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Jun 15
  1. Undergraduate Real Analysis (Theory of Calculus on Abstract Spaces)

  2. Any theorem in Analysis can be fitted onto an arbitrarily small piece of
    paper if you are sufficiently obscure.
  3. — Cambridge professor on analysis, 1987.
  4. Mathematical Analysis I Donald Estep Colorado State University Course Materials     (PG) Estep first made a name for himself as a teacher by writing the highly unusual first course in analysis, Practical Analysis In One Variable.  I've written on the book elsewhere, so I won't waste space rehashing it. I will say here that it's one of the most unappreciated textbooks on the subject and it should be used far more widely then it actually is. This equally beautiful set of lecture notes assumes the material in that book and pitches a sequel course on metric spaces a la Rudin with Estep's own unique touches. In many ways, these notes seems intentionally designed as a direct sequel to the published book. Basic set topology of metric spaces, sequences and convergence, convergence of sequences and series, invertibility of linear maps, continuity and differentiability of functions of several variables, solution of root problems, and nonlinear approximation theory in metric spaces. The development concentrates on functions of several variables. In many ways, this is a more advanced, abstract version of the published book. There are many pictures and examples, along with Estep's amazingly clear handwriting (again, when you're not typing the notes, this makes a big difference), Again, as in the text, the overarching idea here is that the physical applications of calculus and the precise,rigorous structures of analysis are joined at the hip and should not be separated-the main tool which united the theory and the applications is the theory of approximation. (I strongly encourage all students of analysis to read his opening statement, in which Estep vocalizes this principle beautifully. ) This is, perhaps unavoidably, a considerable overlap between these notes and the earlier published text-mainly in the basic definitions and examples. However, it would be a mistake to think this is just a retread of that text since the transition from functions on the real line to metric and normed spaces is taken seriously and developed in depth. Indeed, the topology and analysis of these spaces is given in great detail with many examples, particularly emphasizing the role these spaces play in describing function spaces and uniform convergence of sequences of functions. There are many unusual applications given, including the Law of Large Numbers,  the general Picard uniqueness and existence theorem for ordinary differential equations, the use of ε-nets in the description of sequential compactness on totally bounded spaces and many more. I absolutely love these notes and if you're either a serious mathematics or physics student or professional, you will too. Trust me. The highest possible recommendation.
  5. Honors Real And Complex Analysis Math 55b Curtis McMullin Harvard University 2012 (PG-13) One of a legion of lecture notes by McMullin that has made him a legendary teacher at Harvard-no easy feat given the research load I'm sure he's forced to maintain. These are his own version of the analysis semester of the aforementioned and much feared year long Math 55 for honors freshman and sophomores there.(The first semester is an honors introduction to abstract algebra I review here. ) This is an introduction to the rudiments of real and complex analysis for some of the strongest mathematics undergraduates in the world-and that's how he pitches it. The resulting notes are concise and incredibly deep, covering basic analysis on the real line, metric spaces and the complex plane as well as many other side topics.There are several very original touches the author puts to the notes that allow him to be both challenging and deep and yet human readable at the same time. The first is that not all this material is expected from jump to be covered in class-the students are expected to read much of the notes on their own. Not only that, they're expected to do independent reading in the recommended textbooks and monographs to flesh out a number of these topics themselves outside of class.This is something that most mathematics professors are terrified to even try and do today for fear  the entire class would stampede to the registrar to jump ship on them. Then again, most professors don't have the luxury of the kind of passionate, talented students McMullin is privileged to teach. My point is this pretty much means he has to make the notes readable, otherwise the entire project is doomed. As a result, despite their difficulty level, the notes are shockingly readable!  There are also many sidebar topics that touch on areas of current research, such as fractals and non-standard analysis. My one complaint is the lack of examples-I think the author could have included many more without undermining the formidability ambition of the notes. Still, the author's created a superior set of notes for either very strong undergraduates or weakly prepared graduate students needing to strengthen their analysis skills.These notes will be of great assistance to either group of student or their instructors. Highly recommended.
  6. Undergraduate Real Analysis Anant R. Shastri Indian Institute of Technology Bombay 2010 (PG-13) An intensive, comprehensive set of lecture notes for a year long course on undergraduate real analysis of both one and several variables.  Contents: Review of basic concepts of real numbers: Archimedean property, Completeness. Metric spaces, compactness, connectedness, Continuity and uniform continuity.  Monotonic functions, Functions of bounded variation; Absolutely continuous functions. Derivatives of functions and Taylor's theorem.Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions.  Sequences and series of functions, uniform convergence and its relation to continuity,differentiation and integration. Fourier series, pointwise convergence,Fejer's theorem, Weierstrass approximation theorem. Extremely careful and detailed, with many examples. Shastri makes the notes thorough and sophisticated without making them too abstract for beginners. For example, metric spaces are introduced, but it's largely used as a tool to study distance and subsets of Rn .There are quite a few deep but not too difficult exercises scattered throughout the text. There are also some important topics we don't usually see in general real analysis courses, such as the Hemachandra numbers, the Cauchy product and the Gamma function. Between his wonderful lecture notes and his outstanding textbooks on differential and algebraic topology, Shastri is beginning to make quite a name for himself as a teacher outside of IIT. And the reputation is richly deserved. Very highly recommended for serious students of analysis and their teachers.
  7. INTERMEDIATE MATHEMATICAL ANALYSIS II Stefaan Delcroix University of Southern California Fresno Spring 2013  (PG-13)  Unusually organized second course in mathematical analysis that presumes a standard first course in rigorous analysis on the real line. Pointwise and uniform convergence of sequences and series of functions, convergence of sequences in higher dimensions, continuity and differentiability of functions of several variables, inverse and implicit function theorems, curves, line integrals. The notes are well written and detailed, containing a nice and careful discussion of the differential calculus of several variables and the topology of Rn . But they're somewhat scattershot, with missing proofs. They also completely lack examples. A number of good exercises can be found at the homepage at this link. Students might find them useful as supplementary reading, but I found them too incomplete to use as text. There are far better sources for this material recommended elsewhere on this site. .
  8. Multivariable  Calculus Jerry Shurman Reed College (PG-13)  I commented on the wonderfully original single variable calculus by Shurman in the calculus section. Now the sequel, on rigorous multivariable calculus-turns out to be an equally excellent text.The sequel in its own way is equally original-but the presentation is a bit more standard and structured more like the usual texts. What distinguishes them is the level of presentation and the choice of material.  The prerequisites for these notes are a standard linear algebra course and at minimum a rigorous presentation of calculus-like Shurman's own calculus notes. Even better would be the first semester of a typical advanced calculus course. Mayer's analysis course notes have clearly influenced Shurman's writing here, so that would really be perfect preliminary study. It’s rather interesting that the author mentions John and Jan Hubbard’s outstanding Vector Calculus, Linear Algebra And Differential Forms: A Unified Approach , which is probably the current favorite text for a course at this level. The goal of Reed’s notes is basically the same-that is,a rigorous yet conceptually simple presentation of functions of several variables.His approach is unique in several respects. First of all, as in Mayer's analysis notes, Shurman emphasizes the "sequential" development of analytic and topological concepts more then most other courses at this level. The sequential approach in Rn has certain advantages, such as resulting in a very concise and lucid statement and proof that compactness is topological property i.e. the image of compact subsets of Rn under  continuous functions are compact. (I prefer working with point sets then sequences, but that's me.) Also, although linear algebra is assumed known, Shurman takes special care  to review carefully those aspects which are important in multivariable calculus, such as the matrix form of a linear transformation with respect to a chosen basis and how to express linear transformations of vectors via matrix multiplication of column vectors. He also greatly emphasizes the role of the determinant in analyzing linear mappings and inner products.He also introduces derivatives as an approximating linear transformation (matrix) to nonlinear functions in Rn using the "big Oh"and "little Oh" notation. Outside of numerical analysis courses, students don't really get to see this important notation much anymore, so it's use here is very welcome. I wish more basic courses would use it. Shurman also presents geometry,topology and algebra as essentially interchangable  in Rn -an aspect of n-dimensional Euclidean space which is sometimes obscured in courses at this level.  But without question the best part of Shurman's notes are his wonderful step by step discussion of differential forms, which takes place in the last chapter and draws on virtually the entire machinery developed in the previous chapters. His definition of a differential form-very roughly-is that it is a multilinear function that has the structure of the determinant of the total derivative of the function under the integrand of an integral over a compact subset of Rn . The author very leisurely and with many visual examples, takes his time explaining this very subtle but critical concept to the student-in a simpler and cleaner way then most sources I've seen. Comparing Shurman's with some of the other treatments the commenter has given here, Guillemin's development is equally clear, but it's far more formal and therefore conceptually quite a bit more difficult. Gunning's development is very rapid and also formal, so in some ways, it's even more challenging then Guillemin's. Shurman's is probably the most accessible yet rigorous treatment of differential forms I've seen and well worth students becoming familiar with. It really sums up very nicely the entire work. The book has a ton of great examples and exercises, which also come equipped with many computer generated images.I think this one of the very best sources for a careful development of functions of several variables that currently exists and it may be the most accessible. Here's hoping Shurman continues to make this wonderful textbook freely available for many years to come. Very highly recommended.
  9. Intermediate Real Analysis I Functions of One Variable On Metric Spaces Steve Kaliszewski Arizona State University Spring 2012 (PG-13)
  10. Intermediate  Real Analysis II Functions of Several Variables And Lebesgue Integration on Euclidean Spaces  Steve Kaliszewski Arizona State University Spring 2012       (PG-13/R) These are the notes and materials for a year long undergraduate real analysis course that presumes exposure to some proof-based mathematics as well as a good course in linear algebra.A rigorous treatment of calculus at the level of advanced calculus/ elementary real analysis is recommended,but not necessary. They focus on the presentation of real analysis on metric spaces of one and several variables at about the same level as Rudin. Contents Course I:  Irrationality and axioms of the real number system, Completeness, Consequences,Cantor Theorem,Rearrangements,Limits of sequences and series,Limit Theorems,Monotone convergence,Subsequences,Cauchy Criterion,Series,Double Summation,the Cantor Set, Metric Spaces,Open and Closed Sets,Compactness,Connectedness,Dirichlet Function, Limits and  Continuous Functions in metric spaces,Continuity and Compactness,Intermediate Value Theorem,Discontinuities,Derivatives,Mean Value Theorem,Continuity and Differentiability,Uniform Convergence and Differentiation,Series of Functions,Power Series,Taylor Series,Integration,Riemann Integral,Properties,Fundamental Theorem of Calculus,Lebesgue's Criterion,Abstract Metric Spaces,Construction of the Real Numbers Course II: normed and linear spaces, n-dimensional Euclidean space, differentiation,differential and partial derivatives, the Jacobian, Lebesgue integration,the Inverse and Implicit Function Theorem,convergence theorems for Lebesgue integrals, and the Change of Variables Theorem. These courses are quite impressive in both scope and style. They're very lucid and careful with many examples,mostly on the real line and Rn. There's also a ton of very good exercises,challenging but not too difficult. There are also several deep aspects of analysis that are tackled head on here that usually aren't covered except in the most demanding of courses. For example, in the second semester, multiple Riemann integration is bypassed entirely for a complete developent of Lebsegue measure and integration on the real line, following the excellent concrete presentation in Frank Jones' book. I'm not comfortable generally with jumping into integration theory in an undergraduate course. But it's hard to argue that this simplifies many aspects of integration of functions of several variables as long as the treatment of measure theory is limited to Rn , which Kaliszewski does. He also gives an excellent treatment of the operator norm.Most basic analysis texts steer clear of this concept
  11. and leave it to later courses in functional analysis. I've never understood why since all you really need to understand it in addition to basic one variable analysis  is a good command of linear algebra. In any event, it's done and done well here. An excellent text for a strong analysis course, although it might be a bit too challenging  for self study students trying to learn analysis for the first time. But if they've had a strong calculus course or an introductory analysis course at the level of Kaye or Mattuck, then this might be just what the doctor ordered. Very highly recommended.
  12. Real Analysis Eric T. Sawyer McMaster University  (PG-13) A deep,sophisticated and historically informed real analysis course for undergraduates with strong backgrounds in calculus and some experiences with rigorous proof. Contents:   The fields of analysis  A model of a vibrating string Defeciencies of the rational numbers The real field  The complex field Dedekind construction of the real numbers Cardinality of sets Metric spaces Topology of metric spaces Compact sets Fractal sets Sequences and Series Sequences in a metric  space Numerical sequences and series Power series Continuity and Differentiability Continuous functions Differentiable functions Integration Riemann and Riemann-Stieltjes integration Simple properties of the Riemann-Stieltjes integral Fundamental Theorem of Calculus Function spaces Sequences and series of functions  The metric space CR (X) Lebesgue measure theory Lebesgue measure on the real line Measurable functions and integration Appendix Bibliography A number of things are impressive about these notes: Firstly, they are couched from the beginning in the history of the subject. Sawyer not only includes a host of fascinating historical notes that elucidate a number of facts that are not common knowledge, especially in the early chapters, the notes themselves are structured historically via the 19th century rigorization of analysis.  The notes begin with the physical model of the vibrating string, whose partial differential equation ultimately motivated Fourier's investigation into the foundations of convergent infinite series and therefore calculus itself. The history of the problem itself is discussed beautifully. He then uses this problem to examine the order and algebra properties of rational numbers and demonstrate that the solution space of this and many other differential equations cannot be contained in only. He then proceeds to give the Dedekind construction of the reals and the field properties of the complex field. He then proceeds in this fashion to develop the machinery of analysis on metric spaces, incredibly clearly and with many examples. In many ways, his notes are a much more detailed but equally deep and careful version of Rudin's Principles of Mathematical Analysis. I particularly recommend it as a free supplement to Rudin in courses using that hallowed but absurdly terse text. Secondly, like Rudin and several other sources at this website, he develops analysis on both the real line and the complex plane. It's important to begin to get students comfortable with working with complex numbers in a course at this level, particularly for the study of sequences and series.Thirdly, he develops a lot of modern material not in Rudin, such as fractal spaces, the van Koch snowflake and the Henstock-Kurtzwell integral before developing the Lesbegue integral on the real line. Lastly, he has many wonderful, challenging exercises,the hardest of which come with hints. Overall, this is an outstanding set of lecture notes for an intermediate real analysis course-very readable,informative and it will be a huge asset in self study or in such courses.
