An axiomatic complete ordered field presentation doesn't really give them that understanding unless they already have significant experience with proof. They don't understand inequalities, they don't really understand why complicated limits are true via ε-δ arguments or how to create their own given a limit problem. And this is all because they don't really understand from an axiom treatment why real numbers behave the way they do. Once a student gets their hands dirty building the real numbers from Peano systems, they never have these problems again. At least, I never did. The big objection to doing this has been if you take the time for a detailed development, it'll eat up the whole semester and you'll have no time for anything else. I always thought that was an idiotic response because that's only true if you develop each and every detail pedantically a la Landau. There's absolutely no reason to do this unless you have a group of students who have never seen a careful proof before-and that kind of course is a discussion for another day. If you have students with some background in rigorous proof, then a large chunk of the spade work can-indeed,should-be left to the students. Only several steps-such as the building of the real numbers from the rationals-need to be developed in detail. Fortunately, there are several developments of the number systems freely available as lecture notes online that recognize this and develop the real numbers accordingly-and I've listed as many as I can find on this site. As for these specific notes, Kumar does a wonderful job taking the students from the Peano axioms to the construction of the reals via Cauchy sequences, which really is easier for students to grasp then Dedekind cuts. The presentation is detailed, but still leaves a number of steps for students to prove. A great supplement to analysis courses or just for great reading on the bus. Highly recommended!
  33. Analysis Web Notes John Lindsay Orr University of Nebraska-Lincoln (PG) A very interesting and creative entry in the advanced calculus/ elementary real analysis academic sweepstakes.The author is formerly the chairman of the mathematics department at UNL and is currently a software developer. As a mathematics professor, Orr apparently was fascinated by the development  of modern computer software as a teaching tool to be integrated completely into virtual presentations of pure mathematics.The Analysis Web Notes project was his most ambitious and successful attempt towards this end.  This is a complete interactive online textbook for a first course in real analysis following calculus where the "book" is broken into 3 basic modules: Chapters, Classes and Exercises. The contents can be found here.The most striking part of the notes is that the first 2 modules form simultaneous versions of the notes with identical content, but each with a completely different intent, organization and structure.  The Chapters version is structured much like a conventional textbook, with each topic being covered in relatively verbose,detailed and impersonal prose, with definitions, theorems and proofs, examples interspersed with elaboration of concepts and results.The Chapters version is to be read like a usual text or reference. The Classes version, on the other hand,is broken into much shorter,almost "bullet point" sections with very specific subtopics where the reader or teacher can choose and arrange topics in a much more personalized manner. This allows creating either lectures or self-education according to the level and interests of the desired course. Interesting sidebars on unusual topics in analysis are present in both versions, such as infinitesimals, Zeno's Paradox and the construction of the complex numbers. In either version, the presentation is literate and lively, with many examples and pictures. The Exercises
  34. in the third section run the gamut from softball to truly challenging-and the the student or the instructor can choose as many or as little as they like. Orr is really to be commended here.He's put an enormous amount of thought into how to teach analysis online and he's succeeded in pulling off a very difficult stunt: He's produced an introductory analysis text that is not only well written and very informative, it delivers on the promise of being actually different from the 100,000 other textbooks on the subject that exist either in print or online. It's an analysis text that allows students and teachers to learn or teach actively, jumping around within subsections as they will to produce the path to analysis best for them. Besides-it's a joy to read. I'm not sure if I'd have the nerve to use this to teach a class, but I'm sure going to recommend it as collateral reading in such a course. Very highly recommended!
  35. Fundamentals of Analysis WWL Chen  University of London 2008   (PG) These are Chen's introductory analysis notes. After his wonderful and careful calculus notes here, one expects an equally terrific presentation of the fully theoretical side of calculus. He doesn't disappoint. While the choice of topics is fairly standard-real and complex number systems, sequences and infinite series, functions and limits, differentiation, the Riemann integral and sequences and series of functions-it's done in Chen's terrific style, concise yet incredibly lucid with many examples. Also, there are several aspects of the notes which are unique-for example, throughout the notes, he develops all number and sequence concepts not only for the real numbers, but the complex numbers as well. This allows him to illustrate in many points and examples of basic complex analysis with real valued sequences and functions as special cases.He also develops some aspects we don't normally see in beginning analysis, such as double sequences and the
  36. Cauchy product. There are also many terrific exercises. Another winning set of notes from Chen and a great asset to students and teachers in elementary analysis courses. Very highly recommended.
  37. Undergraduate Analysis Problems Jerry Kadzan University of Pennsylvania 2013 version  (PG-13)  A huge, excellently chosen and diverse set of exercises that range in difficulty level from honors calculus of one variable to a rigorous treatment of functions of several variables in Rn to analysis in metric spaces  at the level of Rudin's classic text and even some graduate student analysis qualifying exam level problems.  All are challenging and some are flat out impossible for mortal men. A resource all serious students of analysis should download and use for preparation for qualifying exams after completing  undergraduate analysis courses. Very highly recommended.
  38. Sequences and Series: An Introduction to Mathematical Analysis by Malcolm R.Adams University of Georgia (PG) Interestingly, Adams' colleague at UGA, Pete Clark, has written and posted a similarly themed set of lecture notes and it's hard not to compare the two. My full commentary for the Clark notes can be found here. Definitions of sequences,  convergence and limits, finite and infinite series, negative and positive terms, infinite series and sums, power series and sequences and series of functions. The choice of material is fairly standard, but it's extremely lucid and lively in the writing. It's a joy to read, with many examples and humorous asides, such as the author's title, "What is reality?" when asking what a real number is. Adams' notes are clearly geared towards a mathematically less sophisticated audience then Clark's notes-they have narrower coverage with a more intuitive and slower presentation. It is clearly intended for students with little or no exposure to a rigorous treatment of calculus. Indeed, it's very similar in content and flavor to the "British" style first courses in analysis based on sequences and series for incoming freshman at university who studied non-rigorous calculus in high school, such as Kaye's or Hyland's notes. But Adams' notes tend to develop the subject matter in more detail and at a slower pace then those notes. The result is an excellent first exposure to rigorous analysis for students without much experience in careful proof, whereas Clark's notes are better suited for stronger, more experienced students. Even with strong students, Adams' notes would make delightful supplementary reading to a more concise text.Very highly recommended.
  39. INTRODUCTION TO ANALYSIS: SUPPLEMENTARY NOTES  Attila Mate Brooklyn College of the City University of New York April 2002 Last Revised: January 2013    (PG)A sharp, nicely written supplement to a standard advanced calculus/ elementary real analysis course taught at one of the hubs of my old alma mater, The City University of New York. Makes me all teary eyed with nostalgia. This course was based on Maxwell Rosenlicht's excellent An Introduction To Analysis. (Since CUNY is a college forever in danger of being shut down or privatized since the wealthy in New York are offended by the very existence of a college that middle or lower class students might be able to afford, it's a university essentially based in effective poverty. So a professor that cares will try and assign a Dover text, which isn't always feasible but a good option. )   The problem with using Rosenlicht is that it's a bit too general for a first course, being done entirely on metric spaces. Mate's notes develop classical single  variable analysis on the real line using Rosenlicht's notation. The notes do an excellent job of complementing the textbook, with many examples and exercises on the real line carefully chosen to illustrate how one goes from the real line to general metric spaces. It also covers some topics either not covered or insufficiently covered in the text, such as total boundedness and a full proof of L'Hopital's rule. A very good supplement to an excellent text and together, they form a very helpful source for elementary analysis courses at both beginning and advanced levels. Highly recommended .
  40. Analysis 1 Jairus M. KHALAGAI African Virtual University   (PG)A standard entry in the AVU series of books, really designed more as a workbook to supplement a text or set of lecture notes then as an actual text itself. Still, there's a lot of good and varied exercises in here that will be of good practice for elementary analysis. Recommended as review and practice.
  41. Functions Of Several Real Variables Richard Barraclugh University of Burmingham  (PG)Another extremely brief, telegraphic set of notes by Barraclugh containing definitions, theorems and terse proofs-and nothing else. I dunno why I include them-maybe just my anal completeness. Nothing here you can't find done better in other sources, but check it out in case you like that kind of thing.
  42. Analysis I Course  T. W. Korner University of Cambridge September 18, 2007 (PG)The lecture notes for the first course in analysis at Cambridge by one of the master analysts of our time. Archimedian axiom, sequences and series, limits and continuity, Riemann integration and much more. Korner, of course, has since written the wonderful textbook, A Companion To Analysis: A First Second and A Second First Course in Analysis-which is referenced in the "further reading" section and by the author's own admission, is more advanced then these notes while covering similar ground. The main difference is that these notes remain on the real line while the finished book focus on metric and normed spaces. For those who can't afford Korner's book-and if you can, I strongly recommend getting a copy-this is a terrific free back up option. It shares all the best qualities of any textbook or notes written by Korner- deep, beautifully and insightfully written with many wonderful examples and enlightening exercises. It also contains Korner's equally awesome wry humor that makes anything he writes an absolute joy to read. Highly recommended.
  43. Sequences and Series Padraic Bartlett California Institute of Technology 2013   (PG) A concise, but very clear and informative set of lecture notes for the second semester honors calculus sequence at Cal Tech that the author has taught there several  times.  The prerequisite is the intensive honors calculus Math 1A course for gifted freshman there-so it's really suitable as a second course in real analysis or honors calculus for students with some experience with careful calculus and the idea of proof. Sequences of numbers, series of positive numbers, series of negative numbers,sequences of functions, power series, Taylor series and complex-valued series. The notes are very pleasant, with clear definitions and proofs of all the major results on convergence and summation in sequences with both real and complex terms. More importantly, they have many explicit step by step epsilon-delta calculations with sequences and series, which are quite important for beginners to see.They are nicely paced for strong students-rapid but not overwhelmingly so.There are also many interesting and well stated exercises, challenging but should be doable for those who master the material. In short , a very good source for analysis and honors calculus courses and their teachers and also a good supplement for more advanced courses to give students practice. Highly recommended.
  44. Sequences  And Series Jay Daigle California Insititute of Technology Winter 2014 (PG)A more recent version of the same course taught by Bartlett in the preceding entry. Daigle covers very similar subject matter, but his notes are a bit more detailed and slower paced, with more examples. He also has more material on Fourier series then in Diagle's notes. He  writes quite well-but so does Bartlett. While Bartlett's notes are excellent, I tend to think the self-study student might find Daigle's version of the course easier going. But really-both are quite good and since both are free, why not avail yourselves of both sources? Highly recommended!
  45. Analysis Piotr Hajlasz University of Pittsburgh
    Analysis:A First Course Piotz Hajlasz University of Pittsburgh Spring 2010 (PG) Another beautiful set of lecture notes by  Hajlasz, this one on the standard one variable advanced calculus/ elementary real analysis course-and even better, these are typed. Basic logic, naive set theory and cardinality, real numbers,mathematical induction,exponential and logarithmic functions,sequences and series, limits and continuity, differentiability. There's very little that's original here, but it's immensely well done with a lot of insightful comments,the kinds of observations that are very helpful for beginners.  For example, in the proof of the triangle inequality, Hajlasz makes the simple but critical observation that despite it's apparent obviousness, the fact needs proof simply because there are ordered fields where it happens to be false. Unfortunately, there's nothing on Riemann integration-I'd recommend the author in the revised version of the notes would add a section on this critical topic. Still, that's easily supplemented from other sources.Again, another set of lecture notes by a master teacher. Highly recommended.
  46. The Foundations of Analysis Larry Clifton Clifton Labs 2013   (?) Ok, this is a truly bizarre set of notes attempting a completely original presentation of classical analysis using a completely different language then is usually used-and it's not the language of nonstandard analysis. From the author's own description: This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of Euclid, and then use these classical results to define the positive real numbers categorically.  Let me tell you something-the notes are interesting reading with a wealth of new terminology and a fascinating approach. I'm all for original spins on standard material-after all, we would still be using Newtonian infinitesimals if that wasn't the case. But does anyone think a standard advanced calculus course could be taught this way? Seriously? That being said-it would certainly make for an excellent basis for undergraduate research seminar and graduate students might want to take a look at it to see if they can use it as a seed from which any publishable research can grow. I know I will.  Kudos for originality and good mathematical thinking-but really can't be used for course material.
    RIEMANN INTEGRAL NOTES SPRING 2007 ANTONR.SCHEP University of South Carolina    (PG)  Exactly what the title says it is-a careful and very brief (12 pg) development of the theory of the classical integral. Nothing special here, but might be good for students first learning or teaching this material for the first time.Worth a look.
