
Basic Calculus (Calculus Without Theory)

Like all good Math books, we begin with a definition.As its
campfires glow against the dark, every culture tells stories to
itself about how the gods lit up the morning sky and set the wheel
of being into motion. The great scientific culture of the Westour
cultureis no exception. The calculus is the story this world first
told itself as it became the modern world. David Berlinski 
Widely used calculus books must be mediocre.Walter Rudin, The Way I Remember It
 Introduction: Mental Foundations  Integrate Your Brain The Basics of Doing Integral Calculus In Your Head (G) This isn’t really a textbook, but a good mathematics recreation book that teaches many very useful mental shortcuts that can cut down on your computation timea skill that comes in very handy not only for exams, but for those aiming for careers in applications like engineering. I’m sorry I didn’t find both these books when I was an undergraduatemy life would been so much easier.
 Multivariate Calculus Jerry Alan Veeh Auburn University April 25, 2003 (G) This is the second part of a strange set of lecture notes, which I have mixed feelings aboutthe first part on single variable calculus can be found here.. Veeh attempts in these notes to give students a discussion of the basics of the calculus of single and several variables that omits proofs completely and focuses entirely on motivation and intuition. That’s accurate, but it’s completely misleading as well because it suggests they’re entirely about freewheeling nonrigorous arguments and that’s simply not so. He assumes the student is familiar with the basics of matrix arithmetic and single variable calculus. The presentation is made up entirely of 2 aspects of calculus: 1) precise definitions (mainly from naïve set theory) and 2) A careful analysis of the geometry of what each stated result means and why it’s true. This allows Veeh to state things carefully and precisely with many examples and exercises, but while developing the strong intuition of the student without the technical difficulty of proofs. While the mathematical purist may find this sacrilegious, for students first learning calculus seriously, particularly those interested in applications, this approach can be much more insightful. Also, for more advanced mathematics students, the notes can supply very needed visual intuition, which is usually lacking in formalized treatments. A good example of Veech’s approach is in the discussion of extreme value problems in 2 dimensions: In order to attack this problem, the nature of surfaces is examined in a bit more detail. The key observation is this. A surface is completely covered by all of the one dimensional curves which lie in the surface. This fact allows the surface to be studied by looking at all of these one dimensional curves. Since one dimensional objects are so simple (at least compared to their higher dimensional counterparts) many computational difficulties can be eliminated by using this fact. He also uses the opportunity to present more sophisticated concepts in this manner, where a purely formal treatment may cause students’ eyes to glaze over. For example, he gives a very nice geometric discussion of tensors and differential forms and their role in multivariable calculus and physics. Personally, I’m more of a purist and I couldn’t really recommend or use these notes as the main text for a course or self study due to the complete absence of proofs. That being saidthey’ll make a terrific supplement for a more formal treatment. For example, I can completely see them being used as a supplement to a 2 year honors calculus class based on, for example, Spivak’s Calculus and Calculus on Manifolds supplying much needed examples and intuition for those very austere and hard core presentations. For that purpose, I like Veech’s notes very much indeed and highly recommend them.
 Calculus I and II by J.M. Heinbockel Old Dominion University (G) This interesting and well thought out online textbook was one of the first that got my attention when I began compiling this list. According to the preface, the goals of the book are to provide “an introductory calculus presentation intended for future scientists and engineers.” The 2 volumes of the book cover all the standard material for both a single variable and multivariable calculus course at the undergraduate level, with several striking unique oddities. For one thing, the book is written so that the course can be extremely versatile on the level of rigor used for the students. Virtually all major theorems are proven rigorously. Simulataneously, applications are given with many examples in great detail from physics, chemistry, biology and the social sciencesincluding a number we don’t normally see in calculus texts, like chemical kinetics, the law of mass action and vibration theory. The proofs and the applications are modulized within the text in such a way that the proofs can be skimmed or skipped without hurting the rest of the exposition. Another fresh aspect of the book is how the author uses nonstandard terminology sometimes to give precise formulations to concepts in a simpler manner then usually given in real variables courses. For example, rather then define relations and functions in the usual manner, he uses the archaic terminology of classical complex function theory, referring to relations and functions as “multiplevalued functions” and “single valued functions” respectively. This has the advantage of using a single defined concept, function, for both and being able to define rigorously the trigonometric functions and their inverses on tbe entire real line. My one concern is this might lead to confusion later in the student’s studiesI would have included a footnote explaining how their terminology differs from that in more advanced texts. A more successful example is how the author very cleverly uses the topological concept of a deleted ε neighborhood to both motivate and precisely define limits. The second volume is equally visual and literate, but in my opinion, not nearly as elucidative as the first volume. This is because of the author’s strange choice to use the awkwardly pedantic classical vector algebra as his framing language, when basic linear algebra would be so much more illuminating and cleaner. Exactly why he decided to do this is a mystery to me. Still, there’s an enormous amount of wonderful material on functions of several variables here, all presented carefully and with lots of graphs in the plane and 3space. Some surprises as wellsuch as the FrenetSerre equations for both plane and space curves. Despite its collective flaws, this is overall an outstanding free textbook that can serve as one of the models for what open source textbooks can be. I’d have absolutely no serious qualms about teaching a calculus sequence from it or recommending it to either calculus students or anyone else who needs to do so. Strongly recommended.
 Calculus, Applications and Theory Kenneth Kuttler Brigham Young University April 14, 2010 Multivariable Calculus, Applications and Theory Kenneth Kuttler Math 214 Brigham
Young University August 19, 2011 (PG) These are the next to final drafts of a 2 volume calculus book the author published last year and has graciously left online for free download and use by the mathematical community at large. Kuttler has included it as part of a myriad of excellent notes available free for download at his website on this material and more advanced topics,each of which we’ll talk about under their subject matter headings. Even more so then Heinbockel. this is a textbook that seeks to combine a rigorous presentation of calculus with its applications. The intent and nature of the book is  very well stated in the Preface:
 I have also tried to give complete proofs of all theorems in one variable calculus and to at least give plausibility arguments for those in multiple dimensions with proofs given in appendices or optional sections. I have done this because I am sick and tired of books which do not bother to present proofs of the theorems stated and either pretend there is nothing to prove, state that the proof is "beyond the scope of this book"say they will omit the proof, or worse yet give a specious explanation. For a serious student, mathematics is not about accepting on faith unproved assertions presumably understood by someone else but \beyond the scope of this book". Nevertheless, it has become fashionable to care nothing about such serious students and to write books for the convenience of those who care nothing for explanations, those people who are forced to take the course to get general education credit for example. It is my intent that this should not be one of those books.
 Any research mathematician or advanced graduate student reading Kuttler’s tirade can certainly empathize with his frustration and rejoice at his courage in writing a book which attempts, to paraphrase John Stillwell, put the guts put back into calculus. When I read his preface, it reminds me of part of the preface to the old classic theoretical text on linear algebra by Kenneth Hoffman and Ray Kunze:
 We have made no particular concession to the fact that the majority of the students may not be primarily interested in mathematics. For we believe a mathematics course should not give science,engineering, or social science students a hodgepodge of techniques, but should provide them with an understanding of basic mathematical concepts.
 The tragedy here is that in today’s textbook market, unless they had paid for the entire publishing themselves, I doubt Hoffman and Kunze could have gotten past the talking phase with their book publication. But I’m getting way off topic. My point is that Kuttler takes the same unyielding ideological stance on teaching calculushe’s giving students the Real Deal, showing them how modern mathematicians and physicists look at multivariable calculus. Kuttler emphasizes the Completeness Axiom as the basis for all of calculus. This is a good way to approach the subject rigorously since a great deal of the axiomatic development of the realsfrom which all the properties of calculus ultimately flowcan be effectively bypassed this way. He does a very good job proving the basic limit theorems of single variable calculus and introducing the central concepts of the derivative, the integral and their myriad of computational formulas and central theorems. Kuttler also emphasizes the algebraic properties of calculus more then most other booksfor single variable calculus, he demonstrates the basic algebraic skills one learned to solve polynomials can be used to calculate limits rigorously and the idea these kinds of calculations are somehow “different” is a fallacy. Even deeper, he establishes multivariable calculus/
 vector analysis rigorously by developing basic linear algebra first. Interestingly, unlike so many textbooks where it’s currently the rage, he doesn’t develop the multivariate material using manifolds and differential forms. Instead, he opts for a completely rigorous treatment of R^{n }as a vector space over the real field with the usual metric. All the same, because of the emphasis on vector space theory, the presentation number of applications, mainly from physics. Unfortunately, Kuttler’s defiance of pedigogy in giving a completely unified calculus course means the book will be very difficult for any but serious mathematics or physics studentsanyone else in a standard calculus class will be utterly lost by page 20. As much as hard core mathematics teachers would love to pretend such students don’t exist, they do and today they make up most of the clientele of calculus classes. Also, the applications are fairly similar, most are drawn from classical mechanics. One would like to see much more diversity in examples of the physical power of calculus. But these are minor quibbles. For an honors course, this is one of the best choices that currently exists and Kuttler is to be heartily commended for his efforts in attempting to give a balanced course in calculus.
 Multivariable Calculus MIT Fall 2007 (G) This is the website for MIT’s OpenCourseWare materials for the standard multivariable calculus/ vector calculus course there and the notes there really don’t contain many surprises. Again, this is the “standard” course and not really a place one would expect to find a whole lot of original or challenging material. There are some nice brief notes and lots of exercises. That being said, the supplementary notes by the legendary Arthur Mattuck do contain some thought provoking sections designed to make the students use their cerebral cortexes a bit more then the average course, such as the second on topological questions on line integrals in the plane. They’re freely availableso why not check them out? Recommended.
 Multivariable Calculus Lecture Notes Kenneth Kuttler Bingham Young University August 29, 2012 (G) This is basically a rewritten version of the second half of Kuttler’s calculus book above. The problem with that book was Kuttler’s dogged refusal to separate the theoretical section from the more practical aspects and applications. I predicted the result of using such a text on a standard calculus class would be a disaster. Apparently, I was right since Kuttler has now made an effort to separate the theoretical material into optional sections, leaving the more practical aspects for standard courses and the theoretical parts for serious mathematics majors. Indeed, Kuttler sounds rather defeated and disgusted in parts of the Preface that he had to do that because the majority of students couldn’t handle his original intent. I feel your pain, brother. Unfortunately, a textbook does no one any good if it's target audience can't understand it and like it or not, this is the academic reality we live in in America. For what it’s worth, even with the revisions, this is a strong book on vector calculus and should be given a long look by anyone slated to teach or take this course.
 Calculus I Compact Lecture Notes ACC Coolen King’s College London Version of Sept 2011 (G/PG) This is a strange set of notes from across the pond that amounts to a 135 page “bullet point” presentation of basic calculus. After an interesting first section on the history of calculus, the author proceeds to the basic precalculus material on complex numbers, trigonometry and functions, then limits and derivatives, Riemann integration and finishes with Taylor’s theorem and series. Does it look like I rushed through my description? Good, because that mirrors the presentation in these notes. Reading these notes, I was reminded of an old commercial I saw when I was a kid. I’ve forgotten exactly what the commercial was advertising, but there was a high school teacher who was teaching a horrified grade school class like this, speaking at auctioneer speed and pulling up a series of blackboards:” Basic algebra! FOIL, quadratic formula, synthetic divisiongot it? Good! Precalculus! Sine, cosine, tangent, cycles of 2II, odd and even functions, got it? Good! Calculus! Limit f(x), sum of the limits is the limit of the sums, product rule, quotient rule, infinite series, got it? Good! Next, advanced calculus....!” That’s how it felt reading through these notes. The material on the history of the subject and the intuitive followed by formal definition approach was nice in places, but the notes to me are just too compact and ridiculously terse to be useful for anything but review.