  13. Beginning Real Analysis S. S. de Silva Lycoming University 2013    (PG-13) Another intermediate set of lecture notes with analysis on metric spaces-this one emphasizes the concrete aspects of order and distance on the real line as a metric space.  Contents: Set theory and foundations  The Complete Linearly Ordered Field R , Metric Spaces & the Topology of R, Sequences, Series & Convergence, Continuity, Derivatives, The Riemann Integral.The notes presume some familiarity with careful proof and the elements of set theory. They are well written, concise and well organized, with many examples and exercises. They're simpler and more visual then most courses at this level. The emphasis on the order and topological properties of the real line are somewhat greater then in other notes at this level. For example, inequalities are explicitly developed as consequences of the linear ordering of the real line. Denseness, compactness and subspace topologies on subsets of the real line are developed in much greater detail then one usually sees outside of a point set topology course, which I think students will find very helpful in their later studies. The author also makes quite a few personal insights into the material that are both useful and unique. For example.de Silva makes a very interesting assessment of the axiomatic construction of the real numbers from first principles and why he declines to do it:  It is possible to construct the set ? as a derivative of the power-set of the empty set ?. This is a fascinating task, which unfortunately proceeds very slowly.  (We disagree with this assessment, by the way-as we've listed a number of free sources where the task is done relatively rapidly.It merely depends on how detailed one wants to make the construction and what steps to leave to the students. If one does develop all the details beginning from the Zermelo-Frankel set theory axioms-then yes, he's absolutely right.)  A very solid, original set of lecture notes for such a course. Highly recommended.
  14. Real  Analysis Brian Forrest Fall 2011 University of Waterloo Notes M. L Baker December 2011  (PG-13)  A relatively brief and concise, but very lucid and nicely typeset set of lecture notes from ML Baker, a mathematics student at Waterloo who's been posting typed versions of the lecture notes of the courses he's either sitting in on or taking himself. Contents: Set theory  Basic concepts  Products Axiom of Choice Orders and Zorn’s lemma Equivalence relations and cardinality Cardinal arithmetic  Sum of cardinals  Product of cardinals  Exponentiation of cardinals Trichotomy Metric spaces Basic concepts Topology Continuity Linear mappings Induced metrics Uniform and pointwise convergence Completeness of metric spaces Completeness Isometries and completions Baire Category Theorem Banach Contractive Mapping Theorem Compactness Compactness and continuity Approximating functions in C(X) . The notes are entirely concerned with the abstract topological and set theoretic aspects of introductory real analysis on metric spaces,particularly function spaces and their topologies. It also to some degree lays the foundations for a follow up course in measure and integration by developing the machinery of the Baire Category Theorem and it's equivalent formulations. The proofs are very clear and detailed and there's a shockingly large number of excellent examples. Forrest (I'm assuming that's who deserves credit for the actual content of the notes, if not, I apologize to Mr. Baker and his active rewriting should have been noted in the manuscript) emphasizes the significant portion of the  framework that abstract metric or topological spaces provide for the theory of calculus-for example, the importance of real valued differentiable functions being defined on compact subspaces of Rn for the Extreme Value Theorem to be valid. He also makes many observations about results in abstract metric spaces that are direct generalizations of important calculus results, such as the Arzelà-Ascoli Theorem being the generalization to abstract function spaces of the Heine-Borel theorem on compact subsets of R.The one major drawback of these notes- and it's a huge one, unfortunately-is that there are no exercises. Still, there's a lot of wonderful material here, beautifully presented. Highly recommended as an advanced  supplement to a standard analysis or honors calculus text.        
  15.  Undergraduate Real Analysis Supplementary Lecture Notes Vaughn Jones Math 104 University of Berkeley Spring 2011  (PG-13) Yes, that Vaughn Jones. And yes, it's every bit as broad, difficult and challenging a set of notes in real analysis as you'd expect from him. The notes were originally compiled by a student in Vaughn's class-he later edited and posted them at his website. But there's more then meets the eye here. First of all, these notes only form half the reading material for the course-probably the most difficult part. The actual text for the course, as stated at the syllabus page here. , was Kenneth Ross' excellent Elementary Analysis: The Theory of Calculus  This choice of text baffles me somewhat, since his goal, as can be judged from the notes, is to achieve a much higher level of real analysis then that book does. So the notes really cover the "higher" part of the course, the part Ross' excellent text either doesn't cover or covers insufficiently. The contents seem to bear this out: Building Up to the Reals Part one
  16. Building Up to the Reals, Part Two Completeness and Sequences Convergence, Monotone Sequences Monotone, limsup, Cauchy , Cauchy Sequences, Subsequences, Defining the Reals, Part One Defining the Reals, Part Two Metric Spaces, Part One Metric Spaces, Part Two Continuity and Cardinality Compactness Set Theory (Guest Lecture by Hugh Woodin) Norms,Separability, Connectedness More (Path) Connectedness, Series and Convergence More Convergence, Power Series, Differentiability Theorems About the Derivative Differentiation of Power Series Uniform Convergence (Guest Lectures by Scott Morrison) Weierstrass Approximation (Guest Lecture by Scott Morrison) The Stone-Weierstrass Theorem Integration The Fundamental Theorem of Calculus. The notes are extremely deep, covering abstract normed and metric spaces at a level that would be considered graduate at a lesser university.They're beautifully written as well, with enormous lucidity. What baffles me is if you're going to pitch a course in analysis at this level, then why pick Ross as the textbook? An outstanding book, no question, but  it's a little like bringing in a 2014 college All American pitcher to pitch to the 1975 Cincinnati Reds. If you're going to go that high a level, then why not use Rudin or Charles Chapman Pugh's Real Mathematical Analysis instead?  Reading the notes, I noticed that Vaughn was very deliberate in using ε-δ  examples and arguments simultaneously with the abstract machinery.  A good example is how he uses such an argument to demonstrate that the interior of the Cantor set is empty.Looking at these wonderful notes,.I've formed  a hypothesis. I think Jones' intent here was to attempt to circumvent having to teach a baby real analysis course before moving on to a metric space based course by teaching them both simultaneously in an interconnected way. He had students that were strong on ability but weak in background-and he knew they'd be killed by a fully abstract approach. Furthermore, even if they got through it, they would still have a mediocre understanding of analysis on the real line. The result was a kind of  hybrid course that was midway in both level and content between Ross' very detailed and concrete approach that emphasizes calculus and Rudin's much more abstract and terse approach. I'm not sure if he entirely succeeded as I'd be nervous even with strong students to try and cram this much material into a semester long course. That being said-the union of these notes and Ross would form an absolutely incredible first course in real analysis for bright students,a course they'd find both inspiring and thrilling. I think using Mattuck combined with these notes would form an even better course. In any event, these are a remarkable set of notes carefully produced by a master analyst and I'd heartily recommend them for all students and teachers of real analysis. The strongest possible recommendation!
  17. Further Analysis W.T. Gowers University of Cambridge Lent 1997 (PG-13) Yes, that W.T. Gowers. It's good to know the generally accepted axiom about giants of the field like Jones and Gowers being above such menial tasks as teaching isn't always true even at top research universities. (Then again, this was 17 years ago, so the truth of that assessment needs some empirical testing...........) In any event, this was an old version of Cambridge's second year analysis course, which was taken concurrently with the third term of the first year analysis sequence. The course covers basic point set topology and the elements of complex analysis assuming this background. In many ways, the course really is old fashioned in that it combines point set topology and complex analysis via the homology version of Cauchy's theorem. This makes pedagogical sense, of course, but the 2 subjects really are only connected (no pun intended) there.So it seems like overkill to do this,which is one of the reasons the mathematics curricula at Cambridge has since been reorganized.    Contents Topological Spaces Compactness Connectedness Preliminaries to Complex Analysis Cauchy's Theorem and it's Consequences Power Series Winding Numbers Cauchy's Theorem(Homology Version). The notes mirror the intertwined historical development of both topology and complex analysis: one the most important problems that lead to the development of homotopy and homology theories was the attempt to give a rigorous formulation of smooth boundary curves in complex line integration. Gowers structures these notes around this development,although not explicitly. Rather concise and a bit dry in prose, as is to be expected from Cambridge notes.But they're also quite lucid with many examples.They're also more sophisticated then most elementary treatments of functions of complex variable due to the strong topological background.  A good second course in analysis with minimal prerequisites for serious mathematics undergraduates and written by a master.  It will be very helpful to students who want to learn either basic point set topology,complex analysis or both quickly so they can move on to more advanced material. Highly recommended.
  18. Real Analysis Danela Oana Ivanovici JA Dieudonné Laboratory 2010   (PG-13?) A handwritten set of notes for a basic analysis course based on Robert Strichartz's book, which I commented on earlier. The title is somewhat deceiving, as it basically covers only differential calculus on Banach spaces and the prequisite point set topology for it. Nicely  written,but very terse and there are almost no explicit examples. Here, the term "example" has the old European definition, which means a concrete exercise. Not bad, but there's nothing here that can't be gotten elsewhere and done a lot better-such as in Hoffman's Dover classic or Bruce Driver's lecture notes. You can check it out, but I'd pass.
  19. Notes On Real Analysis Lee Larsion University of Louisville 2013 version (PG-13)  A substantial and well written set of notes for the undergraduate real analysis course at Louisville the author has been developing for several years. Contents: Basic set theory, notation, Schröder-Bernstein Theorem. Countability, uncountability and cardinal numbers.Axioms of a complete ordered field and some consequences. The most basic properties of R. Sequences,convergence, limits of sequences,Cauchy sequences,Infinite series,An application of sequences. Standard and some more advanced convergence tests.  Various forms of completeness and compactness. Connectedness and relative topologies. Baire category.  Limits of functions, unilateral limits, continuity, uniform continuity. Differentiation of functions, Darboux property, Mean Value Theorem, Taylor’s Theorem, l’Hôspital’s rule.  Development of the Riemann-Darboux integral, Fundamental Theorem of Calculus. Pointwise convergence. Uniform convergence and its relation to continuity, integration and differentiation. Weierstrass approximation theorem. Power series. Fourier series,Dirichlet and Fejér kernels. Césaro convergence and pointwise convergence. There's nothing new or strikingly original here in the choice of subject matter, but Larson does an excellent job of delivering an advanced calculus course at the intermediate level. The course is detailed, extremely readable and informative. Along the way, he points out a number of subtle issues whose importance is sometimes lost on beginners in a standard treatment. For example, early on, Larson demonstrates that the absolute value function's familiar property as a distance function is only valid in ordered fields. Another nice original touch is the coverage of infinite series before the limits of functions,which makes perfect sense but isn't usually done in many courses. Also, Larson covers several unusual convergence tests, such as the Riemann rearrangement and the Kummer test. There are many careful examples and lots of attending graphs and pictures, several of which give substantial intuitive insight into the material. For example, a computer generated graph of a "sawtooth" function i.e. a continuous everywhere but nowhere differentiable function-makes an appearance in the chapter on sequences of functions. All told, this is a very strong, detailed and very readable introduction to real analysis that will be of immense use either as a textbook or study source for students and teachers.Very highly recommended!
  20. Advanced Analysis on Euclidean and Metric Spaces Min Yan Hong Kong University of Science And Technology 2013   (PG-13)   These are the enormous follow-up analysis course to Yan's honors calculus course notes. The notes-all 615 pages of them!!!-cover all of single and multivariable undergraduate advanced calculus as well as some graduate level material. They seem to cover all the analysis Yan has taught in his advanced course over the years. Contents: Limit of Sequence, Limit of Function, Differentiation, First Order Differentiation, High Order Differentiation, Integration,Riemann integration, Riemann-Stieltjes Integration, Series, Series of Numbers, Series of Functions,Multivariable Function Limit and Continuity Topology in Euclidean Space Multilinear Map  Orientation Multivariable Differentiation Inverse Differentiation Implicit Differentiation Submanifolds High Order Differentiation Maximum and Minimum Measure Lebesgue Measure Length in R  Lebesgue Measure in R Outer Measure Measure Space Lebesgue Integration Product Measure Abstract Differentiation  Multivariable Integration  Stokes' Theorem Green's Theorem  Calculus on Manifolds Homotopy Homology and Cohomology Singular Homology deRham Cohomology Poincare Duality The sheer scope of the notes is amazing. It basically begins at the elementary analysis level and brings the student through metric spaces through the rigorous theory of functions of several variables on both Euclidean and abstract n-dimensional manifolds, continuing with a complete course of measure and Lebesgue integration, emphasizing Eucidean spaces and normed spaces with abstract differentiation, then using this machinery to give a completely general presentation of integration of several variables up to the general Stokes' theorem and the beginnings of differential and algebraic topology through de Rham's theorem.  Not surprisingly, there's quite a bit of overlap in the first 4 chapters with his honors calculus course, although these notes are pitched at a higher level and in considerably more detail. Like the calculus notes, these notes are concise yet deep and very smoothly written.  There are many graphs and an adequate number of detailed examples, although  not as many as I'd prefer. The development of measure and integration is quite classical and clear, one of the best and most detailed I've seen. There are many excellent problems woven into the discussion and each section comes with more challenging additional problems which focus on applications and examples  not usually covered in a first course. He also makes several subtle conceptual points, such as the difference between the general concept of derivative and the linear approximation to the function at a given point in the domain: They are very closely related
  21. but they are not the same. I do have 2 concerns with using these notes as a text. Firstly, while requiring measure theory before developing multivariable integration does avoid some of the technical problems one sees in the multiple Riemann integral and the resulting presentation of integration of forms on n-dimensional manifolds is quite (forgive the pun) smooth. But of course, the trade off is that the resulting presentation is much more abstract and challenging to the students. I'm not certain it would work in any but honors courses in average Western universities.Also, the author's non-native use of the English language shows up sometimes in these notes-for example, he names the chapter laying the algebraic foundations for differential forms "multivariable algebra" when it's obvious what he means is "multilinear algebra". But these are really minor quibbles. This is one of the most complete and readable online texts I've ever seen. It really is a striking accomplishment and will be a real asset to both serious students and teachers of analysis from the elementary analysis level all the way through the first year graduate course.  Very highly recommended.
  22. Honours Analysis 2, MATH 255, Analysis in Metric Spaces S.W. Drury McGill University 2014    (PG-13)   Another version of the second semester of the honors analysis sequence at McGill-we reviewed another version by John Labute here. The version by Drury is more advanced and broader in coverage then the version by Labute: they focus on the topology and geometry of Euclidean spaces and analysis of functions of several variables where the other version focused much more on the theory of the Riemann-Stiejies integral and infinite series on the real line. Contents: Complex numbers, lim sup and lim inf, Analysis in Metric Spaces, norms and inner products,Numerical series, Riemann integration, Series of functions, Power series and Elementary functions. The goal here is to lay the foundations for analysis of several variables completely carefully but classically, without manifolds, differential forms or measure theory. Drury develops only what machinery is necessary-for example, the full abstract structure of metric spaces  isn't developed, just what's needed to build the framework of functions of several variables. ( The actual differential and integral calculus of functions of several variables is done on abstract spaces in the next course. ) As a result, the notes are extremely focused and don't digress at all. Very well written and comprehensive, with many examples and  pictures. They are also quite challenging and shouldn't be for any but the very best students. Theorems are proven carefully and in depth and the examples are sophisticated. Several topics which are more advanced then usual, but still very relevant to the material, are introduced. Some examples are the Holder and Minkowski inequalities and the elements of finite dimensional Hilbert spaces.The last topic begins to lay the groundwork for the follow up course.  Drury writes beautifully and extremely informatively-the notes are very readable for a course at this level. It's clear he has enormous passion for teaching the honors courses at McGill and for students who love mathematics, that passion will be infectious from studying these notes. Most highly recommended, as are all of Drury's notes for the sequence.