     Honors Calculus III E.R. Vrscay University of Waterloo 2012   (PG)These are Vrscay's lecture notes for 2 semesters of the 3 semester so-called "physics section" of the honors calculus course at Waterloo. The first semester covers single variable calculus from the real numbers to sequences through the Riemann integral, the other semester covers multivariable calculus with the total derivative, partial derivatives, multiple, line and surface integrals with many applications to the physical sciences. Supposedly, the main difference between this version and the purely mathematical version are the inclusion of many physical examples to motivate students of physics and engineering. Be that as it may, this is hardly a "plug and chug" mindless computational version of calculus (which is why the notes are listed here instead of in the Calculus section of the Library, duh). While there are many wonderful and beautifully presented applications to physics in these notes- gravitation, projectile motion, fluid mechanics, heat transfer, diffusion to name some of the more important-Vrscay does not avoid or omit the theoretical aspects of calculus in his presentation.  Limits, derivatives and approximation theorems are all stated and proved very carefully, with many examples. The notes are rigorous but at the same time, very concrete. There are also many pictures and examples. I can't think of a better and more balanced course for an honors course in calculus for either mathematics or physics majors and it proves a balance can be achieved in such a course. Sadly, the middle semester notes are missing. It's too bad because with that middle section in place, this is one of the best single sequence lecture notes on rigorous calculus that exists either in print or online. Very highly recommended.
  47. Foundations of Analysis Franz Lemmermeyer Bilkent University(Turkey) March 9, 2002  (PG) A very nice and rather unorthodox set of lecture notes. It begins with a full and lucid construction of the number systems from the Peano axioms to a construction of the real numbers via Cauchy sequences. He also mostly avoids full ε-δ arguments in the chapter on continuity in a very original way: He begins by defining the functions f(x) = 1 and g(x) = x and proving they are continuous the usual way. Then, using inverse functions on compact intervals to define a ring structure on R, which he then uses to manipulate continuous functions via sums or products to either f or g, which have already been proven continuous. I'm not sure if this method really simplifies the discussion or makes it any less murky for beginners, but it's quite creative and it certainly cuts down on the ton of epsilon chasing the usual discussion entails. He also covers the Riemann integral in a very lucid way before discussing differentiation! My one complaint is there are very few examples and sadly, the exercises for the course are missing. But by the author's own admission at his website, the notes are unpolished. A very clear and creative set of notes, which I'd recommend all students and teachers take a look at.
  48. Basic Analysis: Introduction to Real Analysis by Jirí Lebl University of Illiniois Urbana-Champlain (PG)  The role of online textbooks in a mammoth self-education guide like this can't be overstated. Beginners, unless the quality control is going to be done by an experienced teacher, are really better off with formal textbooks, which is why the production of such books is critical for the free availability of scientific information in the future. Lebl's book is certainly worth it's status as one of the most popular of the"officially" freely available mathematics textbooks currently online.It's a book for a standard one year course in advanced calculus/undergraduate real analysis of one variable. Contents: 1. Real numbers,sequences and series, continuous functions,derivative,the Riemann integral,sequences of functions and metric spaces While the content is quite standard, the quality of the presentation is quite good and has a number of interesting and subtle original touches. One of the main differences with standard treatments of this material has less to do with content then style: Lebl prefers to prove results either by direct calculation or by contrapositive-he avoids proofs by contradiction nearly completely. As a big proponent of argument by contradiction, I was initially suspect of this approach. But it turns out it works very well indeed for classical analysis. Sometimes it results in lengthy and tedious proofs, such as the fact there exists a real number x such that x2  = 2. But proofs such as this really get students used to using and manipulating the properties of the real numbers in a careful way after taking such things for granted in most calculus courses.  Another inventive touch is the use of the somewhat simpler construction of the Riemann integral using the Darboux sum. I always liked this way of introducing the Riemann integral because it's very straightforward and visual. It also makes more sophisticated integration theories easier to comprehend by building upon the students' intuition.There are also topics present that are not too difficult at this level, but for some reason are usually left out of such courses, such as a careful presentation of the decimal expansion form of the real numbers.There are many pictures, examples and insightful comments. Lebl also provides plenty of exercises, most not too difficult and some are given directly after key examples whose solution is very similar to the preceding example. This allows the student to practice immediately whether or not he or she understood the concepts introduced in the previous section. I'm really surprised how much I like this book and I heartily recommend anyone looking for cheap textbook for an introduction to analysis should give it a serious look. To be honest, I'd be fascinated to know how effective the book, when supplemented with applications to physical and social sciences, would be as an honors calculus text.. Highly recommended.
  49. Gauge Integral An Introduction by Eric Schecter Vanderbilt University 2009  (PG) A VERY brief and lucid introduction to one of the most underepresented topics in modern analysis-the gauge or generalized Riemann integral by an analyst who has done a great deal to develop it. This is a topic I feel most undergraduate students should be exposed to and usually aren't. It's more general then the Lebesgue integral and as long as one stays in R-which there's no reason for undergraduates not to do-it's much simpler and requires no measure theory or function space theory to develop! Unfortunately, the brevity of the introduction really undercuts how useful it could have been to students. Fortunately, he includes a huge number of references for further reading. For that reason alone, it's worth checking out.
  50. A first Analysis course John O'Connor University of St Andrews (Scotland) 2002    (PG) A very nice set of notes for a standardfirst rigorous analysis course in the British style, but more thorough in its coverage. The topic selection is a bit unusual since most of the usual topics of calculus proper-like the derivative and the Riemannintegral are omitted.O'Connor instead focuses on the purely theoretical aspects of analysis that are usually omitted in a first calculus course.I'm not crazy about this idea-it's important to maintain the connection between calculus and the abstract analysis that underpins it or the whole enterprise becomes mysterious to me.But that's really a smallquibble. It's done carefully with a lot of brief commentary and informal discussion that really helps clarify things for a beginner. Also,there's a lot of good historical notes that provide some depth to it beyond the typical  procession of definitions, theorems and proofs.It also helps to compansate for the lack of explicit calculusmaterial by reminding the students of the connection to calculus.There's also quite a few sidebar presentations to additional topics either the instructor or the curious student would love to learn more about, such as Dedekind's cut construction of the real numbers, the golden ratio and space filling curves. An excellent-if flawed-and veryreadable set if notes for either a course or self study. Highly recommended.
  51. The Calculus  William V Smith Bringham Young University (PG) This is another unique and excellent textbook available exclusively online. The author himself gives a terrific description of thecontent and goals of the book at the beginning, which I can’t really improve on: Here  is a free online calculus course. This is essentially an ordinary text, but you can read it online. There are lots of exercises and examples. As we get the chapters scanned in, they will become highlighted so that you can click on themto read. This text is somewhat unusual for two reasons. The text is rigorous. (Proofs are done for nearly everything - eventually.) We do calculus in both one and two variables at the same time. Both points are no doubt controversial, but conceptually the approach gives a kind of clean synergy which generates important examples and unifies calculus to a great extent.    What’s both fascinating and surprising about this book is that it’s the second goal that I think will ultimately proves most problematic for the book’s construction.  The first 7 chapters of the book are pitched at more or less the level of a standard plug-and-chug calculus course, but it organized and written in a manner that is anything but standard. While Smith doesn’t prove things rigorously in these early chapters, he certainly defines everything in a completely modern manner. For example, functions are defined via ordered pairs of real numbers and derivatives are functions defined by subsets of the graph of the original function whose values are the slope of the graph at specific points in the original function’s domain   Another very unusual aspect is that he doesn’t explicitly define integration in these“intuitive”chapters- he only defines the anti-derivative and the area on the real line. There is a method to his madness, as we’ll see. As stated in his preface, Smith defines objects both on the real line and inn-dimensional Euclidean space in the same chapter-although he has the good sense to do the one dimensional case in detail first. For example,partial derivatives are defined after the “ordinary derivative” as the derivative of a function of n-variables for the ith variable where 1, i,£,n and all other variables are held constant. He also makes some interesting choices of definition that I think will help clarify things for a beginner by making the most use of as few set-theoretic objects as possible. For example,he defines a constant as a function f: R ----->{ c } where c ∈R  Applications to the physical sciences are given throughout for both one and several variables, which allows a richer collection then is normally given in a first course.This really is the hidden logic behind discussing one and several variable calculus simultaneously, of course-in the real world,applications are almost always of more then one variable. In chapters 8-11, the hard theoretical structure of calculus is laid bare;first by developing the basic niave set theory needed to precisely render the language of the previous chapters’examples, then defining the Least Upper Bound and Archemedian Properties of the real numbers and lastly,by defining a limit point in R. The topological definition of the limit point allows Smith to give both the definitions of sequential convergence and the limit of function in terms of a limit point-and the idea of a limit point helps to make rigorous the idea of functions “getting close” to a point without actually being defined there.  Once the rigorous definition of limit is given, all the standard limit theorems given in the early chapters are proven rigorously, Smith then develops the theory of the Riemann integral on both the real line and in n-dimensional Euclidean space. I really enjoyed his book very much-I’m sorry Smith left the current draft stagnate since 2001. Not only does it contain hundreds of examples, applications and beautiful graphs in addition to the theoretical chapters-it is superior and entertainingly written with a sardonic wit one rarely sees in textbooks. The construction of the book makes it extremely versatile: a standard calculus course could be taught using just the first 7 chapters (although this would be a tragic shame in my opinion!). An intensive, outstanding year long honors course to students with no prior exposure to calculus could be taught using the entire manuscript by assigning the easier sections as independent reading. My one complaint with the book is that the combination of single and multi-variable functions may confuse students without the assistance of a teacher. That quibble aside, this is a truly remarkable book, one of the best I’ve seen either online or in print. I would be absolutely thrilled to teach from it and students would be equally thrilled to learn from it.  Very highly recommended-more so then almost any other book on this list. Honestly, no kidding. Take a look, please.
  52. CALCULUS INTEGRAL Brian S. Thomson Simon Fraser University (PG) Here's one of the ERA's "sister" books available for free download  at the same site and written by the senior author. After I gave such a glowing review to ERA, I was really eager to take a careful look at the others. This is a book on integration theory for honors undergraduates for either an honors calculus or an elementary real variables course. So the book is intended to be pitched at about the same level as the first part of ERA. That may very well be, but it certainly is nowhere near as reader friendly as that book. The book seeks to revise the standard teaching of integration to undergraduates by using the so-called Newton integral instead of the Riemann integral. The Newton integral is called so because it's an abstraction of the process of integration as understood by Issac Newton in the formative days of calculus-namely as being defined by the fundamental theorem of calculus as an antiderivative. To be precise, given a function f, Thompson defines the definite integral as the change in a uniformly continuous function that is an antiderivative of f at all but at most finitely many points i.e. a set of measure zero. A function is integrable if and only if such an antiderivative exists. As interesting and intuitively clear as this definition is, I'm not sure how comfortable I am with replacing the Riemann integral with it. First of all, the definition requires functions that are even more well behaved then Riemann's-they have to have an antiderivative on the domain of definition.  Secondly, I  think Thompson's disparaging comments about the Riemann integral as a "moldy old 19th century" construct completely miss the point of why the  Riemann integral has held it's ground as the first integral students learn for so long-namely, that in it's conception, it's basically a rigorous version of the ancient Greek "method of exhaustion" for finding areas.Therefore it has a very simple and straightforward geometric interpretation. An antiderivative, despite it's simplicity as a calculational tool, can't really be given a simple pictorial description. But all that's open to debate. It's hard to argue with the fact Thompson has in fact given a beautiful and very careful formulation
    of the integral in terms of antiderivatives- which as he very correctly points out, is the natural way most students outside of pure mathematics think of it. Serious physics and engineering students who take real math courses in particular should find this approach much more appealing then the standard one.The main problem with the book is that it's formulated as a problem course. Except for the definitions, all the main results are given as exercises. Ok, that's not exactly true since all the solutions to the exercises are given in the second half of the book. There are several excellent problem books that are structured in this manner, particularly the topology and set theory texts of Ian Adamson. But obviously, the book is designed so that students can produce all of the results themselves and look up the answers they can't get. With some hard mathematics background, such as topology or algebra, I can see such a book being doable by average mathematics majors. But with this material, I doubt any but the very best undergraduates will be up to the task.  Still, that caveat aside, Thompson has written a beautiful, very instructive and original book that I think those gifted undergraduates will learn an enormous amount from in either a honors course or a seminar. I also think the book will be a gift for entering graduate students before tackling a first year graduate analysis course. Very highly recommended for students with some background in analysis and proof.
  53. Honors Calculus Notes Ambar N. Sengupta Louisiana State University 17th November 2011  (PG) Another honors course in calculus. LSU organizes thier honors sequence in an interesting way: both honors and average students are taught the same basic material in the same class with the honors students being supplemented with more rigorous material. Therefore, the honors students get both theory and applications simultaneously. I'd be interested to know how effective that is and I'd be really curious to try it with my own students. Contents: Basic set theory, infinum and supremum, topology of the real line,limits, derivatives and applications, the Riemann integral and applications.The notes have a lot of unusual touches at this level. For example, the extended real line is used throughout and the definition of limits is somewhat modified to be valid on infinite domains. This is an important concept usually avoided at this level, but Sangupta does an excellent job presenting it. The presentation is very crisp,lucid and smooth, with relatively simple proofs of all results.There's also a surprising number of pictures. One complaint of mine is there are very few examples-however, since the students had a standard calculus text to  work through simultaneously and the notes were basically to fill in the missing proofs, one really wouldn't expect a lot of examples, would one? (In fact, since the students were apparently using James Stewart's text as their "regular" text, I wouldn't expect examples to be a problem at all. Stewart's text has a lot of shortcomings as a mathematics text, but lack of good examples certainly isn't one of them!) Also, the chapter on continuity needs to be lengthened. A much more serious defect is that there are no exercises-but again, given the nature of the supplementary role these notes played, I wouldn't expect any. Unfortunately, this greatly limits their usefulness as a self study text and teachers will have to supplement the notes with exercises. Selected solutions to theexercise sets that accompanied the notes are given sans the actual exercises. I hope the author will collect and put the exercise sets into the notes in future online versions. Still, a very well written set of notes for an honors calculus/advanced calculus course and I like them a great deal. Properly supplemented, they'll be quite useful for several different kinds of courses. Recommended.