 Crowell and Slesnick's Calculus with Analytic Geometry The Dartmouth CHANCE Project 1 Version
 3.0.3, 5 January 2008 Peter Doyle Dartmouth College (PG/PG13) This is the online calculus notes developed for Dartmouth’s CHANCE program, which fosters advanced training and research in probability for both students and faculty. I’m really surprised how much I liked it. It’s about the same level as a standard calculus textbook, but much more rigorous without the subtle technicalities of elementary real analysis textbooks like Ross and Abbott. It reminds me a lot of the old classic calculus textbooks from the middle of the last century, like Moise, Silverman or Lipman Bers’ text. In short, the authors try and do everything carefully, but as simply and with as many examples, detailed explanations and proofs as possible. A good example is on the introductory chapter on integration, which uses the Darboux integral instead of the Riemann integralwhich is equivalent and much simpler to define rigorously. For all that, the book is remarkably shortyou’d expect a calculus book that does that to be a gigantic prawling monster of several thousand pages. The authors stay out of this trap by being very selective about what they coverthe text is mainly about single variable calculus only and some of it’s applications, like curve sketching,the average value of functions work and there’s an entire concluding chapter on differential equations. There is a chapter on the geometry of the plane and vectors, but it’s basically there as prelude to further study in the calculus of several variables. There are also lots of nice pictures, exercises and numerical computations. This is an outstanding online source for calculus and I was very pleasantly surprised by it. As far as I’m concerned, there are only 3 or 4 sources that are superior to it. Highly recommended to both teachers and students of calculus.
 Calculus Online Joel Feldman University of British Columbia (G) An interesting and original set of notes utilizing active Web presentations to explain crucial concepts. This is a standard single variable calculus course, but it has some decidedly nonstandard applications using Java applets for both visual eludication and numerical computations. For example, there’s a section on exponential growth where a calculational applet allows the student to compute the geometric or exponential growth of a population over time. There are also many applet graphs for the student to examine. A nice supplement to any calculus course, but probably doesn’t have enough detail to be used as a course text on it’s own.
 Exercises and Problems in Calculus John M. Erdman Portland State University (G/PG) This is a collection of hundreds of questions on calculus. They emphasize mostly calculations and theory, although there are some applications, particularly to geometry and physics. Erdman distinguishes between problems and exercises exercises are fairly easy, fill in the blank or multiple choice questions while problems are more substantial and require either longer computations or proofs. Because the standard problem collections for calculus and analysis are so expensive, it’s very helpful to have such a collection available for download. That being said, the overwhelming majority of them are far too easy for any kind of real mental workout by students. Unfortunately, in this case, you truly do get what you pay for. Still, it might be useful for beginners as drill or review.
 First Year Calculus by WWL Chen McQuarrie University Multivariable and Vector Analysis by WWL Chen (PG) This pair of notes represents one of the very best introductory analysis lecture notes available on the Web. The first set are Chen’s first year calculus notes from Imperial College in 2008 and they demonstrate precisely why European mathematics students are far superior to those here in the U.S. They stand midway in rigor between a standard pencil pushing calculus course and a baby real variables courselimit theorems are fully proved using epsilondelta arguments, and all the main theorems of calculus, such as the extreme value theorem, are proven carefully. At the same time, there are many graphs, intuitive sidebars, great exercises and applications to the sciences integrated throughout. The second part is an equally careful and well presented course in multivariable calculus. Unlike the single variable course, however, Chen begins to leave some results unprovenand this is entirely deliberate. The real strength of Chen’s notes is that he shows excellent judgement what results are reasonable for hard working beginners to grasp and prove and what concepts are simply too difficult and belong properly in an analysis course. For example, while the first part on single variable calculus proves most results, he’s somewhat looser in the multivariable calculus part, leaving the most difficult results unproven. Moreover, he restricts the development to Euclidean space without differential forms. This is in recognition that a completely modern rigorous development of multivariable calculus requires considerably more sophisticated conceptssuch as manifolds and differential formsthen he’s willing to put into a beginner’s course. That’s not to say he’s not careful in the second parthe proves most of the results of differential calculus as well as Green’s theorem on simply connected regions. But he does avoid a careful proof of Stokes’ theoremwhich it’s reasonable to say is too far beyond this course. In short, this is what every mathematician and serious math major envisions a college calculus course to look like and it’s done with enormous command, clarity and attention to detail. Best of all, when all the PDF files are merged, it comes in at 430 pages, which is positively anemic compared to the average gargantuan calculus textbook.So with the average calculus textbook, you not only have to sell your future children to be able to afford it, you can get a hernia dragging it around campus with you. Chen not only avoids both these problems, he gives a mathematically complete presentation of calculus to boot. One of my favorites and I heartily recommend it to all seriously studying or teaching calculus! Very highly recommended!
 Reform Calculus: Part I Marcel B. Finan Arkansas Tech University Reform Calculus:Part II Marcel B. Finan Arkansas Tech University Reform Calculus Part III Marcel B. Finan Arkansas Tech University (G/PG) Yup. Finan at ATU’s written 3 sets of notes for the standard calculus sequence there. They’re based on the “reform calculus” course materials from the Harvard Consortium, which uses a somewhat controversial approach to calculus that underscores rigor for educational structure. Finan does his usual very solid job of writing clear texts with many examples and lucid exposition. The books cover all the standard topics (functions, limits, derivatives, formulas for derivatives, the major differentiation theorems, the Riemann integral, the methods of washers and slices in computing integrals, techniques of integration, etc.) in a bit of an unusual manner consistent with the Harvard Consortium’s “Rule of Four” approach, which emphasizes building mastery of functions of all kinds before tackling how to manipulate them. As a result, there’s a bit more “precalculus” material then one would like in a calculus book. The only reason this is a bit annoying is because Finan has a very good precalculus lecture note set at his site we've already discussed here and it seems very redundant after that. Other then that, though, there’s not much negative I can say about the “book” these three sets of notes composeit’s wonderfully careful (including one of the best discussions of the precise definition of a limit I’ve seen in a book at this level) , there’s lots of beautiful and informative graphs and drawings, lots of applications and exercises and just a general coolness that makes the book a pleasure to read like all Finan’s other books. There is a freight car full of good to excellent choices of downloadable sources for a standard calculus course now, but this is one of the best at this level. You could do a lot worse for your students using these notes to teach a calculus courseI’d love to teach from them at least once.
 Pauls Online Notes Calculus IIII Paul Dawkins Lamar University (G)This extraordinary resource for both students and lecturers in calculus has become enormously popular among calculus teachers who are heavy into online resources over the last few years and it richly deserves to be Paul Dawkins at Lamar University has spent several years writing up and posting his lecture notes, problem and assignment sets for the 3 term standard calculus sequenceas well as several other basic mathematics courses at Lamarand making them available for download to not only his students, but the entire internet. I don’t know if he’ll continue to make these immense, comprehensive and beautifully written materials available free indefinitely, but it would be an absolute crime if at one point, he shut the whole thing down and published them as yet another bloated, 2,000 page calculus textbook priced at $200. Dawkins writes beautifully and gentlyhe has a detailed and comprehensive vision of what a basic calculus sequence should teach and what level to pitch it at. Most of the text is the usual pencil pushing calculus book, but he gives fully rigorous proofs of each of the major theorems in “Extras” sections of each chapterthis allows the stronger student to study the proofs in addition to the intuitive material, which further increases the versatility of the text. While these sections are very well done, its clear Dawkins expects most of the users of these books to be “proof adverse” or beginners, so most of the book is focused on intuition. The bulk of the book is in examples, more examples then I’ve ever seen in most textbooks. I’m a huge fan of examples and I think mathematics studentsespecially beginnerscan learn more then them then they can from all the careful proofs of theorems or graphs in the world. But it’s more then that,Dawkins has a wonderfully illustrative style that takes the beginner by the hand and leads him or her through each and every concept and shows how they are related in the great web that is calculus. An excellent example is Dawkins’ intuitive explanation of the concept of limit in volume 1:
 So just what does this definition mean? Well let’s suppose that we know that the limit does in fact exist. According to our “working”definition we can then decide how close to L that we’d like to make f(x). For sake of argument let’s suppose that we want to make f(x) no more that 0.001 away from L. This means that we want one of the following: i) f(x) – L < 0.001 if f(x) is larger then L . ii) L  f(x) > 0.001 if f(x) is smaller then L. Now according to the “working” definition this means that if we get x sufficiently close to a we can make one of the above true. However, it actually says a little more. It actually says that somewhere out there in the world is a value of x, say X, so that for all x’s that are closer to a than X then one of the above statements will be true. So just what does this definition mean? Well let’s suppose that we know that the limit does in fact exist. According to our “working” definition we can then decide how close to L that we’d like to make f(x). For sake of argument let’s suppose that we want to make f(x) no more that 0.001 away from L. This means that we want one of the following L  f(x) > 0.001 if f(x) is smaller then L
 All the standard concepts and applications are given with this level of clarity. There are also hundreds of exercises, all given with complete solutions. My one complaint is that these exercises tend to be very easymultiple choice, simple calculations, etc. Those looking for tough, brain busting calculus exercises are going to have to look elsewhere. That being said, as a learning aid and tool, Dawkins has created a pedagogical masterpiece and all students and instructors in calculus need to look at it. Very highly recommended.
 CALCULUS MATH 221 FIRST SEMESTER Sigurd Angenent and Joel Robbin University of WisconsinMadison fall 2009 June 8, 2010
 Calculus Math222 Second Semester J. Lebl Van Vleck University of WisconsinMadison 2013
 CALCULUS MATH 222 SECOND SEMESTER Sigurd Angenent and Joel Robbin University of WisconsinMadison Fall 2010 (G/PG) This is another massive set of OpenSource lecture notes for both semesters of a first year course in single and multivariable calculus from The University of Wisconsin and there are several versions available online currently. The original version written was written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. It is similar in clarity, breadth and depth to the first 2 “volumes” of Dawkins and the number of examples, exercises and graphs. But its main difference is that the presentation is quite a bit more rigorous then Dawkins,the concepts are presented with full proofs and careful definitions. They aren’t afraid to give precise definitions and theorem proofsthey are mostly incorporated into the main flow of the text except for fairly difficult “advanced calculus” level concepts, which are either omitted or presented in optional sections. There are also quite a few physical applications, mostly to mechanics. There are also a chapter in most other calculus textbooks at this level in the U.Son complex numbers and the complex exponential, for use in the chapter on differential equations. It’s a bit drier then Dawkins, but just as thorough and insightfulprobably a better choice as a beginning text for future mathematics and physical science majors. The more recent versions were modified by the instructors, who were J. Lebl, Van Vleck and Richard Kent. These later versions of the notes focus somewhat less on theory and more on applications, but they are equally well done and constructed with care, with many more pictures and computer generated numerical data. Very highly recommended.
 Calculus Chris Tisdell University of South Wales 2011 (G) The lecture notes for a standard calculus course. Fairly run of the mill in terms of subject matter and lots of gaps, but lots of good exercises, too. Useful for drill and review, but not much else.