  23. Honours Analysis 3, MATH 354, Metric spaces. Calculus in several variables and to some extent on Banach spaces by S.W. Drury McGill University 2007   (PG-13/R) The follow up course to the "Honors Analysis 2" course we reviewed here.  This course develops multivariable real analysis on abstract metric and normed spaces, following up the foundations of analysis on Rn in the prerequisite course. Contents: Abstract Metric and Normed Spaces  Calculus in several variables on Abstract Spaces and to some extent on Banach spaces. These notes focus mostly on the point set topology and linear algebra of normed and inner product spaces that's needed to begin to study functions in abstract spaces, such as Hilbert and Banach spaces. The differential calculus  is developed using the Frechet derivative and partial derivatives on abstract inner product spaces.  Riemann integration on abstract spaces using Jordan content rather then Lebesgue measure is developed. The implicit and inverse function theorems on Banach spaces is developed. Again, Drury sets very clear goals for himself from the outset and that allows him to
  24. present the material beautifully. The presentation is very similar in approach, although somewhat more abstract, to the one by Kenneth Hoffman in his wonderful book.  I would heartily recommend Hoffman as collaterial reading to these notes to clarify certain things. Drury's course, while very lucid, is again quite challenging. He also doesn't explain the very important differences between abstract normed spaces and Euclidean spaces, which is so important in functional analysis (Hoffman dedicates considerable discussion to this point.) But there are many examples and insights and it's a joy to read. Very highly recommended for honors undergraduate and graduate students of analysis as well as their teachers.
  25. Honors Advanced Calculus I Xinwei Yu The University of Alberta Fall 2013(PG-13)
  26. Honors Advanced Calculus II Xinwei Yu The University of Alberta Spring 2014 (PG-13)   This is yet another version of the honors advanced calculus course at Alberta, taught by Yu, who's "regular" analysis course we commented on here.They cover advanced topics including multivariable analysis, infinite series, power and Fourier series,   cardinal and ordinal arithmetic, rigorous vector calculus and  applications of calculus in modern science and engineering.  The prerequisites are a year long honors calculus or elementary real analysis course that presents a rigorous treatment of single variable calculus on the real line.(Indeed, Yu's other course contains all the background needed!)  Detailed contents for the first semester can be found here and contents for the second semester here. The notes are very similar in style to the earlier analysis notes-concise, but contain many examples and again, a very large number of exercises Again,they are rather dry and matter of fact,almost in a bullet point style.And again, they are very lucid with many good examples. Yu also sometimes develops topics in a non-standard way that gives some additional insight. For example, the limit of a function of several variables is defined in terms of the cluster point of a nieghborhood in the domain of f : Rm?Rn . This is a bit more sophisticated then the usual definition, but it has the advantage that it makes the conditions for the existence of the limit stingier. A side benefit is that it gets the more advanced student more practice with topological concepts. Indeed, a large portion of the course focuses on the role of point-set topology in analysis, particularly Rn . I approve very strongly of the old school way measure and integration is developed via Jordan measure on Rn first and then multiple integration.By carefully choosing functions and domains in Rn , it's possible to do all possible analysis at the advanced calculus level. The subtleties of Lesbesgue measure can and should be postponed for later courses. Yu seems to agree with me on this. although the second semester ends with a brief treatment of the modern subject, after excellent and detailed treatments of infinite series and the main theorems of vector calculus.  He makes the smart decision to limit the domain of vector analysis to R3, which allows him to avoid completely differential forms and manifolds. Best of all, he includes many applications to classical mechanics, particularly in the second semester notes, such as the use of Green's and Stokes' theorem in kinematics problems based on the Euler-Lagrange equations. Another superior set of notes on advanced calculus which will really benefit strong undergraduates with the proper background. Highly recommended.
  27. Analysis 1 Parts I-III  David R. Wilkins Trinity College Michaelmas Term 2003-2004(PG)
  28. Analysis 2: Metric and Topological Spaces First Semester David R. Wilkins Trinity College 2007 (PG-13)   These are the first and second  year analysis notes taught and written up by Wilkins several years ago. The first year course focuses on a careful treatment of calculus on the real line while the second year course covers the same material in the  general setting of metric and normed spaces.  The material is pretty standard, but Wilkins gives a very example driven approach, which I always thought is very instructive for beginners. Proofs are very detailed,there are lots of pictures and good exercises. Also, several topics are presented in more detail then usual, such as complete metric spaces, normed vector spaces and Banach spaces. All in all, a very good set of notes, particularly for self study due to the lucidity and detail. When you go to the author's website, make sure you get these years' notes-he has several older versions there which are not as good. Very highly recommended.
  29. Advanced Calculus by Lynn Loomis and Schlomo Sternberg  2nd edition online version (PG-13)  For many years, this was the great dark secret textbook on advanced calculus. It was a book that was, in it's own way, scarier then baby Rudin or Herstein for undergraduates. Notorious for it's level of difficulty,it was also the book first year graduate students in mathematics would make secret deals in the night for sums they had to sell their cars to get a hold of.It was the book you had to study to really master calculus of several variables. The 1990 edition was a rare gem that was selling for 300 dollars in fair condition online at one point. (I know-I had to sell my copy to help pay for groceries before my father passed. Still hurting over that.) When a scanned PDF version became available for download for free at Sternberg's website at Harvard, celebrations were held among undergraduates and graduate students.It was a huge gift to all mathematics students of all levels. It is a course on calculus on Banach spaces with examples and exercises that later stimulated research theses for the students who survived it. This book was written for the Math 55 course at Harvard in the late 1960's, an honors course in advanced calculus that justifiably has struck fear into the hearts of mathematics majors for generations as the most difficult undergraduate mathematics course in the United States. As far as I know, this is the only textbook ever written that really approximates the pace and coverage of the actual course. Reportedly, the course has been somewhat reduced in pace and level since this book was written.Apparently, the original version of the course was too much even for 17 year old full scholarship students that learned calculus in grade school that were arrogant enough to sign up. It was an academic meat grinder that left at most 3-4 students standing at the course's end-many of which went on later to become faculty at the top tier universities in the world. Still, it's unimaginable that they actually taught-and to a lesser extent, still do-undergraduates this material at this level and superliminal pace. (I'm told by friends talented or foolish enough to attempt the course that it's really the pace of the course that's lethal more so then the level.If the course was spaced out more over 3 semesters instead of 2, it wouldn't be an academic suicide mission.)  Then again,these were honor students at Harvard University in the late 1960's-argueably the best undergraduates the world has ever seen. In any event,for mere mortals,this is a wonderful first year graduate text and probably the most complete treatment of classical analysis on topological vector spaces that's ever been written. It even ends with an abstract treatment of classical mechanics. It's well worth the effort-but boy,you better make sure you got a firm grasp of both advanced calculus/ honors calculus/ undergraduate analysis of one variable and linear algebra first. Very highly recommended-if you dareUpdate: It's come to my attention that the book has finally(!) been reissued now(Fall 2014)  in an affordable paperback edition authorized by Professor Sternberg, who apparently finally got fed up with the ton of nagging he got over the years to republish it. Kudos to him, but I hope in addition he continues to make the free version available at  his website. Still- very good news for all of us indeed.
  30. A Problem Text in Advanced Calculus John M. Erdman Portland State University Version October 15, 2009  (PG-13) This is an excellent, challenging and very solid problem course from Erdman for either a Moore-method type course on advanced calculus/ undergraduate real analysis or as a source of exercises for such a course. Unlike his earlier calculus problem course, these are pitched at a much higher level and are clearly intended for hard core mathematics majors for a serious undergraduate analysis course. The course sets analysis on metric spaces, which right off the bat tells you it's not for students with weak calculus backgrounds. The course covers both functions of one and several variables in  Rn . Erdman is clearly an advocate of the active learning approach to mathematics-which means to truly learn math is to do mathematics with as little assistance as possible. He strongly encourages in the introductions both students and teachers to try this. More then that, he takes many steps to define and state things in a manner he thinks would make this task as easy as possible.This means things aren't always defined or stated as they usually are in books at this level. For example, he defines connectedness on the  real line in the usual manner, but he uses the fully proven result that a subset A of R is disconnected if and only if it is the union of two nonempty sets mutually separated in  R. The big advantage of using this result on the real line is that it allows connectedness to be expressed in terms of the usual topology on R instead of the subspace topologies of the subsets, which can make the proof of the fact that intervals are connected on the real life-vital to the proof of the intermediate value theorem-a bit confusing. As I've said elsewhere, while I understand and strongly believe in the importance of students getting their hands dirty working problems, I have mixed feelings about this kind of course. That being said, I'm very excited by Erdman's specific approach in this book-he saturates the presentation with worked examples in-between all the main results being left as exercises or  problems.The important difference is that exercises have full solutions and problems don't. This approach excites me because I've always wondered what kind of "Moore method" type course I could seriously consider using as a lecturer.When I discussed the matter with several colleagues, the approach I'd always suggest is an example driven one.After all, aren't theorems merely statements of the common truth of an enormous number of specific examples and constructions?  Imagine my pleasant surprise finding Erdman making the identical observation in his introduction!!!  The result is a deep, broad and far reaching textbook on advanced calculus that strong students will enjoy working through,lecturers will enjoy constructing their lectures from and graduate students will absolutely adore when preparing for their analysis qualifying exams. Very highly recommended!!!
  31. Advanced Calculus II Math 1530 Piotz Hajlasz University of Pittsburgh Fall 2011 (PG-13) This is another absolutely beautiful set of handwritten notes by Hajlasz. (God, I love this guy's notes. I wish he'd give handwriting classes-I'd have passed more classes if I had his secret to magically clear and legible handwriting!! ) This is a rigorous course on functions of several real variables on metric spaces that goes beyond the usual course on this material as it begins with proofs of function space properties of Banach spaces; such as the Banach contraction mapping theorem and the Arazela-Ascoli Theorem. He then proceeds to give one of the best courses of several variable analysis I've ever seen, covering all the main
    expected topics::differentiability, open mapping theorem, implicit function theorem, Lagrange multipliers, submanifolds, integrals, change of variables formula, integration on surfaces: Green's formula, Stokes theorem, Gauss theorem. There are a ton of exercises and beautiful hand-drawn diagrams of surfaces and curves while never abandoning the general perspective. He also includes wonderful sophisticated examples one usually sees in advanced geometry and topology
    courses, such as Hutchenson's theorem, the van Koch curve and the Serpinski gasket. Oh, to heck with it, just download and read them. Then read them again and do all the exercises.Let their beauty and brilliance speak for themselves. The only tragedy here is that the first half isn't available any longer. Highest possible recommendation.
  32. Undergraduate Analysis II Lecture Notes Victor Guillemin Mathematics MIT OpenCourseWare 2005 (PG-13) Strong and very readable set of notes  by a major mathematician for a rigorous course on calculus of functions of several variables presuming an advanced calculus course on metric spaces. They're similar in spirit, level and coverage to James Munkres' classic textbook Analysis on Manifolds, which was actually one of the textbooks used for the course. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. Very readable with careful proofs, but has a big flaw in that it has almost no solved examples. Since the notes are really intended as a supplement and not the main course text, this is disappointing but not surprising. What's quite interesting here is that Guillemin has been teaching two different versions of this course for several years now. He has since 2005 removed and vastly expanded the material on differential forms into an independent,substantial course text which we comment on here. In 2008, he taught a similar version of this course without differential forms using the idea of densities instead in integration and adding material on ordinary differential equations. Those notes are reviewed here.
  33. Theory of Differential Forms Victor Guillemin MIT Spring 2014 (PG-13)  This is a remarkable online textbook draft-and yes, I think it's large and developed enough now that we can safely call it an online textbook draft-that grew out of  Guillemin's analysis and differential topology courses over the last decade. Nowadays, as I said elsewhere, it's all the rage to try and teach strong undergraduate students multivariable calculus from a totally modern viewpoint of differentiable manifolds and differential forms. As I also said, I have  mixed feelings about this approach. This is Guillemin throwing his hat in this ring to produce a totally modern course in vector analysis. Wisely, he decides to require some mathematical maturity from his students-the minimal prerequisites are a term each of honors calculus/ advanced calculus/ elementary real analysis and modern algebra. Contents: Multilinear algebra,tensors and exterior forms,differential forms on Rn,exterior differentiation, integration of forms on open sets of Rn,change of variables formula revisited,the degree of a differentiable mapping. differential forms on manifolds and De Rham theory, integration of forms on manifolds and Stokes’ theorem.the push-forward operation for forms and much more.Applications to differential topology.  This is one of the most careful and lucid sets of lecture notes currently online. The author takes a wonderful middle ground approach which develops forms from both the algebraic and geometric perspective. Both aspects I believe are equally important not only for fully understanding forms, but also for understanding their older and poorly understood siblings, tensors. Most treatments take purely one approach, which is a mistake the author seems well aware of. There are virtually no examples in the first chapter, but  that's to be expected as it's merely developing algebraic machinery out of  structures the student is assumed to know. Indeed, this chapter can be used as a fine supplement on multilinear algebra in either a graduate or honors undergraduate algebra course. The treatment of forms begins in earnest in the next 3 chapters, which abound with rich examples and wonderful exercises.Sadly, there are not many pictures except in the last chapter on manifolds,but that can be forgiven given how clear the presentation is.There are many deep topics not usually covered in first year graduate differential geometry or algebra courses, which is where one usually learns about forms-such as the pull-back operation,the Poincaré lemma, Thom forms and intersection theory. The real shock and joy here is to find Guillemin thoroughly covering applications to physics of this material in a pure mathematics course, such as a terrific discussions of Maxwell’s equations from the differential form perspective!The multivariable analysis text by Shurman would make an outstanding supplementary source,as it contains a somewhat simpler and very different development of forms that's quite a bit more visual then Gulliemen's. These presentations complement each other very well indeed.  I don't know how much longer the author is going to maintain these notes before publishing them as an actual textbook and charging more then the average freshman spends on food in a week for it. I hope it'll remain so for many years to come. I do know that as long as he continues to make it available for free, it will remain one of the very best introductions to multilinear algebra and differential forms that currently exists in any form. The highest possible recommendation!