    Honors  Differential Calculus Problems, Solutions, Handouts Joel Feldman University of British Columbia(PG)
  54. Honors Integral Calculus Problems, Solutions, Handouts Joel Feldman University of British  Colunbia   (PG) These are the honors sections of the calculus courses taught by Feldman commented on above. They don't really add much to the "regular" note sets that Feldman so beautifully produced, as the calculus courses at UBC are rather rigorous to begin with and don't really need much tightening up. (American educators have a lot to learn from this example.....) Indeed, there's considerable overlap between these courses and their "regular" counterparts and the main difference is more focus on epsilon-delta arguments. They have the same wonderful simplicity, directness and clarity that make all of  Feldman's notes such a joy to read. Highly recommended for all students and teachers of calculus at all levels.
  55. Honors Calculus Pete L.Clark University of Georgia complete draft Spring 2014 (PG)  I don't know where to begin with this absolutely beautiful, incredibly well thought out and well written first analysis text by Pete Clark. Well, I guess that's a good start. This text has been evolving at Pete's webpage ever since he taught honors calculus at the University of Georgia in 2011 and decided to write his own notes to supplement the standard texts.  He describes the book as a "deconstructed" version of Spivak's Calculus. He rightly points out that the heart of Spivak is the enormous number of incredibly difficult and daunting exercises in the book-and that truthfully, many of those exercises are too difficult for any but the most gifted mathematics students to solve. Also, he rightly points out that Spivak is unnecessarily concise.(I completely agree. I personally wanted to throw the book against the wall several times during my own reading of it, as much as I liked it. )  So Pete decided to produce a "rewrite" of Spivak that not only preserved the book's best qualities for strong students, but developed and fleshed out the overly terse sections and solved the most difficult exercises. The "book" has no exercises yet-it clearly isn't ready for prime time as a stand alone text yet. A noble project and I'm very happy to report it's an unqualified success. The table of contents can be found here and it's well worth reading through by any prospective teacher and/or mathematically passionate young student. I haven't listed the topics covered in the book because they'd seem very ordinary-it just looks like a conventional advanced calculus/honors calculus class. And you'd be completely wrong.  Clark's incredibly lively presentation, personal digressions, choice of development and overall literary style makes this course not only an absolute joy to read, but one of the most informative presentations of basic analysis I've ever seen. Firstly, he focuses like a laser on the structure of the real line in a classical sense.Topology and norms are almost completely avoided and nearly all proofs, examples and exercises are given entirely in terms of   arguments. This  to me is the smartest decision he makes in the whole book. One of the reasons students in our country are so weak in analysis is because they're assumed to have good skills with "calculational proofs"using  inequalities before moving on to the more abstract settings of metric and topological spaces.Meanwhile, most students are thrown right into abstract spaces with a mediocre at best understanding of rigorous calculus on the real line. This is an absolutely critical skill for students to learn and a basic analysis course that beats them over the head with it will give them the skills they need to really understand later analysis courses.  Secondly, he includes many topics that courses at this level don't discuss, topics that greatly enrich the basic foundation of the students, such as sophisticated looks at polynomials, binary and n-ary sequences, complex numbers and series and much more. Thirdly, the text isn't organized in the usual way-topics are sometimes discussed deliberately out of order to achieve a deeper understanding in the end. For example, he defines and discusses continuity and derivatives before limits via linear functions and developing several important properties before defining the usual definition of limits and then redefining these topics in terms of them. This allows him to use the derivative as a simple example to motivate the  definition of limit by showing the earlier"intuitive" definition of derivative only makes sense in open intervals.  Clark does stuff like this all through the book and it shows he's constantly thinking about the most lucid way to present this material-and motivating his students to do the same. I hate to criticize this book at all, but I'd be remiss as a critic if I didn't mention the few quibbles I do have with the book.I'm not  going to complain that there are no physical applications in the book. I wish it did,but Clark is a pure mathematician, so I'd be surprised if there were any. I do wish the book had more specific calculations of limits instead of just general theorem proofs. Students really need to see a lot of specific examples of ε-δ proofs before they can confidently do their own. While he has some, there are nowhere near enough. I strongly suggest Clark take a look at  Kenneth Ross' Elementary Analysis: The Theory of Calculus  or Arthur Mattuck's An Introduction To Analysis for many wonderful examples he can either incorporate into future editions or serve as inspiration for his own. He should also try and incorporate as many of these into the exercise sets as possible. Spivak does have a number of these kinds of exercises, but they're more difficult then the ones in  either Ross or Mattuck. But these are all very minor quibbles. Clark has written what may be the single best free source on single-variable elementary real analysis that currently exists and I strongly encourage all students and teachers of analysis to download it, read it and savor it. I strongly encourage him to continue drafting and improving the work as a standalone textbook. I also would be so bold as  to suggest he make the finished book available in a cheap paperback or even better, simultaneously freely available as a downloadable PDF. Were he do to so-I could nearly guarantee it would very quickly become one of the most commonly used texts for such a course. And speaking for myself-I'd be proud to publish it in an inexpensive edition if and when my company gets off the ground. The highest possible recommendation.
  56. Honours Calculus I John C. Bowman University of Alberta Fall 2013 (PG)
  57. Honours Calculus II John C. Bowman University of Alberta Spring 2014  (PG) Yet another absolutely beautiful and comprehensive set of lecture notes for the honors calculus sequence at the University of Alberta. (Makes me jealous I was born an American between the education and the free health care up there, seriously.Not that they're Nirvana on Earth,of course,but they do seem to have their priorities in order,don't they?  ) Axiomatic development of the real numbers, sequences and convergence, functions, continuity,differentiation, The Riemann integral, logarithmic and exponential functions, techniques and applications of integration, improper integrals and infinite series. The contents are pretty standard, but it's very well done with many pictures and exercises-including many computer generated 3D images with the high level language Asymptote. There are also many good examples given in full detail-the first 2 chapters don't have many, but once analysis proper begins with sequences, many more are included. There are also many  additional problems, the sets of which are available at each  semester's webpage, which is linked to above.   Of these and the other,classic notes for this sequence by James Muldowney, which I've already commented on here, I have to give a slight edge to the notes by Muldowney since he does such a wonderful job combining theory and applications to the physical sciences. Bowman has some applications to physics in the chapter on integration applications, but there's more with greater depth of discussion in Muldowney. But this is really a minor quibble.  Bowman's notes are deep, very nicely written and broad in scope. I'd have no problem recommending them and their problem sets to any serious mathematics student or teacher of calculus for a one  variable course in analysis. Highly recommended.
  58. Introduction To Analysis Math 112 Ray Mayer Reed College Spring 2006  (PG) These are Mayer's course notes for an elementary analysis/advanced calculus course at Reed College. Like his earlier calculus notes, they are surprisingly inventive and original without being revolutionary. He begins with a standard introduction to basic set theory and logic,albeit somewhat more detailed then usual. The original twist in Mayer's course is that most of the course focuses largely on the detailed structure of the basic number systems-the natural numbers, integers, rationals and the real and complex numbers-without doing an explicit Landau-like construction from sets or Peano axioms. Instead, Mayer first defines a field and then introduces a set of 13 axioms that gradually expands the structure of a "base" abstract field. This converts the base field first to the natural numbers where induction is introduced and it's relation to the order structure of N is given. He then adds 0 and the negative numbers to get the integers and develops the induction principle and recursion on this field,and so forth. So basically Mayer gives a very original "field-theoretic" development of the number systems that is more modern, cleaner and far less tedious then the usual methods.The result is that Mayer gives a very thorough analysis of the number systems as his foundation for analysis without an explicit construction. This provides a strikingly original and fascinating compromise approach between a pedantic construction of the reals and the usual very brief "here are the field axioms of the reals and thier properties-got it? Good,next....." approach. I think it's wonderfully effective and gives the best of both worlds here. The is particularly clear when he proves the basic inequalities of analysis as consequences of the reals being a complete ordered field. Mayer doesn't begin analysis proper-sequences, series and convergence on the complex numbers-until page 131! Another way his approach is original is he defines most of the standard topics of elementary analysis on the complex field, particularly sequences and series, taking the real field case to be a special case-making a note of it when the 2 cases differ significantly. This gives him an important tool, allowing him to use the geometric representation in the plane of complex numbers to visualize critical concepts. For example, he represents sequences of complex numbers by arcs spanning the range of the sequence with it's endpoint at the point of convergence. He also uses some of his own terminology to clarify things, such as the instruction of "dull sequences" and "precision functions" in the definition of convergence of sequences and the proof of it's standard results. The notes have many, many examples and exercises, as well as special problems called entertainments which either cover material not directly relevant or more sophisticated approaches to the current topic. The one very strange omission here is that integration theory isn't touched. Not even mentioned.  This probably has to do with the unusual structure of the calculus training at Reed: Students first take  a careful if not entirely rigorous calculus course based on the idea of integration coming before differentiation and being essentially the area under a curve. (Indeed, Mayer himself has written a strong set of notes for that course, which we comment on here.)  The point here is that students are assumed to have seen a careful geometric treatment of the Riemann integral before this course. Ok, that works well for the students at Reed, but I assume for any outsiders using these wonderful notes, they'll have to supplement it with a treatment of integration to produce a complete first course in analysis. In any event, this is one of the most creative,complete and lucid first courses on analysis at this level I've ever seen. Very highly recommended for all students and teachers of analysis.
  59. Differential And Integral Calculus I Shiri Artstein-Avidan Tel Aviv University, Fall 2009  (PG)Earlier, when commenting on another set of lecture notes from Isreal,we saw how that country is still one of the few remaining that teaches the theory and applications of calculus together in the same course,no exceptions. These notes by Avidan are an even better example of this. Since this is only the first semester, the notes only cover up to and including differential calculus. (The third semester notes on multivariable calculus, written by Mikhal Sodin, are available in English and will be discussed later.) The resulting course notes would only be useful for an honors course or an advancedcalculus course here in the states. The notes begin with an axiomatic development of the real numbers as a complete ordered field with the usual metric, then moves on to give a rigorous presentation of the limits of sequences, followed by Cauchy sequences and infinite series convergence tests and manipulations, then limits of functions via epsilons and deltas, then a discussion of the exponential and logarithm functions via limits and ends with a complete discussion of differentiation of one variable. The notes are very clear and completely rigorous-even containing a number of applications in the differential calculus sections, such as curve sketching and velocity computations.  This excellent set of notes is for the serious mathematics student only,but they’re well worth a look. Highly recommended.
  60.  Analysis I Class notes John Wood University of Iillinios at Chicago  (PG) Very standard and incredibly compressed notes (30 pages!) for a first semester of advanced calculus/ elementary real variables. Rational numbers, consequences of the field axioms, completeness,sequences and limits,series,limits of functions and continuity,functions continuous on an interval,inverse functions, differentiation, integration and integrability, the fundamental theorem of calculus and Darboux and Riemann integrals. It's pretty amazing how clear the notes are given how concise they are-Wood keeps the reader's eye on the ball at all times and there's virtually no expository chit chat or examples. Just the inexorable march of theorems, definitions and proofs in the "merciless telegram style" as Edmund Landau so famously put it. That being said,Wood does write very clearly with nice proofs and smooth writing. I personally don't think these notes would be much help to a student reading on his or her own. If you like these kinds of notes, feel free,but I'd rather use something else.