 Calculus Hugo Rossi University of Utah (PG)These are basic calculus notes to supplement Rossi’s honors single variable and multivariable calculus course. Rossi’s reputation as a master expositor of mathematics is very much on display here. The notes are concise, but not terse as there are lots of nice graphs, examples and clear expositionparticularly with lots of terrific historical insights into the history of calculus. Rigorous proofs are in separate chapters, allowing the instructor or student to have the course be as precise as they wish. Best of all the exercises at the end of each chapter, which are exceptional for online lecture notes. In many ways, the notes remind me of calculus notes from the top universities in Europe, like Cambridge or Oxford. A excellent course, highly recommended, particularly for math and physical science majors.
 Calculus Early Transcendentals First Semester Andrew Koines Orange Coast College August 12, 2010 Calculus II Syllabus Spring 2013 Andrew Koines Orange State University
 Calculus 3 Andrew Koines Orange State University August 26, 2012 (G) These are the notes for a standard 3 semester calculus sequence. They follow Stewart’s text fairly closely, which is to say they’re all about pencil pushing, examples and calculations and leave the theory for the mathematicians. That being said, they’re nicely written and organized and they make a nice low cost option for such a course. I prefer Dawkins or Strang's classic for such a course, though. But that’s just me. Still worth a look for a standard course.
 Integral Calculus University of British Columbia YueXian Li Spring, 2004 (G) I’m a fan of most of the lecture notes available online from the mathematics courses at UBC and this is another good example. These are the notes for the second term of the 2 semester one variable calculus course at UBC, focusing on the Riemann integral on the real line and it’s various formulas and applications, such as the method of slicers and disks, work and potential energy. It includes a lot of topics that are usually just glanced over in most basic calculus texts, like a detailed discussion of finite summation formulas leading to a careful definition of Riemann sums. Also, it includes many detailed computations of complicated integrations in one variable, which are quite helpful to the beginner. Well written, carefully chosen examples and exercises and a more careful then basic courses in the U.S. Highly recommended to beginning calculus students learning integration.
 Understanding Basic Calculus S. K. Chung University of Hong Kong (G) Ever since the beginning of the 20^{th} century, there’s been a vigorous ongoing debate with mathematicians and educators about how much theory we should give in the standard calculus course. Chung has a strange but interesting solution to the central problem of this debate in his calculus lecture notes. The notes are very complete and careful in stating all the main theorems and definitions of calculus, but Chung doesn’t prove everything rigorouslyonly the concepts he thinks beginners can understand. For example, while he defines limits for both sequences and functions rigorously, most of the examples and exercises he gives are of polynomials and polynomial compositions, so epsilondelta arguments never really come up beyond that. The major flaw in the notes that makes very little sense to me in a beginners’ course is there are no physical applications. None. Nada. Frankly, to me, if you’re going to make the notes entirely mathematical with no applicationsthen why not go the Landau route and prove everything rigorously? Very nicely written, but it’s hard for me to see what they have to offer someone can’t find better done in other sources.
 Elementary Calculus and Its Applications Various Authors and Lecturers University of Kentucky (G) This is a gestalt set of lecture notes coauthored by many lecturers at the University of Kentucky for their standard calculus course. They’re for a basic calculus course and those looking for rigorous proofs are going to be very disappointed. But these notes shouldn’t be overlooked by either students or instructors since the exercisesand there are many of themare well thought out and instructive. Nice for practice and drill.
 Calculus Lectures Jerry Alan Veeh Auburn University December 2005 (G) These are notes for a standard calculus course, but Veeh’s notes have much to offer the student in such a course. The main original theme struck in these notes is how Veeh emphasizes how the classical theory of calculus is built up out of purely geometric considerations of real valued functions in the plane, with the primary concept the relationship between the horizontal tangent line to a given point on the graph of a function. There are proofs of the main elementary theorems in both differential and integral calculus, but the main emphasis is on the careful geometry of these results. It also contains many numerical algorithms that can be derived from calculus , such as the bisection algorithm as well as the standard physical applications (the projectile, etc.) The geometric emphasis gives many insights that will be very helpful to mathematics students of all levels. Recommended.
 Calculus II Integral Calculus Miguel A. Lerma Northwestern University: (G) A set of notes for the second semester of a standard calculus sequence emphasizing integral calculus and infinite series. Nicely written and similar to Xian Li's notes above in content, but not as careful and not nearly as many original examples. They follow Stewart’s text closelytoo closely to be useful as anything but review or drill in my opinion.
 Calculus for Business and Economics Kevin Wainwright British Columbia Institute of Technology August 17, 2012 (G) This really should be called “Calculus and Matrix Algebra For……” because that’s really what it is. I generally loathe courses like this because it allows nonmath majors to sidestep actual mathematics, which could actually do them some good in teaching them to think. Then again, considering how our world financial system is structured and how it’s amazingly corrupt infrastructure is always just a heartbeat away from collapsing Western civilization,these sociopaths have aptly demonstrated they don’t need any scientific training. Perhaps the world is better off without them having it.
 Calculus: Early Transcendental Functions Feras Awad Mahmoud Philadelphia University 2011 (G) A set of notes for a standard pencilpushing, problem solving, allergictoproofs course in calculus. Nicely written, lots of nice examples and graphsbut otherwise, just another Stewart clone and nothing special. Nothing you can’t get from an old copy of Stewart.
 Western Mathematics:: Calculus 1501B Rasul Shafikov University of Western Ontario 2010 (PG) This is the second half of the supplemental notes for an “enhanced” calculus course (whatever THAT means) It appears to mean they supplement the coursebased on Stewart’s textwith more theoretical material for the interest of the students, but don’t put any of that material on exams. I guess that makes sense in that without the pressure of needing to learn that material, students might be more inclined to learn it for fun. I don’t know what kind of students these are, but I can guarantee 95% of students this stunt is tried with won’t remember any of it after the course. Still, they’re nicely written and combined with Stewart, they might make a nice gentle honors calculus course. Worth a look.
 Calculus Kevin Hutchinson University College Dublin (G) This is set of lecture notes for a standard one variable calculus course. Hutchinson writes well and his notes are lucid and readable with nice exercises. Not exactly rigorous, though. Not bad, but nothing to be late for class bookmarking it on your PC, either.
 How to succeed in Calculus A nice little 4 page essay. Download it and email it to your students when you’re teaching first year calculus.(G)
 Calculus I Lecture Notes Eleftherios Gkioulekas University of Texas Pan American
Calculus 2 Lecture Notes Eleftherios Gkioulekas University of Texas Pan American  Calculus 3 Lecture Notes Eleftherios Gkioulekas University of Texas Pan American (G/PG) These are the complete set of handwritten lecture notes by Gkioulekas on calculus listed here for the 3 semesters of the standard calculus sequence at U of T Pan American. The 3 sets together make an outstanding complete online calculus text for serious mathematics majors that weren’t quite ready for a fully rigorous honors calculus/ analysis sequence yet. Most calculus notes and texts in the U.S.fall into either of 2 categories: A simple pencil pushing calculus course where rigorous proofs are either avoided completely or at best minimized in favor of physical applications and intuitive geometry or an honors course whose model is Spivak’s Calculus and is the Jungian shadow of the formeri.e. basically an analysis course with complete rigor and careful proofs with no applications or pictures. There are virtually no “middle ground” textbooks that attempt to present the basic material and the physical applications carefully and yet at a level suitable for beginners. Gkioulekas gives a breath of fresh air and ensures the set of middle ground textseither in print or onlineis nonempty. The same care and rigor is present here that is present in all his notes legibility, careful definitions, many examples, exercises and graphs. The Calculus 1 notes cover all the standard topics of single variable calculus up to and including the definition of the Riemann integral and with both precision and applications: basic inequalities, functions and their domains, limits and continuity, limits at infinity, derivatives with their applications and formulas, the differential and finally, integration. There are many graphs and examples and the author gives many proofs and careful definitions, although not with the level of rigor of a full analysis course. For example, he defines elements of naïve set theory and uses basic logic, but he does not give the definition of a function in terms of ordered pairs. Interestingly, he develops the definition of a limit using the topological definition of a neighborhood in R and then gives the usual εδ definition. The second set of notes continues the style of the first, covering techniques of integration ,improper integrals,convergent sequences,series convergence and power series.It ends by laying the groundwork for calculus of several variables by discussing parametric curves The final set of notes is no less careful and well written, it develops basic multivariable calculus. The vector geometry of the plane and 3space is developed carefully with the properties of the cross product in particular laid out in greater detail then usual, with the foundations of the differential and integral calculus laid on top of it. Many of the basic concepts of linear algebra are laid out subtly as well. For example, the orthogonality relation between 2 vectors and their cross product in 3space is used to define linear independence in 3 space. Topology in the plane and 3 space is developed as needed and done carefully, particularly to state the definitions needed for integration. This is an outstanding trilogy of notes for a complete undergraduate calculus sequence, one of the very best I've seen. They will be very helpful for students and instructors in calculus courses who aren’t quite ready for a brutally rigorous course, but nevertheless want a more meaty development then the usual plug and chug course. And the author's handwriting is exceptional, which is surprisingly important in scanned notes. Highly recommended.
 Calculus Spring 2012 (Cohen) University of North Texas (G) A quick, concise rapid standard course in the calculus of one variable. Think of it as a Cliff Notes version of a calculus course. Not useful for anything but review
 Calculus I Alissa Susan Crans Fall 2010 Loyola Marymount University (G) A meaty and detailed set of notes for a standard calculus course in one variable. Very geometric and problem solving oriented, lots of graphs and insightful comments by the author,including historical notes and very original intuitive discussions accompanying proofs, such as geometric motivation for the proof of the product rule of differentiation. It also has many unusual physical applications, such as computation of blood flow velocity. Not as comprehensive as Dawkins, but worth checking out. Recommended.
 Calculus 2 Hamburg University of Applied Sciences Robert Hess Summer term 2013 (G) Very unusual set of notes for the second semester of the calculus sequence at this wellrespected university for applied mathematics students. Looking at the source, you’d think these notes would be purely pragmatic in nature with no proofs. And in a sense, you’d be correct, but that would also be misleading. Although there are no proofs in the rigorous mathematical sense, there are many careful definitions and many geometric pictures motivating why statements are true. Although I wouldn’t use it by itself, I think would certainly be useful as supplementary reading to an abstract treatment such as the notes from Oxford or Yeh’s notes.
 Calculus Refresher, version 2008.4 Paul Garrett University of Minnesota (G) Exactly what the title says
 it is, by a master teacher. Gives an overview of all the main concepts of calculus, as either supplement or reviewthey aren’t really intended to replace a standard calculus course, but to enrich it. The notes really aim to not only review these basic concepts, but to give insights that are usually missing. For example, check out Garrett’s wonderful discussion of the role of the intermediate value theorem in the convergence to a solution of Newton’s Method. Indeed, a great deal of Garrett’s presentation is illustrated by numerical methods and examples in calculus. Since numerical methods are so critical in applications, this provides an informal but very insightful perspective, with Garrett’s usual chatty style. Highly recommended for either students or prospective teachers.
 Algebra and Calculus RM Bryant And C.T. Dodson University of Manchester (G)An applied course for engineering majors. Brief and to the point, but nice and readable. Again, nothing to write home aboutwhich is rather disappointing since both Bryant and Dodson are excellent mathematicians and teachers, judging from their other work.