  34. Undergraduate Analysis II Lecture Notes Victor Guillemin Mathematics MIT Fall 2010   (PG-13) This is a later version of the earlier supplementary notes for the Analysis II course at MIT we commented on here. The main differences are the removal of the material on differential forms (presumably because Guillemin's comprehensive and highly recommended notes on differential forms now exist), the replacement of forms in the treatment of integration of several variables with densities and the addition of a first chapter on ordinary differential equations in Rn . To be honest, I found the material on ODE's to be the weakest part of the notes and better treatments can be found in many other places. The real breakthrough here is in the treatment of integration. A density is a mapping from a product of vector spaces to the real numbers such that the image of any real valued linear mapping is stretched by a factor of the absolute value of the determinant of its matrix. (This is actually a special case of the tensor density in differential geometry.) It's an  interesting alternative presentation, one that emphasizes the geometric interpretation of the determinant in Euclidean spaces. More importantly, it turns out that on manifolds, a density is locally a Lebesgue measure, which of course is why it can be used in place of differential forms to construct a general theory of integration on manifolds. The author does a wonderful job explaining this-many texts at this level completely botch the rigorous explanation of why forms can be used as integrators. This approach using densities is completely clear and works beautifully. As for these specific notes, again, they aren't really complete enough by themselves to function as a full course text.  Together with the aforementioned notes on differential forms and the 2005 "edition", the union of the 3 notes comprise the bulk of a wonderful new textbook on analysis of several variables for either first year graduate students or strong undergraduates. Here's hoping the author eventually combines and revises the notes into such a text. Highly recommended as a supplement.
  35. Undergraduate Analysis Tools  Bruce K. Driver University of California at San Diego May 1, 2013 (PG-13) One of a host of lecture notes in progress at Driver's homepage-this is the latest iteration of his undergraduate real analysis notes. They're intended to supplement a real analysis course based on Rudin's Principles of Mathematical Analysis , so as you can imagine, they're not a trifling treatment of the subject. Indeed, they are more then comprehensive enough to act as the sole textbook for the course.I suspect the reason Driver does not yet do so is because he's still in the process of revising it-the notes have changed dramatically in organization since he began writing them in the early 2000's. (I know this because Driver has allowed  the various draft versions to remain available at his website.It's a fascinating look at the evolution of a textbook in progress.) Basic properties of the real numbers, complex numbers, metric spaces, sequences and series of real  numbers, functions of a real variable and continuity,differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions, differentiation of functions of several real variables, the implicit and inverse function theorems, the Lebesgue integral and infinite-dimensional normed spaces.While there is considerable overlap in content between Rudin and these notes,the notes are far more detailed as well as quite different in perspective and topic selection. Several important original aspects of the notes: Firstly,Driver defines and uses the extended real number system R* = R ∪{∓∞) rather then the usual reals. This simplifies many proofs, particularly those on unbounded subsets which are so critical in delicate limit operations. A much deeper and more significant difference is that although all three equivalent aspects of Euclidean spaces are defined and discussed, the emphasis is much more on normed linear spaces then metric and topological spaces in Driver's notes.He particularly emphasizes the  Banach space structure of the real line and real Euclidean spaces. This is quite critical in developing a very general version of the theory of the Riemann-Stiejes integral in terms of partitions built out of simple functions on Banach spaces. This allows him to express both the classical and modern theories of integration in the same language and allows a much smoother transition from classical integration theory to modern Lebesgue integration via the Daniell approach. Several major textbooks also take this approach, such as the aforementioned book by Hoffman  and the more difficult Foundations of Modern Analysis by Jean Dieudonne. Unfortunately,this results in a theory which is considerably more sophisticated then the standard Riemann or ever Riemann-Stieljies methodology. I think asking the average undergraduate analysis students to work this hard to learn integration theory is asking a lot. Still, it's presented clearly and very carefully.  Driver also uses the same normed space constructions to give a unified and very abstract presentation of functions of several variables on Banach spaces. This is preceded by a chapter on the elements of finite dimensional functional analysis that allow Driver to discuss abstract function spaces and uniform convergence of sequences of functions in a modern manner. It also sets the stage for optional sections on the Lebesgue integral a la Daniell and a brief preview chapter on infinite dimensional spaces. There are many examples and a surprisingly large number of pictures for a relatively abstract analysis course. Driver has written a challenging, original and highly modern undergraduate course in analysis for serious students of advanced calculus of surprising clarity. I don't know if the average analysis student could handle it, but strong students getting ready for graduate school would benefit enormously by working through them, particularly with a good teacher. Highly recommended for serious students.
  36. DIFFERENTIAL AND INTEGRAL CALCULUS III LECTURE NOTES Mikhal Sodin TEL AVIV UNIVERSITY  FALL 2010 (PG-13) These are Sodin's follow up notes to his first semester "calculus" notes at Tel Aviv, focusing on functions of several variables. Again, they demonstrate the different methodology and philosophy of university calculus outside the United States. They presume a year of linear algebra and at least one semester of a theoretical "ε-δ " course in single variable calculus. In other words, in this country these would be considered much more appropriate for a second or third junior/senior undergraduate level advanced calculus or real variables course. Contents: Functions on Euclidean spaces and review of linear algebra, curves in Rn , continuity, differentiability; partial and directional derivatives, Chain Rule, Inverse and Implicit Function Theorems, The open mapping theorem and Lagrange multipliers, Riemann Integral and Jordan content on Euclidean spaces, null (measure zero) sets, Fubini's Theorem,partitions of unity, change of variables, surfaces and differential forms in Rn, line and surface integrals, Green's theorem, the Gauss theorem, the Poincare Lemma, singular chains, integration on chains, Stokes' Theorem for integrals of differential forms (general version). Fundamental theorem of calculus. The level of the notes is about the same as Shurman's multivariable course, somewhat less advanced then Guillemin's differential forms notes and somewhat more advanced then Jones or Nitecki's notes. Sodin's notes are considerably more concise then those sources and many results are shunted to the exercises.But they are surprisingly readable and many helpful hints to the problems are given. He also gives many examples and pictures in the course of the notes, including many applications to physics and geometry such as extrema analysis, volumes, mass distribution, and a careful treatment of work in Rn. He also discusses some important mathematical topics in depth that are usually brushed over or rushed through in the usual treatments, like Jordan content and improper integrals. These notes are superior to his single variable notes in every way-and make an excellent text for an advanced course in functions of several variables. Highly recommended.
  37. Honours  Analysis III Math 354 Dmitry Jacobson McGill University Notes Taken By: R. Gibson and other course materials Fall 2010 (PG-13) Yes, a different version of the same honors course at McGill then the one taught by Drury that we've already commented on here and here. Contents: Introduction to metric spaces. Completeness, Compactness, Connectedness. Measure theory and Integration. Implicit and inverse function theorems.  These notes are much shorter, more concise and drier then Drury's, although they are quite careful and rigorous. They're also somewhat more abstract, as unlike Drury, Jacobson doesn't fully develop the Riemann multivariable integral and instead jumps directly to the theory of Lebesgue measure and integration and makes it front and center in the course. He also develops point set theory in a slightly different manner and emphasizes the aspects which are most important for modern analysis. For example, he covers general convergence in topological spaces via nets and filters. He also focuses on pre-functional analysis material, such as  Baire's Category Theorem, Banach-Steinhaus Theorem and The Open Mapping Theorem. He also gives many excellent exercises at his website. These notes are more sophisticated them Drury's, but they're very clear and well organized. Graduate students and very strong undergraduates in analysis will find them helpful. But personally, I prefer Drury's version as more readable and with broader coverage. Still, recommended.
  38. REAL VARIABLES with BASIC METRIC SPACE TOPOLOGY Robert B. Ash  University of Illinois at Urbana-Champaign  (PG-13) This is the original online version of Ash's book, which he has graciously left available at his website for download. As I said , Professor Ash’s name will occur several times in this list, so get used to it. This Professor Emeritus at The University of Illinois at Urbana-Champlain has made a remarkable contribution to free available advanced mathematics sources by writing a series of textbooks for his courses that evolved over the decades from his course notes-and then subsequently posting those textbooks as open source books at his website. Ash is an interesting guy; apparently he was originally an electrical engineer before going back to school for his PHD in mathematics. As a result, not only does he favor a concrete (though completely rigorous) rather then an abstract approach to mathematics, but he’s given a lot of thought to teaching as he’s taught generations of mathematics students at Illinois ranging in level  from social science freshmen to PHD candidate mathematics students.  (I heartily recommend reading his comments on teaching higher mathematics, available at his webpage here.  ) This is a textbook designed for an intermediate level undergraduate real variables course a la Rudin or Apostol.  It has all the usual topics-basic set theory, real numbers, real sequences and convergence, limits of functions, the topology of Rn  ,continuity, differentiation, the Riemann-Stieljes integral and sequences of functions and uniform convergence. This course is now so standard and the topics usually covered so critical for later studies, you really can’t mess with the contents too much or it defeats the purpose of the course. (This is why some of the critical attacks on Tom Korner’s wonderful A Companion To Analysis completely baffled me. Complaining a book for an  intermediate real variables course was “less original then it appears”? Really? How creative can you be with this course without defeating the purpose of it?)  As I said, Ash’s book, for the most part, sticks to the playbook. What’s different and very helpful here is Ash’s particular style: The book avoids overly abstract reasoning, has many pictures and emphasizes the topological properties of Euclidean spaces in proofs. And of course he has complete solutions to the exercises, as in all his textbooks. It won’t replace the great classics like Apostol or Rudin, but it will function beautifully as a very detailed and illuminating supplement to those textbooks-particularly Rudin, And of course, since it’s so cheaply available, why wouldn’t you want a copy?
  39. Analysis J.M.E. Hyland University of Cambridge Michælmas 1996       (PG/PG-13)   These old and excellent notes are a prime example of what the first semester analysis course at Cambridge used to be composed of before they completely reorganized and restructured the first year course there. In fact, these are the notes from the prerequisite course to Gower's Further Analysis at Cambridge the same year. So combining these notes in sequence will allow one to reconstruct what that year long course looked like. Contents: Real Numbers, Euclidean Space, Differential,Integration, Metric Spaces, Uniform Convergence and The Contraction Mapping Theorem. Hyland's course is-as expected by now-concise, but very clear. It is also surprisingly broad in  scope, covering both functions of one and several variables on both the real line and metric spaces. The proofs of theorems are beautiful and clean and while there are relatively few examples, but the examples he does have are quite good. The one huge flaw is that sadly, Hyland's exercises sets for the course are no longer available. You can try emailing him and see if he still has them available,but there are a number of Cambridge exercise sets-called Examples-for more recent iterations of the course and you should be able to patch together problem sets from
    that. In any event, an excellent source for an elementary or intermediate level  analysis course Recommended.
  40. Introduction to Real Analysis David Kotick University of Waterloo  (PG-13) Concise and rather abstract lecture notes from an analysis course at the University of Waterloo.Contents: Introduction to the Foundations of Math Set Theory  A Formal Symbolic Language ZFC Axioms of Set Theory Binary Relation Functions Systems of Sets Cardinality Cardinal Arithmetic Structured Spaces Inner Product Space Metric Spaces Topological Space Sequences Limits of Functions Continuous Functions Homeomorphism and Isometry Completeness Baire Category Theorem Compact Sets  It's hard to judge the exact level of the course, the author doesn't say so and there's nothing else to give a clear indication. The reason I'm wondering is because these notes seem to be more about axiomatic set theory and point set topology then analysis proper. I guess they could be used for an advanced undergraduate analysis course for strong students, but the contents are pretty sparse-about a third of the results are either stated without proof or left to the exercises. Also,as I said, a surprisingly small amount of the material is actually what most people would really call real analysis .Indeed, analysis doesn't really begin until page 20-almost halfway through the notes! There also aren't many examples. That being said, the notes do contain a lot of important mathematical material for students to learn-such as the Zermelo-Frankel set theory axioms, proof of the uniqueness of completion of metric spaces (which, of course, is equivalent to the uniqueness of the construction of the real numbers) and isometry on metric spaces. A good supplement to an intermediate analysis course, but I wouldn't use it by itself.
  41. ALGEBRA AND ANALYSIS Part 1: ANALYSIS. THEORY OF METRIC SPACES LECTURE NOTES AND EXERCISES  Jim Howie Heriot-Watt University 2008  (PG-13)  A surprisingly good set of lecture notes on analysis on metric spaces.  Contents: Introduction to Metric Spaces Ways of measuring distance Metrics Examples of metrics Metric spaces of functions Exercises on metric spaces Open Sets and Closed Sets Open Balls Open Sets Closed Sets Bounded Sets Exercises on open sets, closed sets, bounded sets Sequences in Metric Spaces Sequences and Limits Sequences in Rn Sequences of bounded functions Cauchy sequences Sequences and closed sets Exercises on sequences Continuity Maps between metric spaces Continuity and sequences Continuity and open sets Homeomorphisms and equivalent metrics Exercises on continuous functions Compactness and completeness Compact sets in metric spaces Complete metric spaces Completion of a metric space Exercises on compactness and completeness Contraction mappings The Contraction Mapping Theorem Applications Approximate solutions to algebraic equations in R Integral equations Differential equations Exercises on contraction mappings. Similar to  de Silva's notes in style, but gentler with many more concrete,detailed  examples and many exercises at the same level. There are also many pictures of various subsets of the plane to illustrate the many aspects of metric spaces. They're also very readable, more so then notes at this level usually are. An excellent set of notes for an intermediate course-in fact, paired with  Kotick's notes, the 2 complement each other extremely well to form the text for a self study or class at this level. I like these notes a great deal. Highly recommended .
  42. Foundations of Analysis Ilia Binder University of Toronto Winter 2010   (PG) Very brief and concise set of notes meant to supplement an analysis course based on Davidson and Donig's Real Analysis With Applications.  This is an excellent book, one of my favorites,so it's understandable why Binder didn't knock himself out writing more notes. You can check it out, they're well written, but really don't have enough substance to be useful as anything but review.