  61. Honors Calculus III-IV Frank Jones Rice University 2005     (PG)   This is a careful and well written set of notes for an honors course in the calculus of several variables. Contents: Euclidean space,differentiation,partial derivatives, higher order (partial) derivatives,symmetric
  62. matrices,manifolds,implicit function theorem,cross product,volumes of parallelograms, integration on Rn,more integration,integration on manifolds,Green's,Stokes' and Gauss' theorem and other coordinate frames. Jones is deservedly well known for his book on Lebesgue integration for undergraduates; it's one of the most accessible introductions to the subject that currently exists. (Sadly, it's also quite expensive.) Like most treatments of multivariable calculus at this level, the notes stay in Rn and emphasize the geometric aspects over the analytic ones. However, Jones emphasizes the geometric aspects nearly to the exclusion of analytic ones. Although linear algebra isn't explicitly a prerequisite, Jones weaves a great deal of vector space and linear mapping theory into his presentation that's developed as needed, including norms, inner product spaces,eigenfunctions,eigenvalues and orthogonality relations.  Unfortunately, the title is somewhat misleading since this isn't really a rigorous course like Hubbard and Hubbard or Sherman online-mainly because Jones avoids proving most of the hard results from real analysis, like the Extreme Value Theorem or the Heine-Borel theorem. But it's not exactly a pencil-pushing, plug and chug course, either. He defines everything carefully and isn't afraid to
  63. use an epsilon-delta argument,although he does make a real effort to avoid them unless absolutely necessary. But he does try and prove things using either geometry or algebra most of the time. So if you're looking for a hard analysis treatment of these topics, look elsewhere. That being said, they're a joy to read and work through. Jones does give an extremely clear and careful presentation of the topics and he almost never takes things for granted without some kind of proof. He also has some original touches-such as focusing on n-1 dimensional embedded "hypermanifolds" rather then the n-dimensional ambient space. I don't know whether or not this approach is less confusing for students,but it works here.The presentation is very visual and intuitive, with many examples and beautiful pictures. He also has many good exercises, none too difficult. They reminded me lot of James Cook's advanced calculus lecture notes-they're careful without being rigorous in the classical analytic sense, but still presents the material in an extremely insightful and informative manner. The lack of analytic rigor might annoy some teachers, but that's easily solved: A more analytic course in the functions of several variables could use Spivak's  classic Calculus on Manifolds for the careful analytic rigor combined with these notes in order to supply much needed examples and intuition. In any case, they're an excellent introduction to the material. I highly recommend it to all serious students and teachers of several variable calculus.
  64. ELEMENTARY  REAL ANALYSIS  Thomson Bruckner Brian S. Thomson Judith B. Bruckner Andrew M. Bruckner Second Edition (2008)   (PG/PG-13) This remarkable textbook has quickly become one of the top go-to texts for teachers and students who need an accessible and relatively cheap introduction to real analysis for undergraduates. Part of it's appeal is the obvious fact that it's available freely online, as one of the first such freely available textbooks.As part of the authors' philosophy, it's also available in a relatively cheap paperback edition. But making it's appeal strictly a matter of dollars and cents really cheapens the true quality of this book as a learning tool. The authors' websites and their reasons behind the writing of this book and it's sequels can be found  here.   It's well worth reading in the authors' words and I won't waste space rehashing it. The table of contents of the book gives the impression,on cursory reading, that this is a very standard real variables text at the undergraduate level. But a closer look shows how utterly wrong headed that impression is. This book is actually 2 courses at the undergraduate level: The first is a standard post-calculus advanced calculus/real analysis course of one variable for students with little or no experience with rigorous proofs at about the same level as Lebl. The second contains material for a more abstract course on n-variable functions defined on Euclidean and metric spaces for either an honors first course or a second intermediate course in analysis at the level of Rudin or Hoffman. Each section also contains "enrichment" sections, which cover material which is either too advanced for a standard course or too specialized, but gives serious students of analysis insights into areas of importance to researchers, such as the Cousin compactness theorems, Dini derivatives and infinite products of series. Lastly, each chapter ends with a selection of challenging problems on the material of that chapter. These aspects and the fact that each section is self contained results in a book that is incredibly versatile and can easily be adapted for a very diverse series of courses at the undergraduate level. Lastly-the book is incredibly well written and lucid, with a very leisurely and conversational style that isn't the least bit rushed, but isn't ponderously wordy to the point of creating a hulking text a student's going to get a hernia carrying around. The best part of this last observation is that the book can be read by students with little or no difficulty. I absolutely love this book,it's one of the best written, most versatile textbooks at this level I've ever read. I wholeheartedly recommend it for all students and teachers of  undergraduate real analysis, if only to use as supplementary reading. I also recommend you check out the other textbooks available at the authors' website and blog, which contains not only this book, but it's graduate level sequel and a half a dozen other equally wonderful textbooks the authors are making available. The authors are to be most strongly commended for beginning and maintaining this project. Analysis is one of the most difficult subjects for most mathematics students and sadly, their understanding of it is usually not as deep or full as we really need it to be before they move on to become either mathematicians or scientists. These resources will be of immense help to students in this regard. The highest possible recommendation for this text and all it's related resources online!
  65. ADVANCED CALCULUS Rudi Weikard University of Alabama at Birmingham 2013   (PG) A brute Moore Method type advanced calculus course in which no proofs are given whatsoever. Like I've said several times already, I'm not a big fan of this method for several reasons. The procession of exercises is very well organized and clearly stated, though. Covers the very standard topics of one variable analysis: axioms of the reals, sequences, functions, limits,derivative,the Riemann integral and infinite series. There are no proofs and barely definitions.Very similar in content to Martin Bohner lecture notes here. The one saving grace is that Weikard gives a very good introduction to mathematical thinking at the beginning to set the stage. I'd pass as a teaching course, but I think students intensively reviewing undergraduate analysis for qualifying or final exams should find them very helpful to test themselves.
  66. Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew H. Nitecki Tufts University August 19, 2012 (PG)    This is an intriguing book-it’s the draft-in-progress of Nitecki’s sequel to his Calculus Deconstructed: A Second Course in First-Year Calculus . That book was an attempt to write an honors calculus course for strong students that was rigorous while remaining concrete and not too abstract. Although I think it's flawed in some ways, it's general approach-trying to balance rigorous proof and computation to produce a single variable calculus course that doubles as an introduction to careful mathematics-is right on the money and the author does a pretty good job walking that tightrope. With this online manuscript, Zbigniew is trying to follow up with a similarly themed course on the calculus of several variables. The main choice which makes this book so interesting is that the author develops the material strictly in R2 and R3  .  The limiting of the discussion to low dimensions has several very good
  67. advantages:   First, it limits both the amount and abstraction level of the machinery that's needed to carefully develop the theory. For example,vector fields and differential forms are used interchangeably depending on which gives a more lucid formulation and/or easier computational tool for a given concept. This really can't be done effectively in higher dimensions. Second and more importantly, it allows a very natural geometric emphasis that makes the presentation extremely visual while sacrificing zero rigor, even more so then the prequel text. Vector algebra, Euclidean analytic geometry and abstract linear algebra are simultaneously developed to give this visual presentation precision-in fact, this book is even more rigorous then the single variable text. As in the prequel, there are many wonderful historical notes of great depth, particularly notes involving the development of the Euclidean and analytic geometry of the plane and space and their roles in the development of vector analysis. As with the prequel, there are many excellent examples and exercises, balancing theory and applications-both to geometry and the physical sciences-in nearly equal measure.Interestingly, the  ε-δ definition of limit, which Zbigniew  downplays in favor of expressing all limit concepts in terms of  sequences, is given much more emphasis here. This is a good decision in my opinion. A purely sequential development of limits for functions of several variables is considerably more cumbersome then a straightforward formulation by limits, see the text by Shurman for this approach. I like this book even better then the prequel-the presentation is smoother and more lucid while carrying that text's strengths into the multivariable presentation. I also think this book will be easier to absorb then Shurman or some of the other free choices on this list. Hopefully, this strong text will remain freely available for several years before it joins it's predecessor on the open market.  When it does,Zbigniew will have completed an outstanding 2 year course for strong undergraduates that will join Spivak, Hubbard and Hubbard, Edwards and a handful of others as the standard texts for honors calculus. Very highly recommended.
    Differential And Integral Calculus I 
    Mikhal Sodin Tel Aviv University Fall 2009  Earlier, when commenting on another set of lecture notes from Isreal, we saw how that country is still one of the few remaining that teaches the theory and applications of calculus together in the same course, no exceptions. These notes by Sodin are an even better example of this. Since this is only the first semester, the notes only cover up to and including differential calculus. (The third semester notes on multivariable calculus, also written by Sodin, are available in English. They are more difficult and will be discussed in the Intermediate Level Analysis section. ) The resulting course notes would only be useful for an honors course or an advanced calculus course here in the states. The notes begin with an axiomatic development of the real numbers as a complete ordered field with the usual metric, then moves on to give a rigorous presentation of the limits of sequences, followed by Cauchy sequences and infinite series convergence tests and manipulations, then limits of functions via epsilons and deltas, then a discussion of the exponential and logarithm functions via limits and ends with a complete discussion of differentiation of one variable. The notes are very clear and completely rigorous-even containing a number of applications in the differential calculus sections, such as curve sketching and velocity computations. They're a bit dry, but this excellent set of notes is for the serious mathematics student only, but they’re well worth a look. Highly recommended.
  68.  CALCULUS - lecture notes Stefan Balint Eva Kaslik, Simona Epure, Simina Mariªs, Aurelia Tomoioaga
  69. (PG) West University of Timisoara  (Romania)  This is a very interesting and original set of lecture notes for first year calculus-and they begin with an amazing statement. These are notes constructed and taught to computer sciences majors.  Ok, that’s not entirely true. The same lecture notes are presented to both mathematics and majors in other sciences-where the courses differ, apparently, is in the oral presentations by the instructors and the difference is best stated in the conclusion of the author’s preface:The written course is presented in a standard form, similar to the course presented to mathematics students. However, the spoken course is full of comments and examples that are meant to illustrate the utility and applicability of the concepts and results at solving real problems.  The reason the earlier statement is amazing is because the written course is entirely theoretical-it reads like a standard American introduction to real variables course. One of my dearest friends is from Romania and she would tell me when we first met how the Romanians would drive their children in the sciences in order to surpass the Western universities-teenagers studying quantum theory and advanced differential equations was not uncommon. More recently, I’ve met Romanian mathematicians who tell me that their native country is one of the few left in the world who still teach calculus “in one stroke” i.e. theory and applications in the same intense 2 years sequence. I hear that recently they’ve revised the second year course at most Romainan universities so that functions of several variables is presented on differentiable manifolds and uses tensor calculus and differential forms! On the other side of the coin, many Romanian scientists trained in this system have told me horror  stories of the calculus course and how non-math students would literally have to memorize all the proofs without comprehending a word of them.  I think a system that forces students to do this completely defeats the purpose of presenting the mathematics carefully. An ideal  system would “sell” to the non-mathematics students on the many advantages of doing things rigorously. Be that as it may, this particular university seems to have come up with an interesting solution that doesn’t water down the mathematics-the written notes supply all the rigorous proofs and all the physical and geometric applications are presented in the oral lectures; with the course pitched at non-mathematics students emphasizing the latter much more then the former. It’s a shame we don’t have an actual transcription of both versions from the oral lectures in English-they would not only be fascinating to compare, there would be more choice for an appropriate audience. That being said-the notes are beautifully written and lucid, although clearly too theoretical for a standard American calculus course. They would make a wonderful and useful set of notes for either an honors course or a baby real variables course-although for an honors calculus course, one would have to supplement it with more examples and applications, for which there’s plenty of free choices on this list. Highly recommended.
  70. Calculus Raz Kupferman Institute of Mathematics The Hebrew University May 5, 2011(PG) Now here’s another good example of how calculus training differs in other countries from our own. The main recommended textbook for the course is Spivak’s Calculus. Anyone who’s familiar with that great classic will know immediately what the focus of this course will be-namely, on the mathematical theory of calculus, what American schools teach in “baby real variables” and “advanced calculus” courses. What’s really impressive about this course that will make it useful to all students studying real analysis  at any level is that it has many explicit ?-? inequality computations, something that all beginners struggle with the beginning and can really only be mastered with as much practice and examples as possible. Kupferman does a good job supplying some geometric intuition to these purely formal concepts and proofs. Unfortunately, it has the same problem as its inspiration-namely, it’s all theorems and proofs and no applications whatsoever.  This really illustrates the problem with the rigid separation of theory and application in many "honors" level first courses: Calculus was first developed and gained its critical importance through its applications to the physical sciences. The response of the difficulty of its formal foundations has been to put them in entirely distinct courses and pretend one aspect doesn’t exist. That’s like teaching music by teaching pentameter and syntax of music in one course and how to play in another and forbidding students in the former to play music while they’re studying it. It’s ridiculous. Tht being said, this is a very strong and well written introduction to the theory of calculus and supplemented with a purely applied set of notes or book-like Strang’s book-it can provide the basis for an excellent honors course in the subject or by itself, a good baby real variables course.
  71. Calculus Notes Math1115, Honours 1 1998 John Hutchinson Australian National University (PG)   Honors Calculus Notes MATH1115, 1999 John Hutchinson Additions and Modifications Australian National University  (PG) These are Hutchenson's notes for the honors sequence in calculus at ANU that is the prerequisite for his analysis courses for undergraduates. They're very well written and organized,but if you're expecting anything original in either the presentation or selection of material, forget it. It's a very standard theoretical presentation of single variable calculus,albeit done with exceptional care and great insight. The real number system via the field axioms, limits, continuity, differentiability, the Riemann integral and differential equations. The 1998 notes form the bulk of the notes, the 1999 " version" is a mere addendum where several topics are presented somewhat differently and some interesting additions are made, such as discussions of the Archimedian axiom, more explicit limits with sequences and a brief introduction to the hyperreals. . He uses simple hand-drawn graphs to illustrate many points of rigorous analysis-the presentation of sequences and convergence is particularly good.  There's some advanced topics in optional sections, such as some multivariable calculus and uniform continuity. It's all presented very well, as I've said. Unfortunately, these notes don't have the exercises for the course present, which is a huge minus.And Hutchinson's style is pretty dry.  Still, they make very nice reading for introductory analysis courses-but I think there are many other, superior sources here. Recommended.