 Multivariate Calculus in 25 Easy Lectures Bruce E. Shapiro California State University Northridge Fall 2006 (Revised: December 6, 2006) .(G) I’ve always liked Shapiro’s lecture notes, which vary greatly in subject matter, from baby calculus to discrete mathematics to advanced ordinary differential equations to plane geometry for teachers. They’re well written, insightful, humorous and lively; he really wants to entertain as well as educate and that’s a great skill for a teacher to have. As an educator, when your students are your willing partners in the task at hand and not your captive audience, that gives you a huge advantage from the outset. These are his notes for his vector calculus course and they’re excellentfor what they are. To use Shapiro’s own words from the introduction: These lecture notes are meant to accompany the textbook, not to replace it.It has been my experience that students who attempt to use lecture notes (even mine, which I have to admit, are practically perfect) in lieu of the textbook receive abysmally poor final term grades. However, properly using the lecture notes can help you. The notes don’t have enough detail to function as a full blown textbook on vector calculus, but they will work beautifully as a study guide and review to Micheal Corral's notes. The standard materialvector algebra in
 R2 and R3 , the dot and vector product, CauchySchwartz and triangle inequalities, vector valued functions and their derivatives and integrals, curves, lines and planes in R2 and R3 , the geometry of surfaces, the total derivative and differential, line and surface integralsis convered very nicely and carefully, with lots of pictures. There’s nothing original or unexpected here, but for a course like this, you really don’t need it to be. Not every result is proven, but lots of examples and computations are given. Some classical differential geometry and applications to mechanics are given. A great exercise for anyone reviewing the material would be to take a pen and paper and try and fill in all the proofs. In any event, a very good set of notes to have on hand when learning or teaching this material.Recommended.
 Applications of Calculus University of Sydney MATH1011 (?) Exactly what it says it isand home to one of the worst, most illegible scribblings that have passed for scanned handwritten lecture notes I’ve ever seen. I sure hope the students in this course down under didn’t miss any lectures to take their own notes because these aren’t going to be of any use except to wipe your butt in the toilet. Run, run as fast as you can………
 Calculus III Angel V. K umchev Towson University (G) A set of lecture notes for a standard vector calculus course. As usual in a course at this level, the more difficult theorems of multiple integration and line/ surface integrals are stated and not proven. Again, nothing earth shaking here, but the material here is very well presented, complete with exercises, lots of examples and lots of wonderful MATHEMATICA constructed drawings of curves and surfaces. Unlike Shapiro, these notes are complete enough to use as a textbook for a vector calculus course and combined with Shapiro as a study guide, they would make a wonderful pair to base either a course on or for self study. Recommended.
 Calculus II Robbie Snellman University of Utah 2012 (G) Some nice handwritten notes on integration theory, L’Hopital’s rule, infinite sequences and series and conic sections. Again, pretty standard “Calculus II” stuff, but nicely written and clearly scanned.
 Vector Calculus Stephen J. Cowley University of Cambridge 2000 Mathematical Tripos: IA (G) You gotta love the Tripos lecture notes. Some of the best written, authoritative and frankly anthropomorphic lecture notes that currently exist on the Web. The interesting and somewhat frustrating thing about the Cambridge Tripos notes is that the Vector Calculus notes, while more advanced then the standard U.S. course, requiring linear algebra backgroundsthey are almost never rigorous courses, only applied courses for physics and engineering majors. Fully rigorous treatments of multivariable calculus/vector analysis is given in the analysis courses at Cambridge and Oxford. Still, the treatment by Crowley is very careful, fully utilizing linear algebra to give a sophisticated treatment and there are so many applications here, all given full treatments with an enormous number of solved examples. For people like me who refuse to distinguish theory from applications in mathematics, such notes are a treasure and its well worth the effort to supplement them with fully rigorous treatments Very highly recommended to all. Multivariable Calculus G.L. Luke University of Oxford Notes of lectures October 14, 2007 (G) Another applied course in this very classical and very important subject, emphasizing vector fields and their applications to concrete physical problems. Similar to Crowley but not as careful, gives fewer insightful examples and has fewer prerequisites. On the positive side, though, they’re nicely written,have many more pictures and strong, detailed discussions of classical applications to physics, particularly to electromagnetism and mechanics. Recommended for physics and applied mathematics courses. It can also be used for pure mathematics students as a good supplement or a purely mathematical treatment
 Vector Calculus Thomas Baird University of Newfoundland 2010 (G) A nice, comprehensive set of notes on the subject with many examples and computations. Linear algebra is assumed and used throughout, so students would need to bring this knowledge. Sadly, many proofs are either skipped or merely sketched. Still, there’s a lot of good stuff here, including applications to partial differential equations and many coordinate transformations. Balanced with a more rigorous presentation, they will be very useful to a student. A friend of mine used it as a supplemental text with Spivak’s Calculus on Manifolds and had great success mastering the subject. Solid if not exceptional. Recommended
 .The Calculus of Functions of Several Variables by Dan Sloughter, Furman University (G) Dan Sloughter has written several useful online texts for his courses, all available at his personal website t synechism.org., all on calculus and analysis. They vary in the level of rigor, but they all attempt to be as careful as possible. This is a standard course in vector calculus set in R^{n} with the usual topics (geometry and functions in R^{n}, limits, differential calculus in R^{n} , parametrized curves, curvilinear coordinates,extrema, etc.) but with modern terminology and rigor. Linear algebra isn’t explicitly used, but linear functions, the usual metric and norm in R^{n }are introduced from the outset and used throughout. The notes are extremely geometric, the lower dimensional case is used throughout to illustrate every concept with lots of examples and all theorems are given careful proofs. Physical applications are present, again, mostly to classical mechanics. The sections on linear functions and multiple Riemann integrals are particularly well done, with computer generated level sets and cross sections of the domains of integration. However, Furman doesn’t just roll over the student with
 brute real analysis the way some courses like this dohe keeps in mind the level of student and instead of sacrificing rigor for a “soft” presentation, he simply omits topics he thinks are too difficult to prove. For example, he spends the entire final chapter giving a careful proof of Green’s theorem and completely omits Stokes’ theorem and surface integrals. This is a reasonable omission in a course pitched at the average undergraduate mathematics major and I’m surprised more people don’t do it. Overall, a very good online text for vector calculus. Recommended.
 Yet Another Calculus Text A short introduction with infinitesimals by Dan Sloughter, Furman University (G) This is a really bizarre text available from Sloughter’s site that frankly baffles me by it’s existence. It’s basically a standard calculus text using Robinson’s formulation of the hyperreal number system and infinitesimals. I’ve never really understood the allure a lot of mathematicians have for this funky construction. I think part of it is they’re incredibly impressed with Robinson using mathematical logic to achieve something most of us were told for 2 centuries was impossible making Newton’s “fluxions” nonsense rigorous enough to serve as a foundation for calculus. To be honest, I don’t know much about it, but every time I’ve seen this stuff, it seems at least as confusing to a beginner as Dedekind cuts,epsilons and deltas and I don’t see any clarity gained. But maybe it’s just the perspective of an amateur. I do try and keep an open mind about such things, but to be honest, I think since most of us live in the WierstraussCantor mathematical universe, anyone learning calculus via Robinson’s alternate universe is going to have to relearn it all using our language, anyway. So what’s gained here? That’s my questionand until I have an answer, I’m refraining from judgement. Hey, it’s my site, sue me.
 Vector Calculus Wayne Hacker Pima Community College (G): A very nice set of handwritten, informal notes that cover the first half of a typical multivariable calculus course. Lots of examples and calculations. Unfortunately, nothing you can’t find in 100 other places. Worth a look and nice to read for beginners, but nothing special.
 Calculus I – Notes Nakia Rimmer University of Pennsylvania (G) Calculus lecture notes, syllabus and exercises for a summer course in freshman calculus. Limits, derivatives, formular for derivatives, definite integrals, blah blah blah. The usual spiel, nothing unusual or original here. But you may like them, so by all means, give them a look.
 First Year Calculus For Students of Mathematics and Related Disciplines MichaelM. Dougherty and John Gieringer (PG) I don’t know where to begin with this beautiful book. According to the preface, the senior author Dougherty was inspired to write this book by the textbook by Richard Hunt. I was completely unawares of Hunt’s text before reading the preface to this book, for which I am quite startled and ashamed. If this draft didn’t exist online for free, I’d fully and heartfeltly recommend finding an old copy of Hunt’s book immediately if you’re serious about calculus. Fortunately, thanks to Dougherty and Gieringer, I don’t have tothey’ve written a calculus book that’s even better. In many ways, the book reminds me of the older classics of calculus, such as Courant or the early editions of Thomas’text, in that both theory and applications are present and students need to be able to prove things to really learn anything from the book. The tragedy of calculus in recent decades is that the “corporatization” of both education and book publishing have nearly removed the theory of calculus from the overwhelming majority of calculus classes and books and regulated it to honors classes only. It’s a delight to see some mathematicians forming a resistance to this forming academic collective hive mind, dismissing the ridiculous idea that calculus has to be presented in an either/or manner when for most of it’s history, it was not presented in that way. What’s more interesting is that the authors’ have constructed their own unique take on how to present a rigorous take on calculus to all students of a scientific interest. You only have to look at the first chapter to see that this is a course that’s mathematically seriousthe first chapter is about the basics of logic. Truth values, connectives, truth tables, negations and how all these elements lead to careful, logical reasoning is done before the authors do anything else. The more I return to logic in my own studies and life, the more useful I realize it is for beginners to have at their disposal until they develop some experience with rigorous proofs. It’s also useful for experienced students who can use it as a tool to check their reasoning. Moreover, it gives the authors a tool for stating results precisely and concisely without sacrificing clarity. The basics of set theory are presented at the same time, using logical language. This will be very helpful to mathematics majors who go on to study axomatic set theoryindeed, if we made this mandatory at this stage, we might not have to separate “naïve” and “axiomatic” set theory. From there, the presentation begins of calculus proper. The authors give a 13 axiom definition of the set of real numbers as the smallest ordered field that contains the rationals. To an analyst, this might seem like a pedantic construction where fewer axioms are strictly needed, but the authors remember these are beginners and they’ve opted for a construction which they deemed simplest. And this is very typical and brilliant of how the authors have designed their textthey take the path through the material they think will be more comprehensible to their students without sacrificing rigor, regardless of how long or unusual that path may look to an expert. For example, they discuss continuity before giving a rigorous definition of a limit, with many calculations,graphs and examples and both done with epsilons and deltas. This makes complete sense as the abstract idea of continuity is considerably simpler then the abstract definition of limit since functions are always defined at the limit by definition. I’m surprised no one’s thought of this approach before. Another example comes later in the book when integration beginsthey study antiderivatives completely and the basic techniques of integration, such as substitution and integration by parts, in detail before giving a careful presentation of Riemann sums and limits of integrationin which they use the many techniques the student has just learned to give many examples of the limits! They then proceed to develop the whole of single variable calculus in equally carefully and rigorously a manner. But this is not a purely mathematical treatment eithermany applications, mostly from physics are considered and done quite well. There are also many geometric aspects of calculus that are discussed, such as translation and rescaling, curve sketching and analytic geometry. Lastly,there are also many discussions of numerical methods,which connect the rigorous definition of limits to the physical applications, showing the 2 are not disconnected, as many mathematicians treat them as. This is simply one of the best all around calculus books I’ve ever seen. I mean that, I’m not exaggerating. Please download it, read it, savor it and use it to teach your students. You’ll be glad you did, I promise. I give the highest possible recommendation to this wonderful work.