  43. Basic Analysis I-II  William Allard Duke University 2009 (/PG-13/R) This surprisingly challenging set of notes form the basis for a 2 semester course at Duke University.The notes cover analysis of one and several variables on metric and topological spaces, including vector  analysis on Euclidean manifolds. By the author's own admission, they're much more difficult and advanced then lecture notes at this level usually are. To quote the author from the course homepage:   We don't have higher level honors courses but if we did these courses would be among them. The syllabi for both semesters' courses can be found here It's important to state the author doesn't intend for these notes to used by themselves, but in conjunction with the course texts, which are the excellent texts by Rosenlicht and Edwards Jr.  Saying these notes are ambitious is a little like saying Josef Stalin was slightly amoral. I'd love to know-just out of curiosity-what the attrition rate for students dropping this course before the deadline was.I'll bet it was pretty high-because  for any but the very strongest students, this course is going to be a meat grinder. For openers, there's virtually no examples. Ok, granted, these are supplementary notes,the main course texts both have lots of examples and some examples are given as exercises and theorem proofs. But not many. The problem here is that the material in the notes is very sophisticated and often has no real counterpart in the texts.Therefore it really needs some of it's own examples. This is particularly important when dealing with subtle concepts like relative topologies-which really need a lot of examples to fully grasp. The author complicates this further by choosing unnecessarily abstract methods of presentation in these notes. For example, Allard freely and without explanation uses the language of basic logic. Allard does the first exercise for the students, by proving in complete logical sentences that the composition of relations is associative. Can he assume his students already know enough logic to follow this? Maybe, maybe not. If they don't, I can guarantee you can tell how many students' eyes glazed over by how many are gone by the second lecture. I'd do a brief introduction to that notation and language at the beginning to make sure-and if not, you could write up a short introduction and make it's mastery the first day's homework. (A great exercise in this case would be to have the  students rewrite the proof in everyday language! ) Also, when constructing the real numbers from the naturals-which I was excited to see him do until I actually saw him do it-he gives a crash course in algebraic structures, much like Gunning does in his notes. Again,unnecessarily difficult! Stating the field axioms after developing the individual number systems step by step would have been so much simpler for students to understand! Save algebra for another course!  His developments of normed spaces,their topologies and the generalization of basic differentiation theorems from R to general normed spaces are all particularly good. He smartly develops multivariable integration via algebras of step functions on Rn by essentially developing the inner measure on intervals of R, before giving a full development of abstract Riemann and Lebesgue integration. Still, I think even strong undergraduates are probably going to find this one of the most daunting sections in the notes.  He also does a good job giving a modern development of functions of several variables on R-although again, it's terse and challenging and there are more accessible treatments available right now.Alright, with those complaints aside, Allard has written a very strong set of notes for equally strong students. If they have the right prerequisites-preferably a strong honors single variable calculus course based on Spivak or Mattuck-and Allard worked to put more examples  into the notes, these notes could serve as the basis for a very good honors advanced calculus course or first year graduate course in analysis. Recommended for the right audience.     
     
  44.   Analysis 3 Boris Tsirelson Tel Aviv University 2013   (PG/PG-13) Another outstanding set of notes in English from Israel, these giving a completely rigorous but classical presentation of calculus of several variables in Rn .  Contents: Conventions, notation, terminology etc. Euclidean space Rn.Appendix: If spaces are not a joy to you. Differentiation  Open mappings and constrained optimization. Inverse function theorem. Implicit function theorem.  Appendix: What is the Implicit Function Theorem good for? (A discussion on Mathoverflow). Integration  Riemann integral in Rn   Jordan Content Jordan Measure Iterated integral Change of variables. These notes presume a good background in rigorous one variable calculus, linear algebra and a good grasp of the basics of point set topology. So lightweight notes  for pencil pushers they definitely are not. However, Tsirelson is determined to keep the notes concrete and relatively simple without sacrificing rigor, something many online notes in functions of several variables sadly, fail to do. For example, he is adamant not to present Lebesgue integration.Instead, he develops multivariable Riemann integration theory first on boxes, then other subsets of Rn with Jordan content and finally, on subsets with boundaries of measure zero. This is how I much prefer to do it, as I said earlier. Tsirelson's own notes are very clear and visual while at the same time very careful. That being said, the notes lack detail. More then a third of the notes are exercises-and many of them are rather difficult, although the author does give a number of hints. He seems leave a detailed presentation to the assigned textbook. ( Ironically,the assigned textbook for the course is Jerry Shurman's excellent online text, which we comment on here!) Nevertheless, the notes are extremely well written and in the parts that are detailed. the theory is developed at just the right level for strong/advanced undergraduates.Tsirelson also inserts many wonderful quotations and observations-many in the footnotes at the bottom of each page. For example, in the review section on topology, he observes,  A set, however, is not a door: it can be neither open or closed, and it can be both open and closed. Also, at the beginning of the section on integration theory, he indirectly states his opinion on the best way to present multiple integration theory carefully and why he enshews developing Lebesgue integration with the following quote from T.W. Korner's excellent analysis text: It is frequently claimed that Lebesgue integration is as easy to teach as Riemann integration. This is probably true, but I have yet to be convinced that it is as easy to learn. I completely agree. Despite being a joy to read-and they are-the fact he leaves such huge chunks of the theory to substantial exercises-makes me hesitate to recommend them as a standalone text. Shurman ,Carlen, Jones, Kaliszewski,  Kovacic, Guillemin  or any of a dozen other sources here would make better material for self study. Still, Tsirelson's notes are wonderful to read, so I heartily recommend them as supplemental reading to a second analysis course on several variables.
  45. MATHEMATICAL ANALYSIS I Gregor Kovacic  Rensselaer Polytechnic Institute  Course Materials And Handwritten Scanned Notes(PG-13)
  46. MATHEMATICAL ANALYSIS II Gregor Kovacic  Rensselaer  Polytechnic Institute Course Materials And Handwritten Scanned Notes (PG-13)  A pair of handwritten set of notes for a very impressive  year long course in analysis on one and several variables. The very extensive contents can be found on the cached course homepages here and  here  Abridged Contents: Number Systems Topology of the Real Axis and the Complex Plane  Limits  Continuity Differentiation Riemann Integral Numerical and Functional Series Metric Space Topology Trigonometric Series Approximation of Continuous Functions Functions of Several Variables Multivariable Integration  Integration on Manifolds  The sheer scope,clarity and level of detail of these notes is incredible-both courses taken together are practically a handwritten textbook. I don't actually know if these scanned notes are from professor Kovacic himself or have been compiled from scanned versions of one or several of his students' personal notes in his  classes. Either is equally likely-in fact, there's an alternate version of the first semester notes from Joshua Sauppe, they can be downloaded here.. I would recommend using the Fall 2013 version of the semester 1 notes-they're much clearer and more legible then the individual posted "chapters" at the website. Korvicic's course covers just about everything you want to know about classical analysis in a year long undergraduate course and does it absolutely beautifully. No, there's nothing new or original here, but for this kind of course, you really don't need there to be. I don't think you'll find better treatments of most of these topics in any other source, either online or in a textbook. The presentation of infinite series is as good as any. He precisely develops one variable analysis on both the real and complex  field, before moving on to multivariable differential and integral calculus on Rn  and metric spaces, and eventually k-dimensional manifolds and differential forms. He limits himself to a careful construction of multiple Riemann integration-Lebesgue integration isn't touched.  As I've said, I consider that a good thing in a course at this level. He doesn't give a fully rigorous treatment of differential forms, which is a reasonable choice I've noted in other authors. (I have mixed feelings about it. I understand they don't want to get bogged down in a huge, possibly confusing digression into algebra. But at the same time, they're a topic of such immense-and growing-importance for later subjects,part of me really feels irresponsible doing that. Maybe the best solution is doing them completely in a separate course like Guillemin does.) There are many,many beautiful hand drawn diagrams, graphs and careful proofs. Also, a real strength is there are an enormous number of exercises compiled by the author and they run the gamut from simple computations to real head scratchers that'll challenge the best students.  I also love the author's commentary on all the textbooks suggested at the webpages. My one serious complaint with the notes
  47. ?  My usual complaint- not enough examples!!! Which is really surprising, given how meticulous and detailed the notes otherwise are! Still-these notes are as good or better then any other on the internet. Seriously, they're awesome. I hope to LaTeX these notes for myself this summer and suggest you do the same. You'll thank me later. The highest possible recommendation.      
  48.   Honors Advanced Calculus Conrad Plaut University of Tennessee at Knoxville 2005-6 (PG-13) Another hardcore advanced calculus course for serious mathematics majors. The presumed background from the contents and style is a serious honors calculus course of one variable, such as one based on Spivak or Mattuck.  Contents:  Logic, Set Theory, and the Real Numbers Basic Logic Basic Set Theory Functions The Field and Order Axioms Completeness Sequences and subsequences Metric spaces Basic Definitions and Examples Open and Closed Sets Sequences in Metric Spaces Limits and continuity of functions Compactness Subspaces and Isometries Product Metrics Euclidean spaces Sequences and compactness in product spaces Connected Metric Spaces Metric Completeness Complex sequences and Series Complex Numbers Complex Sequences Series Convergence tests Power Series Pointwise and monotone convergence Uniform convergence Series of Functions Integration Riemann Integration Borel sets and functions Integration of Nonnegative Borel Functions  Lebesgue Measure Convergence Theorems Simple Functions Fubini’s Theorem Integration of Arbitrary Borel Functions Lp Spaces Differentiation A Little Linear Algebra Derivatives  Basic Differentiation Theorems The Mean Value Theorem and Applications C1 Functions The Inverse and Implicit Function Theorems Real Functions Linear Functions and Integration Change of Variables.  Very abstract and intensive notes,similar in content to the honors version of Pearce or Yu's notes, but less concise and more detailed. There's a surprisingly large number of examples in a course at this level, many of them sophisticated and involve counterexamples to the proven results. He also uses several nontraditional approaches to the material. For example, he uses the sequential definition of compactness in metric spaces.  He does this because in metric spaces, the sequential and general topological definitions are equivalent and the former is simpler to work with. It
  49. also leads more directly to results that are more relevant to analysis then topology. His development of integration theory is interesting in that he begins with the Riemann integral before moving on to abstract measure and the Lebesgue integral-and he also gives this before differentiation in metric spaces. The most striking thing about Plaut's notes is the incredible number of exercises-249 of them!-most of which are straightforward, but require some thought to solve. The result is a demanding but readable and interesting set of notes that will be very useful to both honors undergraduate and first year graduate students in analysis. Highly recommended for the right audience.
  50. Differential Calculus of Several Variables David Perkinson Reed College  (PG-13) The author's own abstract to the notes describes the contents and intent of these notes better then I could, so why not let him speak for himself? These are notes for a one semester course in the differential calculus of several variables. The first two chapters are a quick introduction to the derivative as the best affine approximation to a function at a point, calculated via the Jacobian matrix. Chapters 3 and 4 add the details and rigor. Chapter 5 is the basic theory of optimization: the gradient, the extreme value theorem, quadratic forms, the Hessian matrix, and Lagrange multipliers. Studying quadratic forms also gives an excuse for presenting Taylor’s theorem. Chapter 6 is an introduction to differential geometry. We start with a parametrization inducing a metric on its domain, but then show that a metric can be defined intrinsically via a first fundamental form. The chapter concludes with a discussion of geodesics. An appendix presents (without proof) three equivalent theorems: the inverse function theorem, the implicit function theorem, and a theorem about maps of constant rank. What his introduction doesn't say is that this is a rigorous presentation of the differential calculus, assuming a basic knowledge of advanced calculus of one variable. It's interesting that Perkinson chose to write this notes since his colleague at Reed, Jerry Shurman, wrote the online text for the year long rigorous multivariable analysis course we commented on above. One can't help but compare the two. Of course, Shurman covers a much wider range of topics, including the multivariable Riemann integral and the main integral theorems of vector analysis via differential forms. Perkinson's notes are just as careful and well written as Shurman's, but they're a bit better organized and he emphasizes the geometry of Rn a bit more then Shurman does. Both books have a large number of excellent exercises and lucid examples. Both Perkinson and Shurman tend to skip details in proofs at times and leave them to the reader.But  there's a striking difference in what details they skip. At times, Perkinson is nearly anal in his level of detail,but he makes some weird decisions in the course of his notes of what details to skip. For example, he omits the proof of the equality of mixed partial derivatives for C2 differentiable functions in Rn and refers the  reader to Rudin for the proof(!) . On the other hand, Shurman usually skips small and easily filled in details and is careful not to omit essential and difficult details. Rigorous multivariable calculus/analysis  is by no means an easy subject and it's important to be careful what details you omit. On the other hand, Perkinson is generally more readable then Shurman. While overall, for a serious pass through this subject, you'll get more out of Shurman, Perkinson is very nice collateral reading and shouldn't be skipped by any means. Highly recommended for all it's flaws.
  51. Differential Calculus Course Pierre Schapira Paris VI University, 2011    (PG-13)  A very abstract, "French"-Bourbaki style presentation of differential calculus on normed spaces. Presumes a working knowledge of linear algebra, point set topology and one variable analysis. Despite the high level of generality, Schapira supplies many nice examples on the real line and Euclidean spaces. Differentiable manifolds and commutative diagrams are liberally used towards the end. I think these notes would probably be best used for honors courses in analysis or graduate courses  in this country. If you like this kind of ultra-clean, abstract approach, then by all means, go for it. I'd recommend it for supplementary reading only.
  52. Real Analysis Rasul Shafikov University of Western Ontario Fall 2013 (PG-13/R)  This is an honors second course in undergraduate analysis for students with strong backgrounds including metrics spaces.  Contents:  Functions of several variables: continuity, differentiability, Implicit/Inverse function theorems, Rank theorem, higher order derivatives. Integration: Riemann integral, Lebesgue integral (quick introduction), Fubini's theorem, basic Lp spaces, Holder inequality, integrals depending on parameter. Surface integrals: line and surface integrals, Divergence theorem, Green's formula, integration by parts, spherical coordinates, surface area, partition of unity.Introduction to ODEs and PDEs: basic definitions and examples.Distributions: test functions, basic properties of distributions, convolution, fundamental solutions of differential operators, Magrange-Ehrenpreis theorem. The notes are concise, but quite lucid with a lot of good examples and strong exercises. The study of functions of several variables remains in Euclidean space with vector fields -differential forms aren't used, which simplifies many issues. Still, it's a strange choice given how advanced the last third of the lecture notes and their exercises are. There's a very good development of differential equations in basic physics from a mathematical point of view.These notes are quite readable and the exercises are strong. Recommended as a second course for strong students.
  53. Real Analysis II Michael Filaseta University of South Carolina (PG-13) A very brief and concise set of supplementary notes for the second semester of a  real analysis course based on Manfred Stoll's text. Contents: Some Review Material The Intermediate Value Theorem Uniform Continuity More on Continuity Derivatives The Mean Value Theorem  The Cauchy Mean Value Theorem and L'Hospital's Rule The Definition of the Riemann Integral  Examples of Functions which are Riemann Integrable The Fundamental Theorem of Calculus Miscellaneous on Riemann Integrals Review Problems The Definition of the Riemann-Stieltjes Integral Examples of Functions which are Riemann-Stieltjes Integrable  Computing Riemann-Stieltjes Integrals  A Connection Between Riemann and Riemann-Stieltjes Integrals  Integration by Parts for Riemann-Stieltjes Integrals  Sequences of Functions - Pointwise and Uniform Convergence  Two More Theorems on Sequences of Functions  Sets of Measure Zero  A Necessary and Sufficient Condition for Riemann Integrability  There's less here then meets the eye-they're basically a bullet-point outline review with very terse proofs of the main results in the text. Ok for review,but not much else. And frankly, Stoll's book is nothing to write home about either.............