  72. Honors Calculus of One Variable Fedor Duzhin  Nanyang Technological University 2006-2008(PG) Calculus of Several Variables Fedor Duzhin  Nanyang Technological University Materials of 2007–2010 (PG) NTU has become one of the most prominent technical universities in the Far East in the last decade, as much for the quality of its training as for its research. These notes, written for the 2 year calculus sequence for honors students in mathematics there, are an outstanding example and one of the best available free online sources on calculus listed here. They have quite different levels of rigor,however, and the prospective student or teacher needs to be made aware of the strengths and weaknesses of both. The first is for a highly theoretical honors single variable calculus course at much the same level as Muldowney, Bowman or Clark above. The notes are for the most part rigorous and all the basic concepts of single variable calculus in the first part are proven carefully. Limits in particular are developed in great detail via the epsilon-delta definition with many examples. Many calculated examples and beautiful graphs, many of them computer generated, are given to accompany the rigorous proofs and definitions. There are also many physical examples and they’re usually put in separate chapters from the rigorous proofs and definitions. This gives the instructor or student some flexibility on how much they want to focus on careful proofs and how much on applications-but it would be a real shame to skip the careful proofs since Duzhin does such a good job presenting them at a level suitable for beginners. In fact, the notes are quite similar to Chen’s notes above in terms of content and style-except they are quite a bit more detailed with many more pictures. Like Chen, Duzhin shows excellent  judgement what to prove and what to leave unproven for a rigorous course that beginners can handle. The second set of notes on several variable calculus presume a good course on linear algebra and focus largely on the properties of Euclidean space as a vector space and the differential as a linear map. This allows for considerable rigor and visual geometry as long he doesn't discuss integration in higher dimensions.However, the level of rigor drops considerably when one reaches integration theory and the standard theorems of vector analysis i.e. Gauss,Green's and Stokes' Theorems. Although the multiple Riemann integral in 2 dimensions is discussed very carefully and done well with many pictures and careful proofs, the results on line and surface integrals are done far more loosely.It's hard to criticize Duzhin for this decision. If you want to discuss path integrals carefully, you basically have 2 choices: You can either stay in R2 and R3 using strictly vector fields and Euclidean spaces, proving the major theorems in these special cases only and fudging the details on non-smooth boundary curves or you can bite the bullet and develop the whole general machinery of differential forms and manifolds in Rn from jump. Either approach is really tough slogging for any but the very best  students. This and the great depth and visuals make both of Duzhin’s notes one of the very best sources available online for calculus. Very highly recommended.
  73. Real Analysis I Semester B Steve Shkoller Department of Mathematics University of California at Davis 2011   (PG-13) These are notes for the second semester of the year long undergraduate real analysis course at USCD. Riemann integration, differential maps on Rn ,partial  derivatives, differential in Rn , implicit and inverse function theorem. Their content is close to that of  Stefaan Delcroix's notes for a similar course at Fresno-with several major differences. Firstly, they're broader in coverage then Delcroix's notes, covering Riemann integration in depth. Secondly, unlike  Delcroix's notes, Shkoller  includes many interesting examples and counterexamples.Lastly, he includes many beautiful computer generated diagrams, many of curves and surfaces in higher dimensions, very important in presenting multivariable calculus. Again, sadly, there's no coverage of multiple integration or vector analysis from a rigorous point of view and they only cover half of a typical first year course. But what the notes do cover, they cover very well, with lots of good discussion and exercises. Recommended.  (Skholler is  currently- February 2015-a visiting professor at The University of Oxford-just letting users of this website know in case they Google him at USCD and get goose eggs. )
  74. Honors Multivariable Calculus Eric Carlen Georgia Tech University Fall 2013  (PG) Yet another textbook draft of a rigorous treatment of multi-variable calculus!!! Why is this all the rage now? Well, to be fair, there really aren't that many "official" textbooks for a course like this i.e. a careful treatment of functions of several variables for both mathematicians and scientists that also contains applications. There's Hubbard And Hubbard's wonderful book, Edwards' very unorthodox classic, the more standard and ridiculously expensive book by Shifrin, the nicely priced book by Edwards Jr. , the brutal book to end all books by Loomis and Sternberg- and that's about it. So a lot of mathematicians are trying to create texts for their courses. This entry by Carlen follows the general philosophical trend of most of the current online drafts of such books-it downplays the analytic aspects of functions of severalvariables and emphasizes their geometry and properties as vector spaces. As I've said, this makes perfect sense since the geometric properties of such spaces are really what distinguish them from the real line. Vector geometry and basic linear algebra in RnNewton's laws of motion in  Rn  , limits in R, continuity and the total derivative, the Implicit and Inverse  function theorems, curvature and quadratic approximation, multiple Riemann integration,  integration on bounded regions and parametrized surfaces in R3 ,  change of variables and coordinate systems, linear transformations and the Spectral Theorum in  Rn, line and surface integrals and the theorems of Green, Gauss and Stokes. Carlen's text is rather classical in presentation-the style is wordy while not being overly so. It develops entirely with modern levels of rigor while remaining for the most part in R2 and R3-it's quite similar to Nitecki's lectures in style while being somewhat more advanced and comprehensive. It's also reminiscent of the beautiful text by Bandaxall and Liebeck. Interestingly, unlike Nitecki, he completely avoids the language of differentiable manifolds and forms. It'll be an interesting question whether or not he continues in this vein as the book "evolves" and nears completion. Carlen writes very lucidly and with quite a bit of detail-it's quite pleasant to read. The most impressive and striking aspect of these notes are the enormous number and sheer diversity of developed examples from calculus, linear algebra, differential geometry and physics-probably more so then in any other text draft currently online at this level. Chapter 2 is entirely on classical mechanics and it's relationship to low dimensional differential geometry. Lastly, there are a number of topics not usually covered at this level or in this manner, such as the Householder maps, the squeeze theorem in several variables, the finite dimensional Spectral theorem. In short, this is a very solid, comprehensive and well written work for an honors multivariable calculus course. Very highly recommended.
  75. ANALYSIS  II CONTINUITY AND DIFFERENTIABILITY JANET DYSON UNIVERSITY OF OXFORD HILARY TERM 2013  (PG-13)   The notes and exercise sets for the most recent incarnation of the second semester of the 3 semester analysis course at Oxford.  From the course description:  At the end of the course students will be able to apply limiting properties to describe and prove continuity and differentiability conditions for real and complex functions. They will be able to prove important theorems, such as the Intermediate Value Theorem, Rolle's Theorem and Mean Value Theorem, and will continue the study of power series and their convergence.  Janet Dyson's notes are in the typical Oxford style: Concise but lucid with all essential details,few but extremely well chosen examples and freight car full of terrific exercises. There are also many comments and observations one normally doesn't see in such a course, such as Dyson's comments on the "sensitivity" of limits to a point in asymtotic
  76. functions.In short, the course is efficient while remaining extremely enlightening. Coupled with the first and third semester materials, the Oxford notes form a wonderful self study resource for undergraduate real analysis students. Highly recommended.
  77. ANALYSIS AN INTRODUCTORY COURSE Ivan F Wilde Mathematics Department King's College London  (PG) Another set of notes from across the pond for a standard UK style first course in rigorous analysis.However, these notes are exceptionally comprehensive, detailed and readable for such a course. There are also a substantial number of examples,which are all discussed in lucid detail. Sets, axioms of the reals, sequences, functions, limits, derivative,the Riemann integral,infinite series and special functions (exponential, logarithmic,gamma, etc.)  Of particular interest for students are the many careful explicit ?-? inequality computations, probably more in number and with greater lucidity then just about any other source listed here currently. They are very well written and can serve as a self-study text for any course  at this level. The one sad and major drawback is that Wilde doesn't include any exercises. None. Which really limits the usefulness of the notes as a text by themselves. The very good news in this regard is that Wilde's notes are very much in use at universities all across the United Kingdom and as a result, many sets of exercises specifically designed to accompany these notes are readily available from many websites. At the point of origin university, Yuri Safarov has written a very comprehensive set of exercises for the notes with complete solutions-they can be found here.    Together, the notes and the exercises compile one of the very best self study courses in single variable analysis currently available. Very highly recommended!
  78. Analysis 1 Lecture Notes Written by Vitali Liskevich Slightly revised by Several Others and taught by Misha Rudnev University of Bristol 2013-2014   (PG)Yet another first course in real analysis from the United Kingdom. Contents: Logical propositions and connectives.Sets; finite unions and intersections;differences.Ordered pairs; Cartesian products. Functions and their graphs Very basic introduction to quantifiers; negating quantifiers Injections, surjections, bijections. Invertible functions Proof by Induction Rationals and reals; irrationality of square root of 2 definition of supremum and infimum with example Completeness axiom and other axioms Inequalities Sequences and their limits Theorems on limits of sums, products, quotients and compositions Series. Tests of convergence.Limits of functions (epsilon-delta definition of limit).Theorems on limits of functions. Continuous functions. Definition and properties.Continuous functions on a closed interval.Intermediate Value Theorem; extremal values on closed intervals.Differentiation and its simple properties.Maxima and minima of functions.Rolle's Theorem; Mean Value Theorem and applications.Approximation by polynomials. Taylor's theorem.Inverse function; derivative of the inverse function. Series; alternating series; absolute convergence.Power series.The exponential and logarithmic functions.Trigonometric functions.Riemann integration in elementary terms.Fundamental Theorem of Calculus. Like Wilde's notes earlier, these notes are well written and extremely detailed with many examples.although not quite as many as Wilde's. Also, there's a bit more emphasis on set theory and basic logic here then in Wilde's notes, which concentrate more on the analysis proper. They're also a bit drier to read. But they're still well worth reading. The exercises for the course can be found at the course webpage here and they're quite good as well.   Another very good source for a first course in analysis or self study or a supplement for such a course. Highly recommended.
  79. Dr. Z's Advanced Calculus For Engineers ("Calc 5") Handouts By DORON ZEILBERGER  (PG) A very applied, very classical set of pencil pushing handouts for a course for engineers. Nice and clear, with lots of nice important results for engineers in Fourier analysis and differential equations, but absolutely nothing new, original or special here.
  80. Real Analysis I M D Coleman University of Manchester Fall 2013   (PG) Yet another UK style elementary real analysis course. And that's not a bad things, especially for a student just beginning to learn rigorous mathematics,as most of them are between solid and excellent in quality range. These by Coleman are no exception. A very careful and clear discussion of analysis on the real line, with emphasis on precise limit computations. Contents:     Limits. Limits of real-valued functions, sums, products and quotients of limits.Continuity. Continuity of real-valued functions, sums, products and quotients of continuous functions, the composition of continuous functions. Boundedness of continuous functions on a closed interval. The Intermediate Value  Theorem. The Inverse Function Theorem. Differentiability. Differentiability of real-valued functions, sums, products and quotients of continuous functions, Rolle's Theorem, the Mean Value Theorem, Taylor's Theorem. Integration. Definition of the Riemann integral, integrability of monotonic and continuous function, the Fundamental Theorem of Calculus. Coleman proves many results clearly and concisely, but he delibrately leaves a significant number of holes for students to fill during lectures. A beginner might find this rattling, but trying to fill the holes in the proofs-preferably by themselves,but they could just as easily look them up-would be an excellent exercise for them. And of course, a teacher can easily fill them in. The real strength of these notes is the enormous number of detailed computations with limits solved in great detail. There are also many terrific exercises with complete solutions given at the homepage. Another very solid resource on elementary analysis from the U.K. Highly recommended.  
  81. Introduction  to Analysis Irena Swanson Reed College Spring 2014   (PG) An exceptional first real variables course from Reed College that's of a much more traditional mold then the earlier set by Mayer. However, it too has many original touches by the author-touches that make it one of the best free sources for such a course that I've seen.. It's similar in its purpose to the notes commented on earlier by Klopfenstein- i.e. a thorough first course in elementary real analysis that doubles as an introduction to modern mathematics and methods of proof. Contents: Basic logic and quantifiers, naive set theory, relations and functions, binary operations and fields, order structures, the full construction of the number systems, limits, continuity, differentiation, Riemann integration, sequences, infinite and power series. It's quite well structured and surprisingly detailed,particularly in the early chapters,incorporating what's necessary to lay the foundations for a rigorous development of calculus on both the real and complex numbers. However, it also includes-particularly in the wonderful exercise sets-many unexpected topics from combinatorics, logic, algebra and more to provide a great deal of insight into mathematics for the neophyte.Some examples are summation sigma notation, the Tower of Hanoi and Pascal's triangle. It's organization is also a bit unorthodox-topics are covered in the order that the author believes will make the learning easiest to the student. For example, notice that sequences and convergence are covered late in the notes. As an immediate precursor to the following final chapters on infinite series, it makes perfect pedagogical sense. Notice also that chapter 3 contains a complete and very lucid development of the number systems-the natural numbers, integers, rational, real and complex numbers. Unlike most analysis texts where this construction is given, it's not optional-it's needed to be able to read the rest of the book. This facilitates the other function of the book as an introduction to proof and the careful construction of the number systems is one of the most powerful topics to cover in such a course because it draws on all the tools developed previously. And yet, at 45 pages long, it doesn't threaten to swallow up the rest of the text. I'm really surprised how good these notes are and I strongly recommend it as a text for a course at this level or as a fine supplement to "proofs" course. Very highly recommended!!!