 Calculus Problems Jerry Kadzan University of Pennsylvania (G/PG) A very nice set of exercises by Kadzan on both single and multivariable calculus, running the gamut from calculating derivatives and integrals to epsilondelta limit computations to lengthy proofs of theorems. Most are more challenging then the average calculus textbook and their diversity makes them extremely useful for both students and teachers. Highly recommended.
 Abridged Calculus by Kenneth Kuttler Brigham Young University 2010 (G) This is a standard second semester calculus course Kuttler extracted from from his calculus book aboveremoving all the difficult theory and hard proofs, condensing the discussions,removing many side topics and basically paring it down to what’s essential for the course only. The result is a pale shadow of his other works on calculus that covers basically some differential calculus, the techniques of integration and infinite series. Still, it’s nicely presented, there’s lots of nice pictures and it moves along concisely at a nice clip. Good for review or for a teaching a plain vanilla summer course
 Calculus Revisited: Single Variable Calculus  MIT OpenCourseWare (G) Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshmanlevel introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time.This fascinating resource is one of the creative jewels of the MIT OpenCourseWare site. From the Website: Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshmanlevel introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time.Nothing about this terrific resource has changed in the intervening 4 decades. It basically consists of a series of old school videotaped lectures by Herbert Gross at MITthese lectures in black and inceptionall of them can be found on YouTube as well, incidently.Check it out and prove to yourself newer isn't necessarily better.
 Vector Calculus Michael Corral Schoolcraft College (G) Yet another “Calculus III” textbook online with the usual topics and the usual presentation: Analytic and vector geometry, parametric curves and curvilinear coordinates, partial derivatives, the total differential, multiple, line and surface integrals, applications to physicsyou know, the usual song and dance. There’s definitely a glut of books online about this important subject and it demonstrates the real demand for it among undergraduates. Corral, as we saw with his lower level precalculus textbook, is a very good writer and teacher, the book is careful, nice and readable with good pictures, But there’s nothing you can find in here you can’t find in Slaughter or Crowley or any of a dozen other sources onlineand with the Dover edition of Bendaxall/ Liebeck, you get all this and rigorous proofs, too. It’s ok, but there’s plenty of merely “ok” sources out thereI’d rather have either Cook's notes below or Bendaxall/ Liebeck.
 Calculus James S. Cook Liberty University Fall 2010
 Multivariable Calculus James S. Cook Liberty University Fall 2011 (G) This 2 year sequence is one of a number of lecture notes on various undergraduate mathematics courses available at Cook’s website. I’m a big fan of Cook’s notes because while he believes that rigorous mathematics needs to be an essential part of every mathematics course, the amount of rigor used needs to be considered carefully and tailored to the level of student in the audience. His calculus notes are a terrific example of this. They are supplemental notes to James Stewart’s infamously ubiquitous text, but they are detailed enough to act as a text on their own. ( I’ve spoken in the past at my blog at length about my own feelings about that textI plan on posting an updated and expanded commentary at the new blog in the future. I’ll sum up my feelings as follows: My problem with Stewart’s text is that it’s a textbook on practical problem solving masquerading as a calculus text. It’s not really about calculus, it just uses calculus as the McGuffin It could have been about anything elsecombinatorics, algebra, physics, chemistry, bread baking, doesn’t matterthe number of exercises and the problem solving techniques would have been unaffected. ) They are intended to put the mathematical “guts” back into Stewart’s book, while still preserving the remarkable problem solving and pictorial approach to calculus concepts which is the real and obvious strength of that text. In a lot of ways, the union of Cook’s notes and Stewart’s text form one of the best all around textbooks for a full 3semester course in calculus that current exists in either print or online. But read the preface to the single variable courseCook is a mathematician and he teaches calculus as mathematics. Period. That being said, he doesn’t go overboard on the rigor and turn the class into a real analysis course. That’s the problem a lot of professors trying to put the guts back into calculus makethey go way too far and forget it’s a general purpose calculus class and not an honors course. Cook doesn’t do that. For example, he doesn’t develop the real numbers and instead gives a rigorous review of trigonometry, including advanced sections on the complex exponential and periodic function decomposition. He gives a very lengthy presentation of the concept of limits, both the intuitive definition and a fully rigorous definition, both of which he wants the students to knowbut he accompanies the discussion with many detailed computations with epsilons and deltas. The discussion of differential calculus largely mirrors the one in the text, but it has more counterexamples and rigorous insight. The notes on multivariable calculus are equally excellent and rigorouswhile he doesn’t explicitly use linear algebra, he does develop the geometry of R^{2} and R^{3} in enormous detail with many examples as well as the basic metric topology of those spaces in terms of open and closed balls. He then proceeds to develop the multivariable differential and integral calculus in greater depth then Stewart even dares, even motivating the study of manifolds and topology without giving explicit proofs. In addition, there is a host of calculational examples and applications to the sciences, sometimes developing Stewart’s examples in more depth. The entire presentation is beautifully illustrated as well. The result is one of the very best texts on calculus I’ve ever seenjust the right level of rigor and enormous insights to motivate budding students’ interest, if they’re willing to do the hard work involved. Here’s hoping Cook one day revises and expands the notes into a full blown calculus textbook and it remains available online free for download. Very highly recommended.
 Calculus II John C. Bowman University of Alberta Edmonton, Canada November 9, 2012 (G) This is a course for the second semester of a 2 semester course on calculus designed for first year preengineering students. It covers techniques of integration, parametric curves and curvilinear coordinates, elementary differential equations and their solution methods, infinite series, vector calculus and geometry in space and surface area computations. As a result, you’d expect a minimum of mathematical proof and a maximum of focus on practical methods of solution and computation. And you’d be spot on. Still, there’s a lot of nice discussion and solved problems and this could be very useful for students who need to work on their computational skills.
 The Free Speech Calculus Text Various authors, Gnu Free Documentation License 2004
 User’s Guide to the Free Speech Calculus Text Version 0.5 W.Ethan Duckworth2005 (G/PG) Ok, this is a strange community calculus book project that was one of the first online calculus books begun in 2004. It was put together in a manner similar to a Wikibook over a period of several years by a number of people, including free contributions from other authors over the internet. The goal was to make a calculus text that was all things to all people, but written in a style all students could read and understand. It indeed is written in breezy, patchwork style that coversin one form or anotherjust about all of one variable calculus and some multivariable calculus, with lots of pictures, examples and exercises. It even comes available with a lengthy user’s guide. But to be honest, I’ve never been a big fan of the book. I think the patchwork nature of the book results in an incredibly uneven presentation and some sections are basically a mess. The 2 worst sections are the section on rigorous foundations, which attempt to explain the upper bound property, which I found utterly confusing, and the second on rigorous limits, which gives the formal definition and some nice calculationswhile the whole time explaining most people do calculus without the rigorous limits since “human minds can’t understand infinite processes.” Huh? This would be bad enough, but even the less formal sections can be messy like this. For example, the section on the integration of vector functions may leave students scratching their heads. Some of this is to be expected in a joint effort work like this, but the book’s all over the place at times. A well intentioned but ultimately flawed work that hopefully will get a dramatic revision one day.
 Functions and Calculus Oliver Knill Harvard College/GSAS Math 1a, Spring 2012,(G) This is a concise but clear set of notes for a first semester freshman calculus course at Harvard, covering all the basics with quite a few pictures. Nice and clear, but lacking detail. If you like that kind of presentation, then these might be for you. I like them, but I wouldn’t use them as my sole source for the course. Recommended.
 Calculus I Math 150A Lecture Notes Bruce E. Shapiro California State University,Northridge Last Revised: November 4, 2012 (G)This is Shapiro’s supplementary notes for the single variable calculus course at Northridge. What I said for his multivariable calculus course notes above goes almost verbatim for his single variable coursea wonderfully clear and lively set of notes that while they don’t have sufficient detail to act as a text, they can make a wonderful study aid for students. They’ll also make a terrific review in the basics of one variable calculus for students who need one. Highly recommended.
 Multivariable Calculus Oliver Knill Harvard University Summer 2012, (G) This is the Summer School Second Year Calculus Course taught by Knill at Harvard. Like its predecessor above, the notes are concise and terse, yet very lucid,But in addition, these notes have many unusual additions, such as partial differential equations and the history of multiple integrals. Again, a strong set of lecture notes for any multivariable calculus course, but lacks details to be used solely. Recommended
 Multivariable Calculus Lecture Notes1 Y. L. Wong Department of Mathematics, National University of Singapore, (G) These notes are quite similar to the multivariable second half of James Cook's notes here. They are based closely on Stewart as well, but like Cook, they are more careful then that text. Unfortunately, they are much closer to the original text then Cook and don’t add much to the presentation there except a few definitions and examples. The presentation of Stokes’ theorem is somewhat more careful then Stewart, but not much. Still, they’re quite readable and present the material well. A solid set of notes for a standard “Calculus III” course, but they were fairly disappointing to me in the end because there’s nothing in here you can’t get from some of the other sources listed here.
 Multivariable Calculus Cornelia Drutu, based on notes of Barbara Niethammer Oxford University 2010 (G) This is a fairly short but very rigorous and careful presentation of the calculus of functions of several variables in R^{n} –it has linear algebra, basic analysis and topology as prerequisites.This allows the author to jump right into differential calculus without any preliminary development of the geometry of R^{n } . Differential calculus is developed as the study of the linear transformations from R^{n } into its subspaces that define the total derivative, partial derivatives and the differential. The notes discuss the implicit and inverse function theorems as well as the beginnings of basic differentiable manifolds. There is no discussion of integration theory, which is the main failing of these notes. They are very well written, have a lot of examples and will make a very nice supplement to an honors or advanced calculus course on the subject despite their failings. Recommended.
 Calculus Jason Starr MIT 18.01 Single variable Calculus Fall 2005 (G) These are summaries of Jason Starr’s lectures on single variable calculus at MIT in 2005. They are extremely terse, almost in a “bullet point” style presentation of all the facts. They’re similar to Coolen’s brisk “Compact Calculus” notes from King’s college above, although with somewhat more detail and clarity. Still, it’s hard to imagine these being useful for anything but review in their current form.
 Calculus and Discrete Math Course Materials Jerry Griggs University of South Carolina (G) This is a set of handwritten notes to accompany a standard multivariable calculus course. For the most part, they’re pretty standard and follow Stewart fairly closely. Not bad, but nothing really worth saving on your laptop, either.
 Vector Calculus M. Dorrzapf University of Cambridge Lent Term 2006 (G) Yet another set of Cambridge “ applied vector analysis” notes that avoid hard proofs and instead focus on geometry and applications in vector analysis of R^{n }^{ }It’s pretty amazing how many of these are available at the Cambridge and Oxford sites and how incredibly readable and useful they are to students, even without proofs. Again, similar to Crowley, with the same sophistication, care and ton of examples. But it’s more geometric and in some ways, more comprehensiveit concludes with a full chapter on tensor analysis in R^{n } .Very useful to mathematics and physics majors.