  54. Advanced Calculus Evelyn Silvia University of California at Davis 1995 (PG-13)  An amazing and very substantial set of supplemental notes to accompany Rudin's text in an intermediate level real analysis of one variable course. Contents:  The Field of Reals and Beyond From Finite to Uncoutable Sets  Metric Spaces and Some Basic Topology Sequence and Series--First View Functions on Metric Spaces and Continuity Differentiation: Our First View Riemann-Stieltjes Integration Sequences and Series of Functions Some Special Functions Index These notes were written in 1995, when most universities were still stubbornly canonizing the text for their advanced calculus/real analysis courses in spite of the fact most of their undergraduates were receiving  increasingly watered-don calculus courses and were therefore increasingly unprepared for Rudin's level of difficulty. Silva wrote these notes to enrich the presentation of Rudin and make it human-comprehensible. Basic logic and naive set theory is assumed in addition to regular calculus. Silvia tragically passed away in 2006 after a long battle with ovarian cancer. (Ironically, she passed away a mere 5 months before my father lost a similarly futile battle with prostate cancer.)  In her 33 years on faculty at UC Davis,she had garnered a remarkable reputation as a teacher of mathematics-and these notes are certainly evidence of that. They are remarkably detailed and readable, with many unusual exercises and examples. For example,Silvia uses "fill in" exercises where she leaves specific steps of proofs,computations or deductions empty for the students to fill based on specific axioms or results covered in the notes. This is a wonderful teaching tool I don't think I've ever seen in a mathematics course at this level. There are many other innovative additions, such as a digression into the geometric properties of the complex numbers as transformations of the plane and a presentation of "ε-δ"  limit proofs both on the real line and in abstract metric spaces.  There are many examples, varying from obvious to challenging, all presented in loving detail. Exercises are sprinkled liberally throughout the notes-the exercises within the chapters themselves are usually followed by partial or full solutions, the exercises sets at the end of each chapter do not have solutions and are generally more demanding. This is one of the very best sets of lecture notes for advanced calculus I've ever seen and I heartily recommend them to all students and teachers of analysis. They are the last gift of a passionate teacher and that passion shines on every page. If only all of us had had analysis teachers as dedicated and talented as Silvia was. The highest possible recommendation.
  55. Undergraduate Real Analysis Part 1 Metric Spaces  Lecture Notes Vern I. Paulsen  University of Houston Spring 2012(PG-13)
  56. Undergraduate Real Analysis Part 2 Sequences and Series Lecture Notes Vern I. Paulsen University of Houston March 3rd, 2014 (PG-13)    A superior set of notes for an intermediate level undergraduate real analysis course by a very respected analyst and teacher, each half of which Paulsen taught 2 years apart. Contents: (Part 1)  Metric Spaces  Finite and In finite Sets Continuous Functions The Contraction Mapping Principle Riemann and Riemann-Stieltjes Integration (Part 2)   Behavior of Riemann Integrals with Limits  Uniform Convergence and Continuity  Uniform Convergence and Derivatives  Series of Functions  An Increasing Function with a Dense Set of Discontinuities A Space Filling Curve  Power Series Operations on Power Series  Taylor Series Polynomial Approximation The Stone-Weierstrass Theorem  Fourier Series Orthonormal Sets of Functions Fourier Series, Continued  Paulsen's notes are beautifully written with many deep and clear examples. ( A brief note-he keeps referring to "Math 3333" as the prerequisite course, this is apparently a standard "baby real variables" course that gives a rigorous development of calculus on the real line. Either this or an honors  calculus course would be suitable as background.) The most impressive aspect of the notes is how Paulsen works very hard to make sure the students understand the many differences between the validity of results  on the real line and their analogues on abstract metric spaces.There are many cases where the results generalize from the real line in a straightforward way, but just as often, in many important cases, they do not. The most commonly known example is, of course, the Heine-Borel Theorem.  Understanding these differences are critical for students who wish to go on and do advanced analysis in graduate school. He also has many excellent treatments of topics not usually covered well at this level, such as Fourier series and the multivariable version of Newton's method of approximation.  Combined with Dowd's notes, Paulsen's notes will serve as an outstanding text for self study or as a course text for this level course. Very highly recommended.
  57. Introduction to Analysis in One Variable Michael E. Taylor University of North Carolina 2010 (PG-13) Introduction to Analysis in Several Variables Michael E. Taylor University of North Carolina 2013  (PG-13) Another pair of notes for a rigorous treatment of functions of one and several variables at the intermediate level-this one from UNC's PDE master. The prerequisites are a good linear algebra course, along with the standard 3 semester pencil-pushing calculus course. Contents: (One Variable) Chapter I. Numbers Chapter II. Spaces Chapter III. Functions Chapter IV. Calculus Chapter V. Further topics in analysis (Several Variables) One-variable calculus The derivative Inverse function and implicit function theorem Fundamental local existence theorem for ODE The Riemann integral in n variables Integration on surfaces  Differential forms Products and exterior derivatives of forms The general Stokes formula The classical Gauss, Green, and Stokes formulas Holomorphic functions and harmonic functions The Brouwer Fixed-point theorem A Metric spaces, convergence, and compactness B. Partitions of unity C. Differential forms and the change of variable formula D. Remarks on power series E. The Weierstrass theorem and the Stone-Weierstrass theorem F. Convolution approach to the Weierstrass approximation theorem G. Fourier series First steps H. Inner product spaces  The first link is for a set of notes for a single variable analysis course that can be used as the first half of a year long undergraduate analysis course.  To be honest, I think the stated prerequisites will be insufficient for notes at this level. Both sets of notes are very concise and rather challenging-many results are shunted to the exercises. I think students without some experience with proofs and familiarity with basic logic and set theory may be swamped.  On the plus side,glancing over the contents one can see immediately this is a surprisingly nonstandard such course,as Taylor has a lot of original touches. The first and probably most striking  difference is that unlike most undergraduate analysis courses, Taylor develops a complete axiomatic development of the real numbers.This is a joy to find in an analysis course.Secondly.  many of the original concepts,such as convolution functions and Fourier series,.are of enormous relevance to applications of analysis.  He also includes some subtle but critical results on the real line as a metric space that aren't usually proved in depth, such as the fact every interval on the real line is a connected subspace of it. My one complaint with these notes is that there are virtually no examples and given how concise they are, that's a problem for any but the strongest students.That being said, the first set is very lucid and well written. Unfortunately, the second set of notes isn't as effective. Taylor's intentions with these notes are clear in the preface as follows:   It has been our express intention to make this presentation of multivariable calculus short. As part of this package, the exercises play a particularly important role in developing the material.  The result is a treatment which in some ways  is even more concise and lacking in detail then Dowd despite it's greater length and coverage. The notes are extremely "hard analytic" in flavor and approach-there are no pictures, everything is done via abstract proof or brute calculation. As with the first set. they're extremely precise and for the most part lucid. Unfortunately, in the several variable case, the lack of detail really hurts the presentation in some important places. Unlike in the single variable case, many sophisticated tools are needed for it's formulation that the student is likely to be completely unfamiliar with.The presentation of differential forms is one of the worst I've ever seen,  precisely because of the lack of detail in the development. It's not even that he's somewhat nonrigorous in the formulation-it's that it's so poorly organized.He seems to be brainstorming the development. He defines a k-form as a multilinear function on vector fields with certain algebraic properties.While that's correct, I think most students' eyes will have glazed over by that point. Here's where a few concrete examples and pictures would have immensely helped. There are several other sections of the notes that suffer the same problem. Still, there is a lot of well presented material here from the analyst's point of view. The fundamental existence theorem of ordinary differential equations and the inverse/implicit function theorems are both presented exceptionally well. They're well worth a look, but frankly I think they need further revisions. The second notes' main problem is poor organization and lack of examples-which is much more damaging in the single variable case. I prefer Shurman's text, Perkinson or Runde's notes-or for a briefer presentation, Dowd. And of course, there's the old classic by Spivak.  My hope is that Taylor will continue to revise these notes-especially the second half-for a full blown undergraduate analysis course. As I said, there's a lot of good stuff here and some added examples and better organization would help immensely. I still recommend both sets for strong students, particularly the single variable notes.
  58. Linear Algebra And Differential Forms on Euclidean Spaces David Simms Trinity College Dublin(PG-13) Measure Theory, Integration Theory and Analysis on Manifolds   David Simms Trinity College  (PG-13)These are a pair of lecture notes for a very intensive and abstract year long course on the analysis of several variables that bypasses Riemann integration altogether and jumps directly to the Lebesgue integral. There's also a very good chapter on the differential calculus on abstract normed spaces. Lecture notes for a substantial course on advanced linear algebra that includes tensor analysis and differential forms on manifolds. These concise, very intensive notes are clearly intended for a considerably more sophisticated mathematical audience then the usual first semester of linear algebra and multivariable calculus for undergraduates in the US expects.
  59. Rigorous Vector Calculus Martin Dowd 2012    (PG-13/R) This is strange online textbook from the Hyperion Software site, Dowd's experimental donation supported site which is offering several original online textbooks in modern mathematics, particularly in foundational topics such as set theory and category theory, free for download for students of all kinds who are interested in mathematics. We can only hope such sites become more commonplace in the future.I encourage all who believe as I do in free education and who have the means to make a donation-it doesn't have to be a large one.The book's aim is exactly what the title says it is-a careful and modern presentation of functions of a vector variable.Contents: Introduction Euclidean space Continuity and limits Linear algebra Differentiation Topology Matrix properties Measure and integration. Complex numbers Complex differentiation Power series Transcendental functions Multivariable substitution k-surfaces Tensors Differential forms The pullback operation Stokes’ theorem for cell chains References  The prerequisite level for the book is quite
  60. high-at minimum, a strong background in both "practical" multivariable calculus and "?-?" single variable calculus, Euclidean geometry,linear algebra and naive set theory.  I think the bar is actually higher then this since Dowd develops a considerable amount of point-set topology and some abstract algebra in the axiomatic development of Euclidean space. And yet, Dowd calls it a text for "second year university" students. Clearly, he's referring to an higher educational system far stronger then America's.If the student has no prior exposure to either topology or algebra,  this "need to know" development is going to be very tough slogging. That being said-for the properly prepared reader, the book is extremely readable and presents an abstract and fully rigorous presentation of vector calculus in a very compact venue. The book is extremely parsimonious with words-it has virtually no extraneous chit-chat or exposition beyond what's needed to state definitions or prove theorems.There are virtually no examples in the  usual sense, although there are many substantial exercises with hints.  When the author has a choice of how to present topics, he almost invariably chooses the most abstract and concise method. Interestingly though, he wisely leaves the Lebesgue measure and integral for an optional section and instead develops measure theory via the Jordan measure ,which of course naturally leads to a rigorous development of the Riemann integral in n-variables. In Rn ,this is usually more then sufficient for the purposes of vector analysis. The best part of the book is it's relative brevity.It comes in at 94 pages and really covers just about all the guts of the theory of multivariable calculus. While there are better online and in print stand-alone texts on multivariable analysis, Dowd implies in the preface the text is really intended to be used in conjunction with an intuitive, classical book on vector calculus. I think this would be the best way to use this book-the union of it combined with a more applied but comprehensive treatment, like Corral or Jones, would produce an outstanding rigorous course in the functions of several variables. It could also be used in conjunction with a one variable analysis course text  to produce a full year advanced calculus course on functions of one and several variables. Dowd has produced a beautifully written, very focused text that accomplishes it's goal and nothing more-strangely, that ends up making the text immensely versatile as a supplement. Highly recommended as a supplement for either a standard vector analysis course or a single variable analysis course.
  61. Advanced Honors Calculus I and II Fall 2004 and Winter 2005 Volker Runde University of Alberta August 17, 2006  (PG-13) This is yet another set of lecture notes for the honors advanced calculus course at Alberta. We've already commented on the others by Muldowney, Bowman, and Yu.  Like those others, the purpose of this course is to give a rigorous development of the calculus of several variables. Contents:  The real number system and nite-dimensional Euclidean space  Limits and continuity  Differentiation in Rn Integration in Rn The implicit function theorem and applications  Change of variables and the integral theorems by Green, Gauss, and Stokes Infinite series and improper integrals Sequences and series of functions A Linear algebra B Stokes' theorem for differential forms C Limit superior and  limit inferior Runde is a respected analyst, so not unsurprisingly, the notes focus more on the purely analytic aspects of the material. The notes are concise, but extremely clear and Runde takes the time to develop the necessary details of important proofs, such as Cantor's proof of the uncountability of the reals and the Bolzano-Wierstrauss theorem. There are especially good treatments of arc length,surface area and the classical vector analysis theorems in the plane and space. Also,a very large number of diagrams, many more then one usually finds in courses like this. I found the notes quite readable and well done, but the main problem with using them is that Runde didn't include the problem sets in his notes. You could use Muldowney's problem sets for the earlier version of the course, I suppose-but it's quite disappointing not to use the same exercises as his students and that limits the notes' usefulness as a text. Still, I liked them quite a bit and still recommend them as a supplement for analysis courses.  If  he'd included the exercises, it would have been a homerun recommendation. Oh well.
  62. Multivariable Advanced Calculus Ken Kubota University of Kentucky 1999  (PG-13) This is a course in the calculus of manifolds for undergraduate  mathematics majors and graduate students in engineering and computer science. Contents: Real Numbers  Compact Sets and Continuous Functions  Derivatives Inverse Function Theorem  Integrals Differential Forms Stokes' Theorem on Manifolds It's essentially a set of supplementary notes, exercises and computer programming exercises for Spivak's Calculus on Manifolds. The notes are clear,concise and within the spirit of Spivak. They're nice and readable and will serve a students studying Spivak well.  But these aren't really the high point of the site. What's remarkable here is that Kubota has posted full solutions to most of the exercises in Spivak ,which he assigned as homework. Since as any mathematics student who's done it will tell you, the real meat in Spivak is in doing all the exercises and having the solutions handy means one can self study out of this remarkable text. So bookmark this site-it will come in very handy for serious math majors and graduate students studying differentiable manifolds, trust me.