  82. Calculus And Advanced Calculus Lecture Notes Ng Tze Beng National University of Singapore (PG) These wonderful notes are by the author of a  little known  rigorous calculus text who has taught the subject at both the basic and advanced levels for many years-and the experience shows. It's similar in style to Coleman in that it emphasizes explicit limit computations and proofs, but it's more thorough and sophisticated. Contents: real numbers and complete ordered fields, sequences and convergence, monotone sequences, series, partial and infinite sums, convergence tests and Cauchy criterion, limits and continuity of functions, differentiation and power series expansion, integration, sequences and series of functions, uniform convergence. Most of the notes are handwritten, but they're extremely legible and readable, which as I've said is critical if you're going to post scanned notes. The notes have many beautiful and insightful computations, many using the Squeeze Theorem and creative inequalities. Tze believes very strongly that getting students to be able to do inequality computations is just as important in analysis as proving limit and convergence theorems. There are a host of exercises that practice both these skills beyond simple drill.The result is a deep, beautiful set of notes that I wish I'd known about when I was learning this material. Very highly recommended. Indeed, Tze's entire website on calculus is highly recommended-with many deep proofs, insights and constructions.
  83. Advanced Calculus II Lecture notes Stephen Saperstone George Mason University Spring 2011
    Advanced Calculus II Stephen Saperstone George Mason University Exercises and Assignments Spring 2011 (PG) A set of handwritten notes for the second semester of a year long advanced calculus course based on Robert Strichartz's The Way of Analysis. Contents: Sequences of functions, infinite and power series with convergence tests, limits, continuity and differentiability of functions of several variables. A lot of my teachers and colleagues love this book, but I've always had mixed feelings about it. One the one hand, I love how Strichartz challenges the students with the intuition behind the concepts of analysis and the beautiful history behind it-and he does it in a very original and fresh way. On the other hand, there are lots of times he goes off on wordy and weird tangents-which is one of the reasons the book is so long-and the result is more confusing then enlightening. Personally, I think Korner's book does a much better job with similar goals.  Anyway, I didn't find these notes much of a helpful addition to the text. The handwriting is passable at best and most of the notes seem to either just reinterate what's in the text or solve problems. You can take a look, but I'd pass.
  84. ANALYSIS OF FUNCTIONS  OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 (PG)This online text in development is pretty much what the title says it is. It seems extremely standard in content on first blush-namely, just another book for a first year of undergraduate analysis on the real line-until one takes a closer look. Contents: Axiomatic developments of the real and complex numbers, sequences, convergence of sequences, limit points, functions and continuity, differentiation, integration, integration over smooth curves in the plane,  the fundamental theorems of algebra and analysis.  What makes this book original from all the others on analysis both online and in print is that Baggett takes the view of functions of one variable to mean both one real variable and one complex variable-that is to say, the book is about functions of one variable over both the real and complex fields. This provides a much deeper approach to a first course in analysis without resorting to abstract machinery like metric or topological spaces. The 2 tiered approach to analysis provides a lot of beautiful and subtle mathematics that usually isn't covered until graduate school. Baggett demonstrates how many-although not all,of course, and he makes the student aware of this constantly-many of the basic results of real analysis carry over with little change to the complex plane.This allows him to discuss them both simultaneously, something I don't think most basic analysis books do. He reserves the most difficult and important differences between real and complex analysis until the last chapter, the results revolving around analytic functions and the fundamental theorem of algebra. There are many very sophisticated exercises and Baggett, as the title implies, gives very detailed proofs and interesting digressions on the material.  Unfortunately, as interesting as this take on a first analysis course is, it results in some very strange decisions. First of all,despite Baggett's claim in the preface to exclude functions of several variables, a closer look shows he doesn't-indeed, he really can't. The complex plane is homeomorphic to R2 with the usual topology-the main difference between the 2 spaces at the elementary level are in the algebraic structures on them.  So as a result, he can't talk about complex analytic maps without discussing partial differentiation of their real valued component functions and he can't discuss contour integration in C without discussing line integration over smooth curves and bounded regions in the plane. He even segways into differential forms, a topic I'm sure completely confuses students when it pops up in the chapter on contour integration. I don't mind when authors use differential forms in a nonrigorous way to simplify calculations-lots of authors have done so to great effect. But Baggett does it just to simplify his proof of Green's theorem on smooth bounded regions. He still ends up imposing a lot of conditions on the regions, as he must to avoid problems at the boundaries-so the forms really don't simplify things that much. I think a lot of students will just be scratching their heads. Personally, I suggest he leave it out-if he limits the discussion of integration to R2  ?  C there's absolutely no reason he needs to bring it up at all.  Also, many of the exercises are quite challenging, more so then I think the average good math undergraduate could handle. He gives good hints, but still, the exercises are rather brutal, probably too much so for a first course. Those quibbles aside-he's written a beautiful and unusual text in analysis that with a good teacher and the right students, could serve as the basis for a wonderful course in advanced calculus or real analysis. It'll also make wonderful self-study for students that are very strongly interested in analysis. Highly recommended!
     
  85. Analysis Term 1  Daniel Ueltschi University of Warwick  2011-2012   (PG) Yup, you guessed it-yet another first course in analysis from Europe in the UK manner. Contents: Properties of the real number system. Inequalities.Bounded and unbounded sets. Least upper bound and greatest lower bound. Sequences of real numbers. Subsequences. Limit of a sequence. Convergence and divergence. Convergent sequences are bounded. A monotonic increasing sequence which is bounded above is convergent. Sums, products and quotients of convergent sequences. The sequences {nk} and {rn}. Sandwich rule. Series of real numbers as sequences of partial sums. Sums of Geometric series. Divergence of series. Comparison and ratio tests for series of positive terms. Series with positive and   negative terms. Alternating series test. Relative and absolute convergence. Convergence of power series. Radius of convergence. Cauchy's root test. Functions defined by power series (trigonometric and exponential functions). Similar in structure and content to the first semester analysis notes at Cambridge or Oxford, such as Korner's or Hyland's notes, but they're considerably more detailed. There's also more emphasis on the properties of the real numbers and the convergence  results of sequences and infinite series. Of particular usefulness are the side notes that give insights to the results that aren't obvious, such as the fact upper and lower bounds for subsets of real numbers are not unique, only glb's and lub's are. Many excellent problems interspersed throughout.  One of the better sets of lecture notes for this kind of course and should be looked at by anyone either teaching or taking a first course in  analysis. Highly recommended.
  86. Advanced Calculus I Joel Feldman University of British Columbia (PG)
  87. Advanced Calculus II Joel Feldman University of British Columbia (PG) These are the notes, handouts and exercises from Feldman's year long advanced calculus class at UBC. Not surprisingly, there is some overlap with his earlier calculus courses, especially the honors versions. This really where the rigor and care calculus is presented with in the elementary classes at this Canadian university really differs dramatically from those much softer courses in the US, which are pitched at the lowest common denominator. Here, since the students are required to learn some actual mathematics, the advanced calculus course here is merely the calculus of several variables course done carefully if not completely rigorously. First semester:Vectors in R2 and R3, inner product, cross product, lines and planes,limits, partial derivatives, tangent planes, chain rule, gradient, directional derivatives, implicit functions, higher order derivatives, equality of mixed partials, Taylor's theorem.local and absolute extrema, classification of critical points, Lagrange multipliers.double integrals, iteration, improper integrals, polar coordinates, triple integrals, cylindrical and spherical coordinates.Second semester contents: Curves, velocity, acceleration, arc length, curvature, tangent, normal, binormal, planetary motion.vector fields, field lines, conservative fields, line integrals.surfaces, surface area, flux integrals,gradient, divergence  and curl, vector identities, divergence theorem, Green's theorem, Stokes' theorem, applications,differential forms and the General Stokes' Theorem. Since these notes are really intended as supplementary material, they aren't complete enough to use as an actual textbook. But the material is presented with the same care, detail and lucidity as Feldman's other notes and won't need much in additional material. Rigorous proofs are combined with applications to geometry and physics. There are many computer generated graphs of surfaces and curves. Another superb set of notes and exercises by Feldman and they're highly recommended as supplemental reading for any course at this level.
  88. Numbers  and Functions Alexander Pushnitski Kings College University of London Autumn 2013 (PG)  This is a first course in analysis and rigorous methods of proof in the U.K. style. Which a) is ironic given the last entry and b) the content and level of which should be obvious by now to anyone reading this website. Contents: Sets and functions. Real numbers. Sequences: boundedness, convergence, subsequences, Cauchy sequences.  Pushnitski writes well and emphasizes the properties of the real numbers and computations with inequalities more then usual, as well as giving many clever quotations and pointers to the beginner in the art of proof. There's a good set of exercises, not easy, but not too difficult either.  Another nice course on basic analysis from across the pond. Recommended .
  89. Real Analysis Walter Schreiner Christian Brothers University 2009   (PG) A very clear, immensely readable and interesting set of notes for a gentle advanced calculus course at this small university. Contents: Logic,Set Theory and Math Induction The Real Numbers Sequences and Series Limits of functions Continuous Functions Differentiation The Riemann Integral Infinite Series Extremely detailed and focused on the real line, with many explicit inequality computations,pictures and examples, clear proofs and hints for many exercises. Schreiner takes great pains to present all computations and proofs step by step without any missing details and many additional observations and hints. The course is based on the book by Introduction to Real Analysis, 4th edition, by Robert G. Bartle & Donald R. Sherbert . The notes are very much in the same spirit as that fine text, only with many more details. As such, it's an excellent study source for beginners in analysis who don't have much experience with proofs and the gentle style will be of enormous help to them in learning the discipline of mathematics. Also,because of it's immense readability and many details, the notes are particularly good for self study,particularly for students who are returning to higher mathematics after a long absence and need a little handholding.  Highly recommended for beginners or struggling students in analysis.
  90. Sequences and Series Richard Kaye University of Birmingham 2007 (PG) A remarkable online textbook that's become one of my favs since discovering it. It's available by Gmu Free Documentation License either in full pdf file form or as an interlinked HTML webpage textbook-the contents can be found here. It's a first rigorous course in analysis centered on sequences and series as the main objects, which in and of itself is nothing original, particularly for notes or textbooks on analysis originating in the United Kingdom, as these do. What's original here is the depth and breadth Kaye develops it in-including not only the usual development of convergent sequences,infinite and power series and uniform convergence, he includes a full construction and exploration of the number systems, from the Peano axioms to a full construction of the real numbers via Dedekind cuts.  One of the major themes of Kaye's text/course is to motivate the need for rigor in mathematics and why we shouldn't just accept the elements of calculus-and mathematics in general!- on faith. Kaye does an excellent job of this in particular by showing how "pencil pushing" calculus makes perfect sense as long as you don't look past the surface. A fantastic example is how he demonstrates the limit of the sum of the quadratic series ∑ (1/ n2 )  as n →+ ∞  is far from obvious.  The notes are leisurely, gentle and crystal clear with a very large number of specific examples of sequences, series and their limits.  He also has many wonderful digressions that demonstrate very creative methods of explaining basic concepts, such as a dialogue between a customer trying to purchase a very expensive sequence who needs to be reassured by the salesperson that it will in fact converge via the definition of convergence. This dialogue alone would make these terrific notes worth checking out, but there's so much, much more to offer in them. One last thing worth mentioning are the 2 formats the text comes in-the interlinked HTML and "ordinary" pdf. While the pdf version is more convenient as it can be read like a conventional book, the interlinked version has several advantages-not the least of which is that proofs of main results are hidden from the reader unless clicked upon. This allows the reader to choose which results he or she wishes to read and which they'd like to work out for themselves. They can also decide which sections to cover in self-study or a course. The result is one of the very best introductions to real analysis either in print or online, one that's especially well suited for self study. I love this book and once you've read through it, so will you. Trust me. The highest possible recommendation!
  91. Analysis I Xinwei Yu University of Alberta Fall 2013  (PG) A fairly standard but unusually detailed set of notes for a basic real analysis course.  Content: The basics of logic and set theory, limit, continuity, differentiation and Riemann integration for functions with one real variable. The notes are concise, but contain many examples and again, a very large number of exercises They are rather dry and matter of fact,almost in a bullet point style.That being said, they are very lucid with many good examples.They develop some properties of the real line that are usually reserved for more advanced courses, such as detailed analysis of the open and closed intervals of the real line. The exercises are challenging but rarely too much so-and each section ends with complete solutions to the exercises. It's clear Yu wants to engage the students mathematically and he does a pretty good job here of doing that.The result is a readable but challenging course in real analysis that will be very helpful to serious students and their teachers, as well as for self study. Highly recommended.