 Dr. Vogel's Gallery of Calculus Pathologies Thomas I. Vogel Texas A & M (PG) A beautiful and very useful brief supplement to a 3 semester calculus course, it’s exactly what the title says it is. Each counterexample comes with a MATHEMATICA generated graph in 2 or 3 dimensions, so the student can see visually where these freaks of mathematics go wrong and violate the basic theorems of continuity and differentiability. They won’t replace a comprehensive source on pathologies in analysis like Gelbaum and Olmstead, but you rarely see careful discussions of these counterexamples at the elementary level and it shows a serious calculus course at work. Highly recommended.
 A Summary of Calculus Karl Heinz Dovermann University of Hawaii 2003 (G) Exactly what the title says it is. All the basic definitions and theoremslimits, derivatives, integrals in single variable calculusboth locally and globally (at a point and over an interval) –presented concisely, rigorously and with little chit chat. That being said, Dovermann does go out of his way to make sure that the few examples he does give are deep and interesting and give insight into applications of the material to real world situations. For example, in the discussion of optimization, he gives a lucid and fascinating discussion of the problem of finding the optimal size of an evolved organism in a specific climate as a function of whether then animal is warm or cold blooded. I don’t think it has enough detail to use by itself as a text for a calculus course, but I would certainly recommend it as collateral reading for such a course or a review source.
 A Touch of Calculus Karl Heinz Dovermann Professor of Mathematics University of Hawaii (G)This is an even briefer set of notes by Dovermann which is designed for nonmathematics majors to emphasize the intuitive concepts of calculus. In many ways, it’s the Jungian shadow of it’s longer older brother notes aboveit has careful definitions, but virtually no proofs and lots of graphs and examples of applications of calculus. It looks almost like Doverman performed surgery on a short, rigorous calculus textbook, extracted all the precise, formal content into one set of notes and left most of the examples and applications in the other. I still don’t think their settheoretic union would form a sufficiently detailed presentation to constitute a calculus text proper, but that union would certainly be useful to accompany any such course. And and these notes by themselves could certainly supplement a completely rigorous treatment like Spivak or Kupferman above. Recommended.
 Calculus of one Variable Jerry Shurman Reed College (G/PG) This is a very interesting and original set of notes by Shurman and they should be read by anyone interested in either teaching or learning calculus. They develop one variable calculus in a strikingly original way, based on the original lecture notes for the calculus course written for Reed College’s calculus course by Ray Mayer, which can be found here. The guiding central concept is integration, developed in a purely geometric, almost Newtonian fashion. The central point of the development is to show how integration and differentiation are related geometrically and then to build the rigorous definitions of the limits motivated by these ideas. Shurman explains his reasons for such a development in the preface and in doing so, gives an excellent description of the work that follows:
 The presentation is meant to defamiliarize calculus for those who have seen it already, by undoing any impression of the subject as technology to use without understanding, while making calculus familiar to a wide range of readers, by which I mean comprehensible in its underlying mechanisms. Thus the notes will pose different challenges to students with prior calculus experience and to students without it. For students in the first group, the task is to consider the subject anew rather than fall back on invoking rote techniques. For students with no prior calculus, the task is to gain facility with the techniques as well as the ideas. These notes address three subjects:
 * Integration. What is the area under a curve? More precisely, what is a procedure to calculate the area under a curve?
 * Differentiation. What is the tangent line to a curve? And again, whatever it is, how do we calculate it?
 * Approzimation. What is a good polynomial approximation of a function, how do we calculate it, and what can we say about the accuracy of the approximation? Part of the complication here is that area under a curve and tangent line to a curve are geometric notions, but we want to calculate them using analytic methods. Thus the interface between geometry and analysis needs discussion. The basic pedagogy is to let ideas emerge from calculations.
 In a way, Shurman mimics the historical development of the subject by motivating all definitions by beginning with an “Archimedian “ style of areas and volumes and using that geometric machinery to extract the basic framework of differentiation. All definitions are completely rigorous and motivated by plenty of geometric examples, primarily of “so called” area functions which are then converted to limiting quotients that “stand in” for a formal limit of a derivative. There are a very large number of applications, mainly to geometry and classical physics. Indeed, by its very nature, the notes are much more geometric then most modern presentations of calculus. What’s really interesting is that he shows excellent judgement about what theorems to prove and which to leave unproven. He does this by a really simple and enlightened criteria: he only proves results that are strictly calculus theorems and only require calculus concepts and definitions to prove. 2 great examples that help to illustrate what he means by this are the intermediate value theorem and the extreme value theorem. The intermediate value theorem is an almost trivial consequence of a much deeper result from the full construction of the real numbersnamely, that the real line is connected as a topological space. Similarly, the proof of EVT is a consequence of the Least Upper Bound property of the real numbers. Both of these facts are, of course,equivalent to the completeness of the reals. Therefore, both rely on a full construction of the reals and therefore go beyond calculus. (While it’s true the IVT can indeed be proven using only the ε δ definition of calculus and an argument by contradiction, the resulting proof is extremely lengthy and ultimately does rest on the completeness of the reals in another formnamely, the fact every Cauchy sequence converges on the real line.) The result is a stunningly lucid, original and very informative presentation of calculus and one I’d strongly recommend to anyone. It’s near the top of my list of favorites here and for geometers, this will be a Godsend in your calculus teaching. Very highly recommended.
 Fedor Duzhin Calculus of Several Variables 2007–2010 NTU
 Fedor Duzhin Calculus of One Variable 2006 Calculus I NTU Exam questions 20062008 (PG) NTU has become one of the most prominent technical universities in the Far East in the last decade, as much for the quality of its training as for its research. This pair of notes, written for the 2 year calculus sequence there, is an outstanding example and one of the best available sources on calculus listed here. The notes are forth most part rigorous and all the basic concepts of single variable calculus in the first part are proven carefully. Limits in particular are developed in great detail via the epsilondelta definition with many examples. Many calculated examples and beautiful graphs, many of them computer generated, are given to accompany the rigorous proofs and definitions. There are also many physical examples and they’re usually put in separate chapters from the rigorous proofs and definitions. This gives the instructor or student some flexibility on how much they want to focuson careful proofs and how much on applicationsbut it would be a real shame to skip the careful proofs since Duzhin does such a good job presenting them at a level suitable for beginners. In fact, the notes are quite similar to Chen’s notes above in terms of content and styleexcept they are quite a bit more detailed with many more pictures. Like Chen, Duzhin shows excellent judgement what to prove and what to leave unproven for a rigorous course that beginners can handle. This and the great depth and visuals make Duzhin’s notes one of the very best sources available online for calculus. Very highly recommended.
 Calculus 1st edition by Gilbert Strang MIT (G) The great classic by one of the greatest living applied mathematicians and teachers, available free for download. If you dream of writing a calculus textbook for good beginning students that’s careful but not rigorous and emphasizes the real world significance of the subject in a way that connects them to current researchyou absolutely need to read this book because no one has written a better version of such a calculus textbook then Strang. Beautifully written,illustrated and there’s probably no better source for applications in calculus for any student or instructor.There’s simply nothing more that can be said about this jewelexcept to warn you not to spend money on the more recent 2 nd edition. It bears virtually no difference to the 1 st except for a summary chapter on calculus, called Highlights of Calculuswhose original live lectures are available at Strang’s webite at MIT. Don’t botherjust go there and watch Strang’s awesome lectures on calculus and linear algebra and download the first edition.
 Mathematics3 Calculus African Virtual University 2012 (G)This is another of the interesting course note packets from the African Virtual University and has no real analogue in the traditional Western courses in calculus.It takes, in addition to a good background in school algebra and geometry, basic naïve set theory and some exposure to mechanical calculus in high school. It isn’t designed as a full course text, but rather as as supplementary workbook for the calculus course. Unfortunately, the original recommended online textbook G.S. Gill’s The Calculus Bible, seems to have disappeared recently from the internet after several years online. Still, there are even better sources listed here that can stand in easily. This packet gives precise definitions as the expected background for the course and defines everything rigorously, with many exercises and original examples that are hard to find in other textbooks. Interestingly, virtually all proofs are missing, apparently to be filled in by the students. An original and intresting work, but due to it’s unusual organization and choice ofmaterial, it’s probably best used as a workbook for students. As such, I suspect it would be very helpful, particularly to applied mathematics students who need to sharpen their basic proof skills
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 Calculus CK12 Raja Almukkahal, VictorCifarelli, ChunTak Fan, and Louise Jarvis (G) This is the calculus book in the previously mentioned STEM series for grade and secondary school students in honors courses. This is the calculus course for advanced students in the United States. It’s interesting to note that in many other advanced countries whom we’re trying to catch up to, this is the standard material that would be required for all final year secondary school students about to enter university. In short, this is a “practical” calculus course that emphasizes problem solving over theoretical proofs,with all the basic concepts of calculuslimits, derivatives and integrals in both one and several variablespresented mostly informally, but intelligently and with very substantial connections to the geometry of the plane and 3space. The book isn’t devoid of rigor, but clearly the emphasis in on solving problems. There are literally thousands of diagrams and fully solved problemsas well as a host of applications. What’s interesting is that the emphasis seems to be on numerical applications, such as the method of least squares, which usually aren’t found in standard calculus courses.This increases the utility of the text with both computers and programmable calculators. There are some strange things in the book,though for example, they leave proving the intermediate value theorem as an exercise. This would be a challenging problem for a course at any level in calculusbut even more so, it would be impossible without at least stating the Completeness property of the reals in some form and as far as I can tell, it’s not stated in this book at all. Thinking maybe it was mentioned in the prerequisite books, I did a throrough check of the CK12 site. Goose eggs. Nada. Zip. Somehow, I doubt it was a mistakefor that to be a reasonable problem, the authors need to make certain that subtle property of the reals is covered somewhere in the cirriculia and reviewed.This unfortunately is a common failing of such collective author efforts where each part is connected to the othersthe lack of a comprehensive index for all results. Still, that quibble aside, a very good textbook for rank beginners. It not only can be used to teach a high school honors course, it can also be used to supplement an entirely theoretical treatment in an honors calculus course such as one using Spivak’s classic text. Strongly recommended.
 Difference Equations To Differential Equations:An Introduction to Calculus by Dan Sloughter Furman University (PG) This is a calculus book with a fresh approach and for that alone, the author needs to be commended. The central theme of the book is considering the limiting process as moving from the discrete i.e. finite or at most countable sums subdividing sets to the uncountable i.e. analytic sums. A natural way to express this transition is to posit differential equationswhich one can very simply argue as the central objects of calculusas the limit of a convergent sequence of difference equations. This is the guiding principle of Sloughter’s presentation of calculus and it’s quite interesting. It’s also both very intuitive and rigorous simultaneously. It also has many solved examples and graphs illuminating the convergence of the given “discrete” sequence, giving the desired analytic property. For example, derivatives are throughout considered as the limit of a sequence of tangent lines or planes to a specific point in R or R^{2 }. The definite integral is motivated by the method of exhaustionwhich very naturally leads to the Riemann sum passing from finite to infinite sums of rectangular or cubical areas under a curve or a surface in the limit. There are also many numerical methods given, such as the ubiquitous bisection algorithm and the method of best affine approximation. There are many applications to geometric problems, such as optimizing the area of a fence of length l, but relatively few applications to the physical, social or economic sciences until the last few chapters. That being said, there are several excellent and detailed applications to physics in the closing chapters, including an original chapter on the 2 body problem in mechanics. There’s also some original topics like hyperbolic functions and complex valued calculus that would be of great use to physics and engineering students. I was quite impressed with these notes. It’s very well written and this approach allows one to unify the geometric and analytic aspects of calculus precisely in a way I’d never seen before. It’s similar in this sense to what Shurman tries to do in his single variable calculus notes above, but Sloughter’s approach is much more systematic. I dunno if I’d use it for a calculus courseit’s a bit difficult for the average course.But I’d certainly consider using it for an honors course or a student seminar in calculusor even supplementary reading for serious calculus students or a real analysis course. Highly recommended.