  63. Introduction To Mathematical Analysis John E. Hutchinson 1994 Revised by Richard J. Loy  Australian National University 1995/6/7(PG-13)
  64. Analysis 1 John Hutchinson Australian National University 2002 (PG-13)  These are 2 versions of the introductory analysis course at ANU. Not surprisingly, there's considerable overlap with his honors calculus course, which we commented on earlier. The newer 2002 version is similar but shorter-it is meant to be used as a supplement to the textbook Fundamental Ideas of Analysis by Micheal Reed. By comparison,the older version is longer and more substantial, but has fewer pictures and exercises. It's an undergraduate real variables course set on metric and normed spaces for functions of several variables, with the theory of the real line assumed known. It's another source I was completely unaware of before compiling this list. The problems for the course can be found here. ; the solutions can be found here. This online book, while it covers the standard topics for such a course-limits and derivatives of functions of several variables in Euclidean spaces, uniform convergence, etc.- in other respects it's quite different from the conventional books and lecture notes on undergraduate real analysis-it differs even significantly in some respects from the 2002 version. For example, it doesn't cover integration. Not at all, not Riemann, not Lebesgue, doesn't cover it at all. This would seem to destroy the value of the notes as a text if you didn't understand that several topics, including Riemann integration on the real line, are generally covered in standard American analysis courses are covered at the Australian National University rigorously in the calculus sequence for mathematics majors.Therefore, in the USA,these notes would be better suited for an intermediate level analysis course after either honors calculus or baby real variables. Hutchinson's notes for that course can be found, along with my commentary, here. Also, the follow up course to these notes, Analysis 2 at ANU, covers general topology, measure theory and the Lebesgue integral. Therefore, how integration is taught depends on the level the student or instructor wants the course to be. The notes can either be supplemented with the calculus notes (or the section in the 2002 notes) to supply Riemann integration material or one can jump directly into measure theory. (Indeed, Hutchinson recommends using the notes in tandem with either the 2002 version or his older calculus notes in the introduction!)  I know, it's a bit unusual, but the author makes all the necessary material freely available at his web site here.The 2002 version rectifies this by having a chapter on the Riemann integral-probably to make it more palatable to students.In general, I think the newer version, although shorter and covers less, is more readable and covers functions of several variables in a clearer and more visual way. The second thing that's impressive about these notes is the highly unusual topics that are discussed in lieu of standard presentations of integration, such as much more thorough introductions to set theory and logic, ordinary differential equations  then is usually present in real analysis courses at this level. It also includes a wonderful introductory chapter on fractals with many examples and pictures. Between the maturity and depth of the 1997 version and the readability and visual nature  of the 2002 version, the union of both of these notes forms one of the most unusual, mature and masterfully written online introductions to mathematical analysis at this level that currently exists. Both of the notes are deep,informative and quite well written. If organized and supplemented appropriately-with Rosenlicht's text for integration, for example-they can serve as a wonderful foundation in real analysis for strong undergraduates and a great teaching tool for teachers of analysis. If you're going to use the notes for self study,.make sure you use both versions together-neither by itself is going to cover the material as effectively as the pair together will.
  65. Honors Analysis I Robert Sharpley University of South Carolina Fall 2008 (PG-13) These and the next 2 links are to 3 courses on undergraduate real analysis that Sharpley has given over a decade and it's interesting to see how they vary in level and presentation. Contents: Countable and uncountable sets, the real numbers, order, least upper bounds, and the Archimedean property Metric spaces: topology, open and closed sets, convergent sequences, completeness, compactness and the Heine-Borel Theorem for the real line, connectedness. Limits; Continuous functions and their properties, limits, continuous functions on a compact metric space, continuous functions on a connected metric space; intermediate and extreme value theorems, uniform continuity, monotone functions and inverses.Derivatives and their properties, the chain rule, Rolle's theorem and the Mean Value theorem, Taylor's theorem, L'Hospital's rule.The Riemann integral, its properties, and the Fundamental Theorem of Calculus.  The notes are all relatively brief and vary in terseness of details-but all are really intended as supplementary notes with the details to be filled in in class.So they're not really intended as written to serve as full classroom texts, but as outlines to be filled in by the students. The first course is an accelerated honors course for very strong undergraduates based on Rudin's book. As as result, they're the most brusque of all-partially to force students to fill in details. The contents are very similar to Rudin's book with some additions, but mostly they're basically to add some examples and needed details to proofs help soften the breakneck pace and merciless conciseness of Rudin's treatment. They're sharp and nicely written-but they're really too brief to be that helpful. Maxwell Rosenlicht's Introduction to Analysis is cheap and would make a far more useful supplement to Rudin. By far, the best supplementary notes for Rudin online are the beautiful companion notes at USCD by the late Evelyn M. Silvia. Of course, you could sidestep writing notes or needing a supplement altogether while still preserving the brute difficulty level of the course by ditching Rudin for Charles Chapman Pugh's outstanding Real Mathematical Analysis or the much cheaper but nearly as excellent Analysis in Euclidean Space by Kenneth Hoffman -but that's my opinion.
  66. Analysis  I Robert Sharpley University of South Carolina Fall 2004(PG)
  67. Analysis II  Robert Sharpley University of South Carolina Spring 1998 (PG-13) These are Sharpley's notes for a more standard 2 term course in analysis with the 2 terms separated by 6 years-they are surprisingly gap-free between them. The first semester gives a fairly standard "advanced calculus" course detailing the theory of functions of one variable on the real line. Contents of the first semester:Real numbers, convergence and sequences, limits,derivatives and the Riemann integral. The content of the second part is very similar to that of the honors course above, but the pace is much slower and more detailed. It also includes some topics not covered in either the honors course or the first semester, such as special functions in calculus and detailed topology of normed linear spaces. Contents of the second semester: Special functions, topology and completeness of metric spaces, compact subspaces and total boundedness, connectedness,  Riemann-Stieltjes integration, infinite sums and sequences in metric and normed linear spaces and sequences of series and functions. I find it rather interesting that Sharpley's lecture notes improve in quality in direct  proportion to how old they are. The most recent notes from 2008 are incredibly brusque and nearly useless, while his older notes for a gentler course in 2004 are much more readable and helpful, with many examples. His notes for the second half from 1998, a decade earlier then the most recent notes, are the best of the set-detailed, well-problemed and very insightful. I suspect it's a reflection of his increasing promotion in the faculty that, of course, results in an increasing importance on his own research, which left him far less time ( and let's face it, encouragement) to focus on teaching. If so, it's a very sad proof of what I've long suspected about the current state of American academia. Be that as it may, I found the 2 sets of notes for this year long course, separated by over half a decade, to be excellent and are very highly recommended.
  68. Analysis Yawp! Interactive Video Real Analysis Lectures With Transcript  Francis Su Harvey Mudd College   (?)Now here's something truly remarkable-a complete set of videotaped lectures and their accompanying lecture notes for an real variables course based on Rudin given by Su in 2010. I'm sure most self study students love taped lectures and fortunately, there's a growing number of them currently available online.  I'll bet you'd love to know my opinion of them. Well, I'd love to give it to you-but the night I was to work on this review, my computer's speaker died. Still, I strongly encourage students with working speakers to give them a try if only for the reason that professor Su and his students went through all this effort to make the lectures freely available online for everyone! (I hope at some future date I can give a commentary on these lectures and their corresponding notes. Damn technical difficulties............)
  69. Analysis Notes James Cook Liberty University 2006 (PG-13) These are another one of Cook's very nice scanned notes, these for an intermediate level real analysis course on R  partially based on Rosenlicht's text.They have quite an unusual selection of topics that make them well worth reading and having.They are handwritten in markers of diverse colors, which are very old school and legible with many pictures.There are no formal table of contents,but the notes cover Rn   as a metric and/or normed space, review of matrix and linear algebra, metric and normed spaces, sequences in abstract metric spaces, the matrix exponential (a critical topic many textbooks on analysis omit for some mysterious reason), Cauchy sequences and completeness, applications of the matrix exponential to differential equations and the Jordan form, sequences of functions and uniform convergence, inverse and implicit function theorem and finally, a long final section on multilinear algebra,tensors and differential forms. Essentially, this course is a cross between a very traditional semi-rigorous advanced calculus of several variables course-like Cook has given for several years at Liberty-and a standard intermediate level analysis course. The cross is pretty effective and the 2 aspects of the course balance each other very nicely. There's also a number of concepts you don't normally see in either course-such as the multivariable Newton's method formula and it's relation to the contraction mapping theorem ,the Einstein convention for tensors and an introduction to Hodge duality. They're very readable and clear. Unfortunately, the one down side is the usual one you'd expect with notes like this-no exercises. Still, there's a lot of good stuff here and they're well worth having.   

    Elements of Real Analysis I David K. Neal Western Kentucky University Spring 2013    (PG-13) The  first semester lecture notes for an intermediate level real analysis course. Contents: Real number system, metric spaces, limits, sequences, functions, and continuity. Fairly standard and concise, but Neal develops several aspects in more depth then usual in books and notes at this level. For example, he develops a complete constructions of the real numbers beginning with a sketch of the traditional development with the natural numbers and carrying it through the rationals,and then a detailed construction via rational or irrational decimal expansions. Yes, this is the most cumbersome and awkward way to do it, but it's also the method that's simplest in terms of the machinery developed. There aren't many examples, but the ones he gives are good and instructive ones. For example, as a as an example of a metric,he fully develops the properties of the norm on Rn . Many proofs are merely sketched or left to the exercises, of which there are many. Also, as you can see, the scope of the notes is pretty limited. That being said, the notes are readable, well organized and most of the exercises are straightforward. These notes will appeal to the student or teacher that likes to learn actively. If read with a pencil in hand, a student will be able to learn quite a bit. Personally, I prefer Paulsen or Silvia's notes or Rosenlicht's text for a course at this level.  Recommended.
  70. A Modern Introduction To Analysis Robert Gunning Princeton University July 2014 version (PG-13/R) Honors Calculus/ Analysis 2 Functions of Several Variables Robert Gunning Princeton University 2012 (PG-13/R) Yes, that Robert Gunning. The famed analyst at Princeton University.He also has a reputation among the students there for being an exceptional teacher in some very high level courses, something most faculty at that famed university rarely get. These are apparently 2 versions of a textbook Gunning has been developing for the very strong honors calculus students at Princeton for several years-the 2012 version is a shorter version covering functions of several variables only, while the 2013 version is a full blown analysis/honors calculus text on functions of both one and several variables. I've read through them both and I can honestly say I like the shorter version better. The longer book is very similar in content and style to the shorter one but differs considerably in organization.  As one might expect for a course developed by a top-notch mathematician for gifted students, it's a very intensive and advanced course. Seriously, it's ridiculously intense and it's amazing that Gunning could teach freshmen out of it, even at Princeton. It's basically an undergraduate analysis course pitched at a higher level then baby Rudin! Of course, there's a precedent for a whirlwind course like this-namely Harvard's Math 55 in the 1960's and the resulting text by Lynn Loomis and Schlomo Sternberg.  A student that's able to finish a year out of Gunning's own version and master most of it would be qualified to enter a graduate course in analysis at most other universities. It's not so much the level of the course that's so daunting-it's the speed of it. To give you an idea what a student taking this course would be getting his or herself into, Gunning covers naive set theory and an axiomatic development of the basic number systems, followed by a crash course in abstract algebra defining groups, rings and fields strictly in order to be able to give a completely rigorous and modern definition of vector spaces and their linear maps-all in 23 pages. Which,as well done as it is, is insane.  Ok, granted most students at this level are probably going to be simultaneously be taking either an honors algebra course or a standard undergraduate algebra course. But that's not guaranteed and even if it was,  considering this is a course aimed at students with relatively little hard mathematical training, wouldn't a rigorous treatment of linear algebra independent of the larger scope of abstract algebra make more sense? Hell, even Loomis and Sternberg didn't try and do that! To me,it's really indicative of what's wrong with the longer version of the notes-they're too ambitious for their own good. The rest of the course is an equally breakneck pace course on functions of one and several variables in metric spaces that presents single and multivariable analysis simultaneously with the one variable case being presented as a special case. On the plus side, Gunning writes very well and both versions of the book are quite lucid and informative.They're also surprisingly readable given the amount of material that's covered.  But in the longer version, the warp-speed  presentation really hinders it's depth. This could be alleviated considerably by the addition of many more explicit examples. But to be honest, I just think any but a gifted student is going to be bowled over by this course. In a sense, the author himself acknowledges this by the fact a slower version of the course spread out over 3 semesters using the text for "average" Princeton math students-who are obviously better then all but the very best students at most universities-was necessary for most students there. This is basically why I like the shorter version  a lot better,although again I think it needs work.It's on functions of several variables and basically assumes the student has a strong command of functions of a single variable and some linear algebra.  As a result, the notes are much more focused, smoother, organized and doesn't try to do too much in one course. Gunning also has an original  presentation of differential forms that emphasizes an explicit construction of the exterior algebra in an arbitrary vector space V where the members of the algebra are permutations of  an ordered set of basis vectors of the original vector space V. He then applies it to the special case of V= Rn where the elements of the exterior algebra are defined on open subsets of Rn as multilinear maps of the differentials of the coordinate functions, which are of course basis vectors for  Rn To be honest, I'm not sure if this construction is any easier or less confusing then the full monty algebra construction using dual spaces, such as in Guillemin's notes. I'm not sure there really is a simple way to introduce forms, unless you want to just use them as a calculation tool and to hell with theory altogether!  See, this is where a lot of explicit computations and examples would be very helpful. This is the main flaw in both sets of notes and until Gunning develops more examples, either version of these notes aren't going to be anywhere near as helpful or informative to students as they can be.  Still, Gunning's to be commended for attempting to create notes that will challenge the very best mathematics majors and I hope he continues to polish and expand them until they have sufficient intuition to balance the rigor. Until then, I'd recommend using either version in combination with either Rossi's online book, Sjamaar's notes for an honors course on calculus of several variables or as a second semester for an undergraduate analysis course. These sources will supply the much needed intuition and visuals
  71. for the course.( I'd also recommend to Gunning he look at his original inspiration. As concise as Spivak's original text is, he gives plenty of examples and pictures! )
  72. Theory of The Integral by Brian Thompson 2013   (PG-13/R) Another fine analysis book from the co-authors of Elementary Real Analysis  (ERA) and it's brethren texts on real variables for undergraduates and graduate students. This book is pitched at a strange "in-between" level for either strong undergraduates or first year graduate students-it can be seen as sequel to either ERA or the author's earlier The Calculus Integral  In other words, the prerequisite knowledge is a solid grasp of one variable theoretical calculus.As with all the books from Thompson and his co-authors, a free PDF copy is available here. This book gives a self contained presentation of all the modern versions of the integral-from the so-called Newton integral to the traditional Riemann integral to a concrete and classical presentation of measure and Lebesgue integral on the real line only  and the so-called gauge integral or
  73. Henstock-Kurtzwell integral, which is more general then the Lebesgue integral on Euclidean space. Thompson writes extremely well, the book is very lucid with many penetrating observations, clear proofs and many exercises. The book combined with it's prequel would serve both students and teachers very well as a book for self study to master integration on Euclidean space in it's many manifestations before tackling a graduate course in analysis. It would also make for a wonderful seminar for honors undergraduates to strengthen their grasp of basic integration theory before graduate school. Very highly recommended.