  92. Analysis II Xue-Mei Li with revisions by David Mond University of Warwick March 8, 2013 (PG) A second course in analysis for those who have had a U.K style first course in analysis i.e. the elements of proof, basic logic and set theory, sequences and series and convergence theorems.Contents: Continuity of functions of one real variable,the Intermediate Value theorem, continuous functions on closed intervals,Continuous limits,Extreme Value Theorem, Inverse Function Theorem,Differentiation,Mean Value Theorem,Power series,Taylor's Theorem,L'Hopital's rule. A very readable and detailed course with a lot of good pictures, examples and insights for undergraduates in analysis.  Li and Mond focus on why we need to prove results in analysis-in other words, why just the nice plug and chug manipulations of calculus aren't good enough for mastering the subject. Indeed. since the notes are very similar in methodology to Kaye's notes, those notes and this one would form  a very natural pairing for a year long analysis course. Also,considering that Ueltschi's notes for the Analysis I course at Warwick, these notes and the follow up integration course by Zaboronski, form a single cohesive unit of introductory analysis from Warwick that will be of immense help to students studying analysis. Highly recommended.

  93. Analysis I Sequences and Series Hilary Priestley Mathematical Institute University of Oxford 2013 (PG) Yawn. Yet another version of the first semester analysis course at Oxford. Again, the same basic contents: The real number system,sequences, convergence theorems and limits, infinite series and power series expansions. Not only are there detailed summaries of Priestley's notes, he also posts the similar but more detailed 2011 lecture notes by Richard Earl, which are quite readable with many examples. Priestley also includes many exercises to supplement the notes. Nothing special, but very solid and well written material that will help any beginning analysis or honors calculus student. Recommended.

  94. Real Analysis I  H. Bruin Department of Mathematics University of Surrey 2011  (PG)  You know, by now I can basically

    tell you this is a UK style first course in real analysis and that will pretty much tell you what these notes contain. The real number system,sequences, convergence theorems and limits, infinite series and power series expansions. They are quite concise and with very few examples. However, they are nicely written, quite careful with many nice exercises and slick proofs. Still, I think there are many better lecture notes and texts online to use that are better for students. You can take a look, but I'd pass. 

  95. Introductory Real Analysis J K Langley University of Nottingham 2004       (PG)  This is a second course in real analysis for undergraduates, again, in the UK style-it assumes familiarity with the axioms of the real numbers, sequences and series and their basic convergence results. Contents: Review of Sequences. Functions,Limits and  Continuity Differentiability Power Series Representing Functions by Power Series Indeterminate Forms. Integration. Improper Integrals. They're quite similar in style to Yu's regular analysis notes-dry, concise and telegraphic in style. However, Langley's notes are somewhat more readable and  manage to pack a lot of examples and side observations into them. In particular, his discussion of the natural exponential and logarithm functions is very thorough and deep, more so then usual in notes like this. There are many good exercises scattered through the notes. Not my favorite, but a solid choice for self study in real analysis. Recommended.
  96. Introductory Real Analysis Notes Ambar N. Sengupta Louisiana State University November, 2008  (PG) Notes for a first course in real analysis given at LSU in 2008. You might expect some overlap with the author's honors calculus notes, which we've already commented on-and you'd be right. That being said, these notes are more sophisticated, quite a bit more detailed and generally more developed and capable of being used as a text independently then the supplementary honors calculus notes. A strange defect in these otherwise very comprehensive notes is that it omits a chapter on continuity, from which the course referred to the original draft of the excellent advanced calculus text by Leon Richardson. Still, that's a minor quibble since that's a hole easily filled in from other sources. Contents: Ordered fields, completeness of the real line, the extended real line, topology of the reals, limits, Bolzano-Weierstrass theorem and Heine-Borel theorem, the major results of continuous functions including uniform convergence and completeness of C[a,b],  differentiability, Riemann integration and the Darboux  Criterion. As with the earlier notes, they're well written and informative-they have quite a few more examples then the earlier notes, as would be expected from an online text that's more then a supplement. The author also adds his own original marks to this otherwise standard material. For example, completeness is expressed via the supremum and the infinium properties of R throughout the notes and consequently the Riemann integral is expressed in terms of lower and upper Riemann sums. At the end of the notes are a good and diverse selection of exercises. All told, this is quite a solid set of notes for an advanced calculus/ introductory analysis course and I hope the author continues to polish them in the future. Even better would be if he could keep them available as an online text. In any event, a very good online source for analysis. Highly recommended.
  97. A Primer of Real Analysis by Dan Sloughter Furman University 2009 (PG)  A very nice,somewhat unusual and relatively short book on undergraduate real analysis that can be very effectively used for self study. The contents indicate the originality of the book's structure. Contents: Sets and relations, functions, rational numbers, sequences, real numbers, upper and lower bounds, real sequences and series, cardinality, topology of the real line, limits and continuity, derivatives, integrals and special functions. Notice Sloughter develops many aspects of calculus within the rational numbers first. This is done intentionally so that he can slip in the construction of the reals as Cauchy sequences of rationals without having to begin with the usual Peano axioms. This is a very clever way of getting around the usual lengthy development of the reals while still getting to the essentials. There are many clever touches like that throughout the book, where the essentials are presented very carefully with many exercises. The book is very readable and clear, it could also be used for excellent honors calculus course.There are also many pictures and side observations. All told, an excellent text from  which to learn basic analysis, one that's both very readable and yet engages the student actively. It will also serve as an outstanding refresher for students who have been away from real analysis and need to brush up before graduate school. Very highly recommended, especially for self study.
  98. Real Analysis An Interactive Textbook Bert G. Wachsmuth Seton Hall University 2010  (PG) A clear and interestingly organized online textbook on real analysis that's been floating around the internet for several years. Contents:  Sets and Relations Infinity and Induction Sequences of Numbers  Series of Numbers  Topology  Limits, Continuity, and Differentiation  The Integral  Sequences of Functions Historical Tidbits  The book is fairly sophisticated, using topology on the real line freely-but the choice of topics is fairly standard and not dramatically different from other online notes or books in print.  One can't help but draw comparisons to the other major online "interactive" html formatted book on real analysis, that by Kaye.. Wachsmuth's text is more sophisticated and has,in some ways,broader coverage. However,it shares with Kaye the selectively interlinked structure that permits students and teachers to not only choose their own material, but to choose which specific results they wish to prove themselves and which they wish to see the proof of. All the examples in Wachsmuth  are stated without proof or explicit construction, which is hidden behind a GUI link. Therefore, one can either try and verify the result yourself or click on the link to see the reasoning. Interestingly, Wachsmuth has funded the site entirely himself with AdSense, using minimally intrusive ads. I certainly can empathize with his corundum, with the near vanishing of funding from public sources for such a site from traditional academic sources. And of course, this site is partially funded by AdSense. As long as we have an open internet-something we take for granted in America and is long-term very much in danger right now- independent sources of funding for both individual profit and public service will form an integral part of the availability of sites like his and mine. But I digress, my apologies. I like Kaye better as an introductory textbook. Wachsmuth's writing is clear and careful if somewhat dry, and  I wish he'd included more examples. Those quibbles aside, this is a solid textbook for real analysis and here's hoping Wachsmuch keeps making it freely available to students and teachers alike. Recommended.
  99. INTRODUCTION TO REAL ANALYSIS William F. Trench   Trinity University (PG)  This free online textbook for an elementary analysis course has become very popular.  Preface The Real Numbers Differential Calculus of Functions of One Variable Integral Calculus of Functions of One Variable  Infinite Sequences and Series of Real Numbers Real-Valued Functions of Several Variables Vector-Valued Functions of Several Variables Integrals of Functions of Several Variables  Metric Spaces Introduction to Metric Spaces Compact Sets in a Metric Space Continuous Functions on Metric Spaces Answers to Selected Exercises  Index The author's goals and approach are best summed up by himself in the preface: Without taking a position for or against the current reforms in mathematics teaching, I  think it is fair to say that the transition from elementary courses such as calculus, linear algebra, and differential equations to a rigorous real analysis course is a bigger step today than it was just a few years ago. To make this step today’s students need more help than their predecessors did, and must be coached and encouraged more. Therefore, while striving throughout to maintain a high level of rigor, I have tried to write as clearly and informally as possible. In this connection I find it useful to address the student in the second person. I have included 295 completely worked out examples to illustrate and clarify all major theorems and definitions.  Towards this goal, the resulting text is extremely readable, detailed and gentle with many examples. Indeed, the book seems directly intended for self study. Many textbooks in analysis make this very appealing claim-and end up being very hard slogging indeed for any but the very best students without the assistance of a teacher. Trench's text is one of the few whose claim actually stands up. Because of the level of detail and the gentleness of the exposition, a determined student armed only with pencil pushing calculus and some exposure to proofs shouldn't find this book a struggle at all. Trench has an interesting tool for clarifying results: He states major theorems of analysis in what appears to the expert to be an overly wordy manner  in order to introduce terminology rather then the usual converse process. He then uses the new terminology to restate the result more concisely. For example, he gives the Heine-Borel theorem as stating that every open cover of a closed and bounded subset of the real line has a finite open subcover- which is, of course, equivalent to saying every closed and bounded subset of the real line is compact.The development of functions of several variables is particularly good, one of the best I've seen for beginners. He uses the idea of Jordan content to give a comprehensive treatment of multiple integrals, which I've always thought was completely appropriate for beginners. Yes, it does give more complicated constructions and proofs, but the result is conceptually a lot easier then using some formulation of measure theory.There are a ton of excellent exercises, many of which either develop aspects of analysis not covered explicitly but are closely related to material that is discussed or give concrete examples of principles or theorems. The gentle nature of the book also makes it an excellent choice for the theoretical part of an honors calculus course-although the course would have to be supplemented by applied topics and examples from standard calculus, which is easily done. Strong students and their teachers will probably find it too gentle and should probably avoid it in favor of a more challenging source. I basically have one small complaint about the book. Trench downplays the role of inequalities and specific ?-? type calculations with limits in favor of more general topological results. The idea is to make the arguments more intuitive. I don't know if it does that,but the ability to understand and construct these kinds of proofs is so important, I dunno if the clarity gained is worth it. That quibble aside-this is a wonderful analysis book for complete beginners-especially for self study. I'mreally glad the author's decided to make it freely available online. For all students struggling in a first analysis course, this book will be a tremendous assistance. Highly recommended!
  100. Introduction to Modern Analysis Birne Binegar Oaklahoma State University Spring 2000 (PG)  A fairly standard and concise, if well written first course in real analysis, at about the same level and similar in spirit to Wachsmuth's text. Contents  Lecture 1: Proofs and Logic Lecture 2: Methods of Proof Lecture 3: Methods of Proof, Cont'd Lecture 4: Methods of Proof, Cont'd Lecture 5: Review of Set Theory Lecture 6: Relations  Lecture 7: Functions Lecture 8: Cardinality Lecture 9: Integers,  Rational Numbers and Algebraic Numbers Lecture 10: Fields and Ordered Fields Lecture 11: The Completeness Axiom Lecture 12: The Completeness Axiom, Cont'd Lecture 13: The Topology of the Reals Lecture 14: Compact Sets Lecture 15: Convergence Lecture 16: Limit Theorems Lecture 17: Monotone Sequences and Caucchy Sequences Lecture 18: Limits of Functions Lecture 19: Continuous Functions Lecture 20: Properties of Continous Functions Unfortunately, it doesn't cover nearly as much material. Instead, it develops methods of basic logic and proofs in the  early chapters. This will make the notes more useful to a beginner who hasn't had much experience with rigorous mathematics, but it seriously undercuts it's quality as a first course in analysis. Still, it's well written with many good exercises. Weak students will benefit much from working through them. Recommended for weak students.
  101. Real Analysis Last Updated: University of Texas Faculty Co-Authorship July 2011   (PG/PG-13))Surprisingly good and informative set of notes to supplement the U of T undergraduate real variables course. Contents: The real number system, of real sequences, and of  limits, continuity, derivatives, and integrals of real-valued functions of one real variable.These notes are co-written "by committee" i.e. there are 9-10 coauthors. Also, they're very terse and concise You'd expect such a set of notes to be a train wreck. Rather shockingly, the notes are very instructive and readable. As I said, they are exceedingly concise and many results are given as exercises. What prevents these notes from becoming a uselessly curt skeleton of results leaving students to prove in a miasma of confusion is despite the lack of detail, what discussion there is focuses on definitions and their clarification. This is something that sadly, a lot of books and notes for beginners in rigorous mathematics don't emphasize/ They really should because without mastering definitions, students can't really master the art of proof. Proof really is the art of deriving results from the simplest and most minimal assumptions that need to be completely understood and internalized before the truth of a proposition becomes clear. These notes really do work to get the students to do exactly that.  A good instance of this occurs when the notes are discussing the basic intervals of the real line and after defining them, discusses at some length the importance in mathematics of being able to construct your own examples and counterexamples. Appropriately, the exercises of that section give some requests for constructed intervals with hints-and whether or not some suggested intervals are "legal". Make no mistake-they are demanding and will require students with some familiarity with proof-absolute beginners will probably find them too difficult.  I had a lot of fun working through these notes and I think many serious mathematics majors and teachers of real analysis will as well.  Highly recommended for the right students.