 Elementary Calculus: An Infinitesimal Approach Online Edition H. Jerome Keisler, revised February 2012.(G) A basic course in calculus famous for using Abraham Robinson's infinitesmals approach. Kieser gets big points for originality,but I've never really been comfortable with the language. Again,maybe that will change as I get more familiar with it.
 Topdown Calculus S. Gill Williamson (G) This is a textbook in calculus designed for computer science majors above all and students in the physical sciences. As one might expect, the emphasis is squarely on problem solving and techniques of calculation. But that wouldn’t explain the very unusual structure of this book. The book’s overall design emphaisizes not only an algorithmic approach to calculus beginning with the “big picture” intuitive content through graphs and concrete examples, it also focuses more on specific topics that would be important to specific real world computational problems, such as the logarithmic and exponential functions in numerical modeling. So we have here another very creative, interesting take on the basic calculus course. The mathematically bastardized thinking of some of the passages may make the hard core analyst cringe, but the many beautiful and instructive pictures and examples give readers strong insight into the geometry behind many of the deep analytic constructs. I particularly like the author’s blanket description of calculus in Euclidean spaces: He describes the Central Idea of Calculus as the fact that in “small” enough places, functions behave linearlynamely that the derivative function of a function can be thought of as identical to the original function under a microscope at a specific point. This very enlightening perspective on calculus was first told to me by Josef Doziukin a junior level vector analysis course and I wish with all my heart some of my early calculus teachers had shown it to me earlier. Williamson lauches his presentation with this concept from jump and I think a lot of beginners are going to find it immensely useful in understanding the logic behind a lot of the computations. There is a host of applications, particularly to numerical analysis and geometry. I think the mathematician may find it a bit too loose for their tastes, but I wish I’d had this book as a high school student before beginning a serious study of calculus. And of course, it can be used as a good supplement to a rigorous course to supply intutition. Recommended.
 Calculus for the Intelligent Person Lee Lady: University of Hawaii (G) I remember seeing the old Oliver Stone movie, “Any Given Sunday”, about pro football. It turned out the only thing more forgettable and uninteresting then a pro football game is an Oliver Stone movie about the game. But one line in this utterly forgettable movie stuck with me: There’s a scene with a veteran football player talking to a rookie about his life in the game and he laments about the legion of forgotten players who aren’t stars, but who are every bit as important as them in the popularity of the game. For every Lawrence Taylor, there’s a hundred brothers you never heard of. And they’re as big a part of this game as he is. In this sense,academia is a lot like football. For every Terence Tao, Mike Artin, Noam Elkies, S.P. Novikov, Allen Hatcher and Peter Woitthere’s thousands of people who get their PHDs from elite schools and who quietly get their first job at a small college, teach dozens of courses for decades, give seminars, publish 2 dozen or so papers, quietly retire with a small pension and die without anyone in mathematics even hearing their name. And for the thousands of students they’ve had, some of whom you will eventually hear of, they will have an impact beyond calculation as mentor, teacher and in a handful of cases, friend. (The author of this website you are reading, to this point, has been one of those shadows. Whether or not I remain so, the next few years will determine.But I digress.) Lee Lady was for many years one of those unsung trench warriors of mathematical academia. After decades of teaching at The University of Hawaii, hequietly retired and allowed his website to remain online with his personal reminisces, commentaries on mathematics and teaching and above all, his lecture notes on the various subjects he taught there. These are his wonderful supplementary notes on calculuslectures on various topics such as applications of integration, convergence of infinite series, curvature, discontinuities for functions of several variables, Green's Theorem, Kepler's Second Law, etc. These notes emerged from Lady’s decades of contemplations on calculus and how to make its structure less opaque to both the nonmathematics student and the rank beginner. These notes not only have many innovative applications and insights, they’re also as wonderfully opinionated and discursive. A terrific example is Lady’s explaination of the reasoning behind the classical definition of the Riemann integral, which he motivates through the use of an approximating sequence of step functions that “covers” the given region and whose area can be added to approximate the total region:
 If we now consider all possible step function approximations to a given function, we can see that the Distance = Area principle is true up to any conceivable degree of accuracy. In other words, the principle is just plain true, period.This reasoning is completely different from what one sees anywhere in precalculus mathematics and it is the very heart of what makes calculus different from high school algebra. It goes back to what Archimedes called the Method of Exhaustion. Namely, in calculus one uses the idea that by taking a sequence of closer and closer approximations one can finally arrive at a limit which is exact, even though none of the approximations themselves are exact.
 This very simple but extraordinarily powerful historical insight is then used to reformulate many of the standard applications of integration in this context, such as work, area and volume. And did I mention his wonderful commentary? Oh forget itjust read them if you’re either taking or teaching calculus. Highly recommended.
 Calculus Lecture Notes S. Tryphonas P. Hill University of Toronto Scarborough (G/PG) This is a diverse
set of calculus notes from the University of Toronto and they are a reminder of how good a standard calculus course can be when it’s taken seriously. There are 3 sections, “Goldbook” notes, which contain material on rigorous calculus for honors students, such as mathematical induction, rudimentary logic and approximations by rational functions, “Redbook” notes, which contain mostly practice exercises on standard topics for average students, such as limits, derivative, Rolle’s and The Mean Value Theorum and a final section containing many old exams for students to practice with. The “Gold” notes contain a lot of important material for mathematics majors, written briefly and concisely. But the real treasure here are the hundreds of terrific exercises in the second 2 sections for calculus students and instructors to mine for practice. Recommended as study material.  A Reform Approach to Business Calculus Marcel B. Finan Arkansas Tech University (G) Think of it as an abridged version of the earlier calculus texts by Finan, with the guts of the proofs taken out and lots more examples specific to business. Finan’s terrific teaching style is on hand here as wellbut it is what the title says it is. Good for it’s intended audience, but for the sake of future generations, let’s keep that audience as small as possible, please?
 Vector Calculus Richard Barraclugh University of Burmingham (G/PG)Part of collection of notes Burraclugh has posted at his webpage from his student classes. Rather brief and dry, but very well written and rigorous, assumes a good command of both single variable calculus and linear algebra. Remains in the plane and 3space, which allows a careful presentation of all the major results without differential forms, even Green’s and Stokes’ theorem. Makes a very good supplement to the nonrigorous Cambridge notes on the subject. Surprisingly good and recommended.
 Dr. Z's Calculus Handouts DORON ZEILBERGER Rutgers University (G)Supplementary notes to a standard calculus course, consisting mainly of solved problems and hints. Useful for self study practice and for instructors for additional problems. Good but nothing special.
 Vector Calculus lecture notes Andrew Monnot University of California Riverside (G) Andrew Monnot is a graduate student in mathematics and he has been kind enough to post his class lecture notes at his website. These are his notes on vector analysis.The good news is the notes are very well constructed and written, very clear and complete in terms of topics and definitions. Unfortunately, Monnot omits many proofs and in a junior level vector calculus course, this can be fatal for any but the very best students. It’s a shame,toothe notes are rigorous and sophisticated, they assume linear algebra throughout. If he’d been careful to fill in all the proofsor at least give lucid outlines of proofs this would be hands down one of the best sources available for free on the subject. Stillsince they’re so well written, I’d still recommend them to students and the serious math majors should try and fill in all the blanks. I’d also strongly recommend Monnot expand and revise them into an online textbook someday.
 Calculus (Multivariable) Kiyoshi Igusa Brandies University Spring 2012 (G) Igusa's supplementary lecture notes for the vector calculus/multivariable calculus course at Brandies,done with his usual thoroughness,rigor and clarity. Lots of good examples and worked out calculations. Recommended.
 Multivariable Calculus George Cain & James Herod Georgia Tech 1997 (G/PG) This is an experimental calculus text Cain and Herod wrote proposing to teach the calculus of several variables first to beginning calculus students. It’s an intriguing idea and it has some motivation from a applied standpoint: namely, in the “real” world, most physical phenomena are functions of more then one variable and the “applications” one sees in one variable calculus. Ok, that’s true, but asking students to swallow the calculus of functions of several variables before understanding how calculus works for functions of one variable is asking a lot even of very talented freshmaneven if they’ve seen calculus before. The authors’ approach is basically that of the typical “Calculus III” multivariable calculus course, using the oldfashioned method of vector calculus and analytic geometry of R^{3 }. The coverage is the usual spiel (geometry and functions in R^{n}, limits, differential calculus in R^{n} , parametrized curves, curvilinear coordinates, extrema, etc.) But at the same time, the course is considerably more careful in some ways then the typical “Calculus III”course For example, the authors use the terminology of naïve set theory throughoutfor example, using the ordered pair definitions of functions and relations and defining vectors in R^{2 }and R^{3} as eqivelance classes of directed line segments. While they enshew the direct use of linear algebra, the authors do develop coordinate systems and the basic language of linear mappings and matriciesfirst in R^{2 }and R^{3 }, then higher dimensions where it’s impossible to draw conventional graphs. There’s a particularly good and complete discussion of Lagrange multipliers, which often isn’t done well in most multivariable calculus texts. While full proofs are given of most results of differential calculus, they’re a bit looser with the more difficult subject of integration on R^{n }–for example, stating but not proving Stokes’ theorem. Of course, this is one of the problems with giving a course like thisa full proof really requires the machinery of manifolds and forms, even in low dimensions. When the authors skip a proof, though, they give many computations and examples. They tend to give geometric proofs rather then formal ones, which in a course at this level is mostly helpful. There are also many of the usual applications to physics and geometry. While the book is quite well written, informative and has many wonderful pictures, I do have 2 minor complaints. First of all, I think a lot of the more awkward passages and discussions could have been simplified and clarified without losing any spatial intuition by developing the basic elements of linear algebra. If you’re going to go so far as to use naïve set theory, open and closed sets, linear maps and matrix algebra, you may as well go the distance and introduce the framework of vector spaces.This wouldn’t have resulted in a book markedly different from the one here as long as one limited it to what’s essential. Secondly, it’s not clear what the level of preparation of the students using this book needs to be. It’s fairly obvious the authors assume some familiarity with basic calculus from high school, but how much and at what level of sophistication isn’t apparent. For example, read the following passage from the chapter on limits in R^{n} :
 Recall from grammar school what we mean when we say the limit at t0 of a realvalued, or scalar,
function f is L. The definition for vector functions is essentially the same. Specifically, suppose f is a vector valued function, t_{0} is a real number, and L is a vector such that for every real number e > 0, there is a d > 0 such that  f ( t) –L < e½ ever 0 <t  t_{0} < d and t is in the domain of f .
 I think it’s very safe to say that the authors are being facetious with the “grammar school” schtick. But do they really expect high school calculus courseseven at very good schoolsto cover the εδ definition of a limit? This is a strangeto put it kindlyassumption to make of even honors undergraduates. Still, these are minor quibbles. This is a very readable, interesting and deep course in multivariable calculus. As long as you have good students and decide what aspects can be skipped, it will make a very good course indeed. Highly recommended.