  74. Calculus on Manifolds Simon Rubinstein–Salzedo from a course given by Martin Scharlemann at Berkeley University of Albany Consort Spring 2004 (PG-13)  Another rigorous course on the functions of several variables. They cover approximately the same ground as Micheal Spivak's Calculus on Manifolds and in the same concise style. They're readable, but don't add much to Spivak's presentation. You can take a look, you may like them-but I'd rather have Shurman or Gunning.
  75. Advanced Calculus I-II Scott McCollough University of Florida 2013-2014   (PG-13)  An intermediate level advanced calculus text under revision at McCollough's website. In fact, the working draft has a huge watermark "DRAFT" on all it's pages, so you know he's planning on publishing it someday soon. Contents: Review of Sets and Functions The Real Numbers Metric Spaces Sequences Cauchy Sequences and Completeness Compact Sets Continuous Functions  9. Sequences of Functions and the Metric Space C(X)  Differentiation Riemann Integration Series Complex Numbers and Series Derivatives of Mappings Between Euclidean Spaces The Inverse and Implicit Function Theorems Mappings Between Matrix Algebras Fourier Series First Order Initial Value Problems Index The text is quite standard in content,no real surprises here. But the standard material is done quite well here. The text is readable and informative if somewhat dry. McCollough is very careful, there are many good examples and an even larger number of exercises-many of which are excellent and challenging-are peppered throughout the text. A good solid set of lecture notes for an intermediate level course without all the bells and whistles. If you like your advanced calculus served traditionally, as I do, you'll like this book. I think there are better sources covering the same material-but if I was forced to use this book as a course text for advanced calculus, I'd have no qualms using it. Recommended.
  76. Multivariable Calculus Krishna V. Kaipa Imperial College Spring 2011  (PG-13) The title is somewhat deceiving, since it would lead one to think this is a standard vanilla “Calculus III” course on polar coordinates and vector algebra, partial derivatives, vector valued functions and the Dynamic Duo of theorems on vector calculus, Green’s Theorem and Stokes’ Theorem-all presented with all the pretty pictures in R3 ,applications to physics and lots of things stated without proof.   And you’d be completely wrong. This is a completely rigorous course in multivariable calculus using the language of manifolds and differential forms in Euclidean spaces.  Kaipa has written a very clear and informative set of handwritten notes to supplement the course-the “official” textbook was Spivak’s Calculus On Manifolds. As anyone who has studied Spivak can tell you, it’s basically a problem course with definitions and a lot of examples presented-most of the course is in the exercises, which are substantial at best and brutally impossible at worst. As a reference, the course cites James Munkres’ Analysis on Manifolds, which is a much more comprehensive and thorough text on the same subject and one of my favorites. But many instructors find Munkres far too wordy and pedantic-and it’s also much more expensive than Spivak. Reading the notes, I think Kaipa was trying to compose a supplement for the course that was less terse then Spivak, but more concise then Munkres, while covering all the major points clearly and efficiently. I think he’s mostly succeeded-the notes hit all the critical points yet are relatively brief. A very good resource for serious mathematics students. Highly recommended.
  77. Calculus IV Functions of Several Variables Dima Pasechnik Draft Nanyang Technological University April 24, 2006  (PG-13) This is a higher level version of the functions of several variables course at Nanyang. We listed and discussed a more intuitive and less rigorous version of the course as taught by Fedor Duzhin here. This is a completely rigorous version of the course, building multivariable calculus on embedded manifolds in Euclidean space and using the language of differential forms. Functions of more than one variable, limits, continuity, partial derivatives, differentiability and total differential, chain rule Directional derivatives, gradients, Lagrange multipliers, Double integrals, area of a surface, triple integrals line integrals, Green's Theorem, surface integrals, Gauss' divergence theorem, Stokes' Theorem. The notes are all buisness:  No pictures or examples, ruthlessly concise and brief.In fact, no exercises. Indeed, they're too brief and incomplete to form a complete text and I think looking at Nanyang's mathematics program supplies a large clue as to why. There are 2 versions of this course-a regular one for ordinary mathematics students, which use the earlier comprehensive but far less advanced lecture notes of Duzhin-and an honors version of the course which requires much more background and is supposed to be a theoretical treatment. I suspect these are the "theoretical" part of the honors version of the course, which would be used in tandem with the notes by Duzhin.  This makes complete sense since Pasechnik's notes contain the rigorous"guts" of several variable calculus and nothing else-while Duzhin's notes contain everything else. Together, these 2 lecture note sets can form the basis for an outstanding honors course in calculus of several variables and I recommend they be used that way.  But honestly, any good non-theoretical book on functions of several variables could be used in conjunction with these notes to create a good course text. Jones, Corral or Herod and Cain could all be used in conjunction with these notes. But frankly, if the notes can't stand on their own at all, they're really going to be of limited use. I'd rather use Edwards Jr., Perkinson, Yu, Shurman, or Carlen, frankly.
  78.  Introductory notes in analysis Stephen Semmes Rice University  (PG-13)   A very condensed, almost "cliff notes" version of an intermediate level analysis course on metric spaces. Contents:  Metric spaces Least upper bounds Open sets Closed sets Complements of subsets Unions and intersections Compactness Relatively open sets Compact sets Totally bounded sets Closed intervals Compactness properties Countable sets Separable metric spaces Connected sets Sequences, series, and functions Sequences Complex numbers Subsequences and sequential  compactness Cauchy sequences Infinite series Alternating series  Power series Extended real  numbers Upper and lower limits Sequences of sets Root and ratio tests  Continuous mappings Continuity and compactness Continuity and connectedness Uniform continuity Compact spaces . Uniform convergence Complex-valued functions The supremum metric The supremum norm The contraction mapping theorem Limits of functions Limits and sequences One-sided limits Monotone functions III Some additional topics Appendix Three homework assignments A 1 Maximizing distances A.2 Minimizing distances A.3 Positive lower bounds There are mostly careful definitions and observations-there are virtually no examples or even explicit theorems and their proofs. I don't think these notes can really be used as anything more then a review or a supplement, due to the lack of detail. I suspect they aren't really intended to be used any other way. I'd pass.
  79. Honors Introduction to Analysis Erin P. J. Pearse University of Oaklahoma May 2, 2007 (PG-13)   This is the other version of the real analysis course for undergraduates by Pearce, the stronger version for honors students. A very challenging, terse, but interesting set of lectures for a strong undergraduate real analysis course. Logic and naive set theory, the rational numbers, Axiom of choice, a construction of the reals via Cauchy sequences of rationals, topology of the real line, continuity, differentiation, integration, sequences and series of functions,uniform convergence, power series and polynomial approximation. The proofs are very effeicient with virtually no chit chat and there are almost no examples-all the examples are shunted to the many, many exercises. It's clear this is an intense course designed to challenge the strong student and will appeal to the Rudin-liking crowd. It's well written and does contain some unusual and important material, such as analytic convergence and applications of the Stone-Weirstrass theorem to differential equations.  Personally, though-I'd advise all the strongest students in advanced calculus to avoid this version in favor of the low octane version here. Recommended for the right students.
  80. Calculus on manifolds Fall 2003 Math In Moscow lecture note  (PG-13)   Amazingly brief but very deep set of lecture notes on the calculus on differentiable manifolds. Contents: Smooth curves in the plane and in the 3-space Submanifolds Smooth mappings of submanifolds, diffeomorphisms, and manifolds Abstract manifolds, tangent spaces Vector fields Frobenius Theorem Differential forms Exterior (wedge) product Gelfand-Leray and area forms Differential Integration of differential forms Stokes formula Cartan Identity L = id + di  Poincare Lemma. De Rham cohomology Short review of principle formulas The amount of material the authors manage to pack into a mere 25 pages is incredible. As one would expect, there is virtually no fat-there are very few examples and most of the smaller results are shunted to the large number of exercises. Still, they're wonderfully written and I can't think of a better supplement to a textbook or course on this material then this. Highly recommended as a supplement.
  81. Calculus on Vector Spaces 1 Massimo Marinacci and Luigi Montrucchio Collegio Carlo Alberto, Università di Torino January 2009 (PG-13)   Yet another example of how other countries leave the current United States school and university cirricula in the dust. These notes-which hail from my ancestral homeland, Bellesimo! –have a somewhat deceiving title. At first glance, you’d think this was a course for either a standard “Calculus III” type multivariable calculus course or a somewhat more rigorous “Vector Calculus” course which assumes a good background in linear algebra. The first part (chapters 1-4) of the notes is indeed that, but it is presented far more rigorously then either
  82. of those courses are presented in the U.S. The first 3 chapters give a very abstract treatment of vector spaces and linear operators in R  , with some interestingly unusual terminology ( a linear functional is defined as a real valued map in   Rn , a  map from  R to R where  m  n is called a linear application, etc.) . Chapter 4 is on the differential calculus of R and its subspaces, including excellent presentations of the various kinds of differentials and derivatives that exist in  R (real valued, partial, total, Gauteax and Frechet) . This part is particularly impressive since a lot of textbooks don’t clarify the subtle differences and relationships-most pick one and stick to it. The authors do a fine job here. But the emphasis here is on pure theory, there are almost no applications in the usual sense (except one or 2 to economics, which is bizarre). There’s no integration theory either, which I presume is covered in the second semester course. I couldn’t find any lecture notes corresponding to it online, so it’s  mythical…….The rest of the notes focus on the classical theory of the calculus of variations and abstract metric and normed spaces, material that one usually sees in a graduate analysis course.  The presentation is wonderfully careful and lucid, with literally hundreds of examples. If you’re looking for a good presentation of multivariable calculus, you’re better off with one of the more standard presentations here like Corrall or Bandaxall/ Liebeck. But if you’re serious about analysis, there’s a lot of wonderful, hard to find stuff in here to treasure. Highly recommended to analysis students.
  83. Stokes' Theorem A Rigorous Course in Mulitivariable Calculus Benjamin McKay University College Cork 2014  (PG-13)  A very recent textbook draft that in my humble opinion, will very quickly take it's place as one of the premier sources for this kind of course. A very recent textbook draft that in my humble opinion, will very quickly take it's place as one of the premier sources for this kind of course. As the title indicates, the purpose of these notes is to present a careful course in the calculus of functions of several variables, utilizing differentiable manifolds and forms. The requisites are a good course in linear algebra, some elementary topology and a rigorous course in calculus of one variable.  McKay presents the material very visually, with all manifolds embedded in Euclidean spaces and with many examples. What’s unique to the course is that McKay takes a largely “intuitive” approach-while he doesn’t “handwave” his way through the hard definitions, he doesn’t always give fully precise ones either. Instead, he often states what a definition means and makes his students restate his loose definition precisely as an exercise. For example, this is how McKay states the definition of a convergent sequence in Rn  : Notice-it’s not exactly the precise definition, but all the essential elements of the definition are present. It’s a very straightforward task for a mathematics student with the right background to rewrite it precisely in terms of either ?-? arguments or metrics. And indeed-exercise 1.9 asks the student to do precisely this. The author does this to great effect throughout the notes-including similarly intuitive proofs of the Hiene-Borel theorem in Rn  ,the contraction mapping theorem and the Inverse function theorem, to give just a few examples. This balance between rigor and geometric intuition is not only extremely informative for a beginner, it really smoothes the sharp edge that a completely rigorous treatment of this material often has that confuses students. There’s an emphasis on linear algebra rather then hard analysis, which not only will beginners will be more comfortable with at this point,but it allows  McKay to precisely define and emphasize orientation in the presentation.  McKay further  assists the reader’s understanding with many, many exercises and nearly as many pictures in the plane and 3-space. Lastly, the relationship between k-differential forms and vector fields is  emphasized rather then a purely algebraic approach. The result is a superb and extremely reader-friendly presentation that will serve students and teachers very well in preparing for more advanced courses in differentiable manifolds and topology. Very highly recommended!Contents: 1 Euclidean Space 2 Differentiation  3 The Contraction Mapping Principle 4 The Inverse Function Theorem 5 The Implicit Function Theorem 6 Manifolds 7 Integration 8 Vector Fields 9 Differential Forms 10 Differentiating Differential Forms 11 Integrating Differential Forms 12 Stokes’s Theorem 13 The Brouwer Fixed Point Theorem 14 Manifolds from Inside 15 Lagrange Multipliers 16 The Gradient 17 Volumes of Manifolds 18 Calibrations 19 Sard’s Theorem 20 Homotopy and Degree 21 Fubini’s Theorem 151 22 Homotopy and Integration 23 The Lie derivative 24 Open Covers and Partitions of Unity 25 deRham cohomology 26 Homological Algebra and the Mayer–Vietoris sequence 27 Cech–deRham cohomology 28 Currents 29 The Moving Frame 30 Abstract Manifolds 31 Abstract Manifolds with Corners 32 Tangent Vectors 33 The Grassmannian Hints Bibliography List of Notation Index
  84. Several Variable Calculus: Josef Dick University of South Wales 2014 .  (PG)A strange, concise little set of notes for an advanced calculus course of several variables at The University of South Wales.  The contents are unusual, but the presentation is fairly standard. Contents: Chapter 1: Fourier series  Section 1: Background information   Section 2: Inner product and norm   Section 3: Fourier series and pointwise convergence   Section 4: Examples and general periodic functions  Section 5: Convergence of sequences of functions  Section 6: Heat equation \Recommended reading: Motivation of formulae for Fourier series and a comparison to Taylor series   Additional Material: Mean square convergence of Fourier series Chapter 2: Vector fields and the operator Δ Section 1: Vector fields  Section 2?Divergence and Curl Chapter 3: Line integrals  Section 1: Scalar line integrals  Section 2: Vector line integrals  Section 3: Fundamental theorem of line integrals Section 4: Green’s theorem   Recommended reading: Line integrals and orientation  Additional Material: Rectifiable parameterised curves   Additional Material: For which curves and regions does Green’s theorem apply? Chapter 4: Surface integrals Section 1: Parameterisations of surfaces Section 2: Surface area and surface integrals of scalar fields  Section 3: Surface integrals of vector fields Chapter 5: Integral theorems  Section 1: Divergence theorem Section 2: Stokes’ theorem    Recommended reading: The fundamental theorems   Additonal Material: Differential forms and the general Stokes’ theorem The notes are quite careful but extremely terse-there are few examples and exercises and many links to other sites, particularly the relevant Wikipedia pages,  where proofs and fine details from analysis and topology can be found. They are readable and give a nice brisk presentation of the material, particularly on Fourier series, line integrals and differential forms. But because of the lack of detail, I doubt they could be of use as anything other then review.
  85. You're much better off with Shurman or McKay for functions of several variables and one of the many good sources in the Harmonic Analysis section for Fourier series. I'd pass.