  102. Calculus for Mathematicians, Computer Scientists, and Physicists An Introduction to Abstract Mathematics Andrew D. Hwang University of Holy Cross 2001 (PG/PG-13)  This is a text for yet another attempt to combine the foundational and practical aspects of calculus into a single course. It’s interesting to note that while all the attempts to do so I have encountered in compiling this bibliography are all markedly different from each other in both style and underlying philosophy, they all stem from remarkably similar concerns and goals-namely, as John Stillwell so brusquely put it, the desire to, “put the guts back into calculus.”  There’s a sense among pure mathematicians-and even mathematicially inclined physicists and engineers-that the removal in its entirety of the careful foundations of
  103. calculus from most standard courses has removed something essential from it that greatly diminishes the entire project. I tend to agree, but we seem to have trouble making the case for pure mathematics to those who are more practically minded. Hwang’s attempt to rail against the tide of pragmatic “do this” calculus classes begins with the preface, which he very appropriately uses as a “mission statement” for the book. So this means the author has a very ambitious goal here: He wants the basic first year calculus course to double as an “introduction to proofs and mathematical reasoning” course. I have to admit, I’m a bit surprised because I’ve never seen anyone attempt to combine both of these types of courses  before. I’ve seen calculus textbooks and notes that attempt to be completely rigorous, but they usually presume the students have some experience with proofs and logic. I’ve also seen authors attempt to combine the basic calculus course with the linear algebra course, which is rather natural due to the relationship of both to Euclidean space. But I’ve never seen anyone try and use the calculus course as an introduction to proofs and abstract mathematics before-all such courses on the latter I’ve seen have always assumed at least a pencil pushing calculus course. Thinking about it-I’m surprised no one’s ever tried it before. All technical students,whether mathematics, physics or computer science majors, need to take calculus. Also, most computer science majors need to take both calculus and a discrete mathematics course, which is rather similar in content to what mathematics students take as a course in introduction to abstract mathematics.  So it seems like a natural thing to try and hybridize both courses for this general audience. In fact, the result in many ways reminds me of a good discrete mathematics course. Quickly, the contents:The language of set theory,construction of the real numbers from the natural numbers,functions,limits and continuity,convergence of sequences,infinite series and continuous functions on compact intervals,differentiation and integration, Riemann integral, differentiation, critical point theory on the real line and the mean value theorem,fundamental theorem of calculus,sequences of functions,the natural exponential, the logarithm and the trigonometric functions  the Taylor expansion and numerical approximation, introduction to complex analysis. The Appendix from the author-explaining the origin of the book and the various influences on it’s writing-is an absolute joy to read, it’s almost worth skipping the rest of the book to get to it. The text is beautifully and deeply written with so many wonderful examples anf graphs-so much so, it’s eminently quotable, which I found irresistible while writing this review!  It is also somewhat metaphysical in approach, as the preface suggests.Hwang not only wants to teach students how to do abstract mathematics, he wants the students to think about what they’re doing and to understand why they’re doing it. What the benefit of this level of logical rigor is to not only calculus, but mathematics in general. In that sense, the book is in some ways an extended essay on mathematics itself and why we shouldn’t separate the rigor from the applications, especially in calculus. It’s this aspect of the book that really makes it unique and absorbing to read and makes it different from all the other rigorous treatments of calculus. It’s hard to imagine this compelling argument will make instant converts of anyone who reads it. I doubt this argument will convince premeds, prelaw or business majors, who usually think the math professor is a sucker they need to teach them what they need to get through the course before going out and getting a real job making real money. But hopefully, it will convince the curious young student who is undecided about a career and has never considered mathematics seriously before. There are several other points to be made about Hwang’s methodical construction of the text: First, he uses the analogies from computer science to introduce, motivate and explain aspects of logic and pure mathematics to students.This leads us to my second observation about the book-Hwang uses each “stage” of presenting calculus to introduce and explain new concepts of abstract mathematics in a concrete, specific task manner. For example, in the aforementioned lengthy chapter 2, he uses the basic language of set theory and logic introduced in the first chapter to introduce induction, recursion, the binomial theorum and the precise definitions of relations and functions-all with the explicit purpose of presenting in detail the construction of the real numbers as Dedekind cuts beginning from the Peano axioms. Lastly, the book always follows a logical development that never divorces the pure mathematics from the applications. I think I’ve fallen in love with this textbook and may never use another one again. Seriously. I’ve never seen a book that more closely matches my own personal philosophy about the subject of calculus and how to present it to students. The book is sadly far too difficult for average students who just want to learn how to differentiate on their calculators. But for serious students in the technical sciences-I can’t think of a better book to send them on their paths with. Run, don’t walk, to your computer and print out a copy for yourself before Hwang sends it off to a publisher to sell it for 200 dollars a pop.  Please. There’s lots of good online textbooks and lecture notes in calculus on this list for the cash strapped student-but I dare anyone serious about it to read this book and tell me they’d rather have another. Go ahead-I dare you. The highest possible recommendation.
  104. Introduction to Real Analysis John Hunter University of California at Davis Fall 2013   (PG)   A thorough set of notes for an advanced calculus course after calculus. Contents: The properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. Hunter writes well with some nice pictures and careful proofs. There are 2 big drawbacks to these notes, though. Firstly, there aren't many examples. There are some, but not nearly enough for my taste. But this is really a matter of personal taste more then a quibble. A much more serious flaw for those looking to use them for self study is that they have no problem sets. This is a huge problem when considering using them as a course text. Still, a nice,solid set of notes by someone who clearly has great command of the material.  Recommended as supplementary reading.
  105. Analysis III Oleg Zaboronski University of Warwick 2013-2014  (PG)Yes, these are the notes concluding course for the 3 semester undergraduate analysis sequence at Warwick, following the semester I notes by Ueltschi and the semester II notes by Li and Mond. The syllabus of the course can be found here. These notes concentrate on Riemann and Riemann-Stieljies integration via step functions as well as sequences of functions and norms. The emphasis on step functions rather then  partitions is very European and I prefer working directly with partitions. That being said, the transition to the Lebesgue integral is easier this way. They're handwritten scanned notes, unlike the others which are typed. I prefer typed notes for online scanning for obvious reasons.But Zaboronski has good handwriting, so it's not that much of a problem in this case.  In any event, the notes are a perfect capstone to the previous 2 sets. They are, like the others, concise but extremely lucid and conceptual-again, the purpose is to make students understand why the machinery of integration is developed the way it is and not merely give them a taxonomy of miscellaneous facts. There are many insightful drawings, careful examples and good exercises. Together with it's predecessor notes, it forms as good a free online textbook in introductory real analysis as one can find. Highly recommended.
  106. Honors Calculus part I Min Yan  Hong Kong University of Science and Technology January 7, 2014  (PG/PG-13)
  107. Honors Calculus Construction of the Real Numbers Min Yan  Hong Kong University of Science and Technology January 7, 2014 (PG) An extremely well written,lucid and focused honors calculus course on functions of one variable-which is part of a larger, comprehensive set of notes covering all undergraduate analysis of one and many variables. The larger notes can be found here and we'll be commenting on them there. An interesting twist here is the course  begins with an optional but complete construction of the real numbers from the Peano axioms to the natural numbers to the integers to the rationals and the construction of R via Dedekind cuts. Technically, this construction isn't part of the course notes proper, which is why they're posted separately. It would be a shame not to cover these notes, though.Yan gives one of the most lucid and complete presentations I've ever seen-and he manages to keep it's length down to 37 pages. The content of the actual honors calculus notes: Sequences, limits and continuity, differentiation, Riemann integration, infinite series. The notes are very similar in spirit and level to Spivak's text or Muldowney's notes-they emphasize the real line and give many computations, examples and an enormous number of diverse exercises. There are also many graphs and pictures to illustrate concepts. He also motivates many things using real life computations, such as explaining the rigorous definition of a limit by the error in an approximation calculation, such as the population of an area. Even better is how Yan describes the derivative as a linear approximating function to an unknown function at a point. This conceptualization of the derivative is  not only very accurate, it generalizes to abstract spaces very readily in more advanced treatments.The very large number of exercises within the text that are challenging but not too difficult are the real strength of the notes. Yan comes down hard on the side of pure mathematics-despite the motivation of concepts,there's very few actual applications to geometry or physics. All told, it's a very solid and well written set of notes for an honors calculus/elementary real analysis course and will serve students and teachers very well for such a course. Highly recommended.
  108. Honors Calculus of One and Several Variables Matilde Marcolli California Institute of Technology 2012  (PG) This is CalTech’s idea of a  one semester calculus course for freshman-which is to say it’s an undergraduate real variables course for mere mortals. Concise but very well written and have all the basics of the theory of calculus: the real numbers, limits, sequences and series, derivatives and the Riemann integral. The real find here are the very good supplementary notes written as additional optional “deeper” sections for students who want more detail-the notes by D. Ramakrishnan on the construction of the real via Dedekind cits and sequences are very good indeed and should be checked out by all students taking real analysis for the first time. There’s a lot of good stuff here despite the overall brevity and dryness  of the notes.I think they're too brief to act as a main course text, but they certainly can and should be used as supplementary material. Recommended for all students taking theoretical calculus for the first time.
  109. Introduction to Real Analysis Liviu I. Nicolaescu University of Notre Dame 2014 (PG) These fairly new (at this writing,  April 2015)  lecture notes,apparently for an honors calculus course at Notre Dame, came as a real surprise to me as I was surfing the web and assembling this website. Nicolaescu is well known as geometer and topologist, so it was rather interesting to see what his "take" on the first rigorous course in calculus/analysis would be. Contents: The basics of mathematical reasoning, the real number system and special classes of real numbers, sequences and their limits,limits of functions, continuity, differential calculus and its applications, the Riemann integral and it's applications. While the material is very standard for this course, it has Nicolaescu's great care, attention to detail and clarity that's present in his other lecture notes and books. Proofs are detailed without being too pedantic, there are many examples and exercises and pictures to illustrate main points.  Nicolaescu also develops aspects of the theory we don't normally see done in detail, as well as including many footnotes that assist the student in digesting critical results. For examples of the former, he uses Fibonnaci sequences to introduce recursion in the chapter on convergent sequences as well as a full proof of the binomial theorem using induction. For the latter,  Nicolaescu describes in the footnote to the proof of the Archimedian property of R it's description as the principle that "an ocean can be filled with grains of sand." He also includes several alternate proofs in small text inserts, many of which are interesting and useful.  A very solid and well written set of notes that I hope Nicolaescu will continue to eventually develop into a full fledged online book on undergraduate real analysis, bringing his unique style and superb teaching to the subject. Highly recommended.
  110. Introduction to Analysis( Non Honors) Erin P. J. Pearse University of Oaklahoma 2007(PG)  This is one of 2 versions of the real analysis course for undergraduates by Pearce-one for regular math majors here and a much stronger version for honors students This is the  slower version of the undergraduate real analysis course for non-honors students.The much more intense honors version can be found here. Contents: Logic and naive set theory, the rational numbers, Axiom  of choice, a construction of the reals via Cauchy sequences of rationals, estimates and approximation, topology of the real line, continuity, differentiation, integration, sequences and series of functions. This version is far more detailed with many examples, discussions and clever asides, including a very good introduction to inequalities and their role in analysis. . Pearce also includes many mathematical applications, such as error analysis in numerical approximation,the Fibonnaci numbers and the derivation of the number e. And of course, Pearce includes many good problems, most challenging but not impossible. An outstanding set of notes I would heartily recommend to all analysis students. Very highly recommended.
  111. Further Topics in Analysis Lecture Notes Vitaly Moroz and Julia Wolf University of Bristol 2014  (PG)  This is a second course in analysis in the U.K style. The original notes were written by Moroz in 2007 and this updated version adds some material and exercises by Julia Wolf. The prerequisites at Bristol is the Analysis I course, which is represented here by the lecture notes by Vitali Liskevich. In other words, the prerequisites are an axiomatic development of the real numbers, sequences and basic convergence theorems, limits and continuity of functions, differentiability and infinite series. Contents: Relations and functions,Finite,infinite,countable and uncountable sets. Cardinality,Power set,Hierarchy of cardinalities,subsequences,accumulation Points and The Bolzano Weierstrass Theorem,Limit Superior and Limit Inferior,Cauchy sequences,uniformly continuous functions,pointwise and uniform convergence,the Riemann Integral,criterion of integrability,classes of integrable functions,inequalities and the Mean Value Property of integral,further properties of integral, Fundamental Theorem of Calculus. The notes are concise, well written and organized with an unusually large number of well thought out examples for this kind of course, mainly on specific subsets of the real line or Euclidean space. There's also an unusual focus on more set theory and topology of the real line then is usual in a course like this, including a relatively deep study of cardinality and the Continuum Hypothesis as well as counterexamples to the Bolzano-Weierstrass theorem. They can certainly be used for an honors calculus course or basic real analysis course.