 Calculus II Micheal Hill University of Virginia Fall 2008
 Multivariable Calculus Micheal Hill University of Virginia Fall 2008 (G)This is a pair of note sets to supplement a fairly standard, “plug and chug” pair of calculus courses. While they’re well written and have lots of nice solved problems, there’s really nothing special about them. Check them out if you like, but don’t expect magic.
 Calculus and Analytic Geometry 1 Marta Hidegkuti Truman College Spring 2012 (G) This is a very weird set of “lecture notes” for a standard calculus course. Mostthough not allof them basically consist of solved problems and their solutions. The solutions are very detailed and informative, but they clearly assume the student actually learned the techniques elsewhere. The strangeness here is that there’s no official textbook for the course, just a list of recommended onesand Hidegkuti suggests, “The more textbooks you use, the better!” Ok,but then you need to be a little more suggestive on what books to use since all the ones you suggested are very different and will serve to confuse the rank beginner. Still, these supplementary examples are quite nice and instructive. Combined with one of the online sources listed herelike Strang, Dawkins or Cookthey could be a very useful study source for students. Recommendedfor what they are.
 CALCULUS KEN KUNIYUKI SAN DIEGO MESA COLLEGE 2012 (G/PG) The follow up to his Precalculus course above and it's every bit as good if not better. This is baby calculus from a mathematician's perspective, trying to keep the course at a general audience level while refusing to make any special concessions for nonmathematics students. It's a very difficult tightrope to walk and Kuniyuki does it better then just about anyone I've seen. The seriousness of this textbook is clear right from jumpas he spends several pages laying out in pedantic detail the notation for functions, sets and other objects that will be used in the book. He then moves on to the standard topics in a calculus course. The mark of this online text is not its’ creativity or originality, but the enormous care and clarity with which the author explains each concept and drills it into students. While maintaining the level appropriate for beginners, he proves everything clearly and precisely from several different viewpoints simultaneouslycombined with many examples and graphs. It’s all done in loving detailand the author works very hard to ensure both geometric and algebraic methods are used to explain why things work in calculus. 2 excellent examples are how he explains the rigorous definition of limits and the definition and discussion of the derivative as instantaneous velocity. In the rigorous discussion of limits, Kuniyuki draws the analogy of a lottery to give the following wonderful description of the precise definition of limit: Imagine a lottery in which every x ∈ Dom( f ) represents a player. However, we disqualify x = a (here, x = 4), because that person manages the lottery. (See Section 2.1, Part C.) Each player is assigned a lottery number by the rule f (x) = 7  (1/2) x. The “exact” winning lottery number (the “target”) turns out to be L = 5 . In this lottery, more than one player can win, and it is sufficient for a player to be “close enough” to the “target” in order to win. In particular, Player x wins (x¹ a) Û the player’s lottery number, f (x), is strictly within e units of L, where e > 0. The Greek letter e (“epsilon”) often represents a small positive quantity. Here, e is a tolerance level that measures how liberal the lottery is in determining winners. Symbolically:Player
x wins (x ¹ a) Û L  e < f (x) < L + e e < f (x)  L < e Similarly: Player x wins (x ¹ a) Û this distance is less than e .We only care about players that are “close” to x = a (here, x = 4), excluding a itself. These players x are strictly between 0 and d units of a, where d > 0 . Like e , the Greek letter d (“delta”) often represents a small positive quantity. d is the halfwidth of a punctured d neighborhood of a.  This may be the best explaination of the concept for beginners I’ve ever seen. He proceeds to give many explicit examples of the “lottery” and how these computationsare done.The idea of a punctured niebhoorhood occurs repeatedly in this text, it gives a very powerful and pictorial way of explaining the idea of εδ distances and computations.The author develops the whole of calculus in this manner, both single variable and multivariable (although the multivariable part of the book looks like it’s much less polished then the single variable part, much of it is still in handwritten scan form) . This is a book I’d be proud to teach a course with and I heartily recommend it to all students and teachers of calculus. Very highly recommended.
 Calculus II Lecture Notes prepared by R. Tavakol Queen Mary, University of London 2001–2003 (G) These are the notes for the second semester of a very standard applied calculus course for science and engineering majors. They cover a fairly complete course in ordinary differential equations and multivariable calculus with all but the simplest proofs omitted. Still, it has many excellent graphs and solved examples. Supplemented by a more rigorous source, it can make for very good study material, particularly for applied students.
 Complete Course Calculus Text 2012 Edition developed by Leah EdelsteinKeshet, et al The University of British Columbia (G)This is another applied calculus text developed specifically for students of theThe University of British Columbia life sciences. It has a complete lack of rigor, so mathematics students expecting limits and careful deductions are in for a letdown. What it does have , which is of immense interest to anyone serious about calculus, is a remarkable number of applications of the basic concepts to modeling in the biological sciences and which you won’t find except in specialized textbooks or monographs. Just some of the treasure trove of applications are LineweaverBurke plots in chemical kinetics, optimization of cellular size and shape, optimal foraging in animal population models and exponential growth of microbes. As a former student in biochemistry, I found these notes a pleasure to read and wish I’d known about them when I was constructing enzymatic models in my labs. Highly recommended for all.
 Calculus Problem Solving Padraic Bartlett CalTech Math 8 (PG) These are the lecture notes for the first year “boot camp” in calculus at Cal Tech; focusing on diverse methods of proof, calculation and problem solving. A lot of nice insights and examples into the science and art of problem solving and proof and all done in a rigorous calculus context. It’s really supposed to complement the notes above on Cal Tech’s first year calculus course, but these notes stand very well on their own as an introduction to proof for students making the transition to careful mathematics. Readable and quite useful. Highly recommended. Calculus II 2009 S.R.Srinivasa Varadhan NYU (G) Yes, that Varadhan. The author of the classic textbook on Lie groups that taught a whole generation of graduate students beginning in the mid1960’s, that Varadhan. The one that taught,via some office chatter, the basics of probability theory to a confused neurological biology graduate student at Rockefeller University named Daniel Stroock and made such an impression on him he switched to mathematics (at least, that’s how Stroock tells it in the preface to his graduate probablility theory book) –that Varadhan. The legend at the Courant Institute that got his start as one of “The Famous Four” at the Indian Institute of Technologythe other 3 being R. Ranga Rao, K.R. Parthasarathy, and Veeravalli S. Varadarajan that Varadhan. And what does this giant of mathematics do in these notes? Teach the second semester of a pencil pushing calculus course on techniques of integration and infinite series. That’s so awesome, that he doesn’t feel himself above such menial tasksit’s a shame more academics of his stature don’t feel that way. Still,it’s a pretty standard course with formulas to memorize and nothing really special.
 Calculus III Problems, Solutions, Handouts Joel Feldman University of British Colunbia (G/PG) More of Joel Feldman's wonderful supplemental calculus lecture notes, this time for multivariable calculus.They're somewhat less rigorous then the single variable casebecause of the complexities of multivariable calculus theorybut they're no less careful and well written. The discussions emphasize the geometry and physics of classical vector analysis, which are critical for understanding it's significance. The sections on vectors, quadratic surfaces,the equality of mixed partial derivatives and complex numbers are particularly excellent. Highly recommended.
 Calculus Robert Heffernan University of Conneticuit Spring 2013(G) These are lecture notes for a basic “Calculus II” course on the techniques of integration, infinite series, polar and spherical coordinates and some differential equations topics. They’re actually pretty goodapplications to physics and geometry are discussed in depth with lots of nice diagrams. The discussion of the “method of washers and shells” in integrationwhich tends to confuse beginnersis particularly nice. Not comprehensive, but what it does, it does well. Recommended as a supplement or study aid.
 MULTIVARIABLE and VECTOR CALCULUS Joel Feldman University of British Columbia (G) More of Joel Feldman's wonderful supplemental calculus lecture notes, this time for multivariable calculus. The discussions emphasize the geometry and physics of classical vector analysis, which are critical for understanding it's significance. The sections on vectors, quadratic surfaces,the equality of mixed partial derivatives and complex numbers are particularly excellent. Recommended.
 Calculus Problems, Answers, Handouts Joel Feldman University of British Columbia (G/PG) Joel Feldman's excellent supplementary lecture notes for the calculus courses at the University of British Columbia. Canadian universities don't coddle thier students with plug and chug coursesinstead, they try and explain things to the average student in a way that makes it accessible to them. While Feldman does include the standard mechanical aspects and they're done well, he also includes a great deal of more careful analysis as well. The "Warnings" section is a particularly good examplethe subtlties of the validity of L'Hopital's rule for derivatives with limits at infinity are very rarely explained this well and concisely even in analysis courses. Welll worth a look for not only serious students of calculus, but thier instructors. Check out his notes for the subsequent sections here as wellthey're just as masterly. Vector Calculus Padraic Bartlett Caltech Math 1c, Section 14 (G)Here are the notes to Bartlett’s undergraduate vector calculus course at Cal Tech and they appear to be more for engineering majors then mathematics majors, as most proofs are missing. That being said, he does have a lot of excellent discussions of the common errors students make when dealing with the fairly ornerous notation that exists in multivariable calculus, such as the subscript nightmare partial derivatives can be. Lots of nice pictures in R^{3 }^{ }as well. I’d certainly recommend them as collateral reading to a standard vector analysis text for all Bartlett’s nice pointers on calculation, but they’re not substantial enough to be used on their own.
 Calculus II for Mathematical Sciences JingJing Huang University of Toronto (PG) These are the handwritten notes for a very careful and rigorous course in calculus based on Spivak’s classic text. They’re brief and don’t really add much to Spivakand they’re sloppy in places, too . You’re much better served using Cook or Clark’s lectures if you’re looking for a hard core course online.
 Integral Calculus with Applications to Commerce and Social Sciences Lecture Notes Robert Klinzmann University of British Columbia (G)This is another applied course in multivariable calculus, with most theorems not even stated, let alone proved. It does have many computations and pictures, particularly of applications to the social sciences, as it’s title suggests. Again, nothing special and ok for review or practice. This may be the best explaination of the concept for beginners I’ve ever seen. He proceeds to give many explicit examples of the “lottery” and how these computations are done.The idea of a punctured niebhoorhood occurs repeatedly in this text, it gives a very powerful and pictorial way of explaining the idea of εδ distances and computations. The author develops the whole of calculus in this manner, both single variable and multivariable (although the multivariable part of the book looks like it’s much less polished then the single variable part, much of it is still in handwritten scan form) . This is a book I’d be proud to teach a course with and I heartily recommend it to all students and teachers of calculus. Very highly recommended.
 Calculus Z. Qian Mathematical Institute, Oxford November 22, 2012 (G/PG)Yet another applied calculus course from across the pond at Oxford. Still,they’re pitched at a higher level then most western calculus courses, assuming basic linear algebra, and most of the basic theorems are proved and many examples and discussions are given. There’s also many, many terrific problems at the site. Recommended.
 Calculus: early transcendentals by David Guichard and friends Whitman College 2012 Draft (G) This is an extensive draft of an online calculus textbook. The book is written entirely in interactive JAVA code where applets form the basis of a dynamic online presentation. There are many excellent computer generated graphs, a number of which take the form of animated applets to demonstrate the uses of calculus in physics such as projectile motion. So this is a good application of modern computers in the teaching of calculus available free online and it certainly livens up and adds visual depth to the text. Otherwise, in terms of actual content, this is a pretty standardif more comprehensive then usualpresentation of the topics of single and multivariable calculus. There are a number of simple theorems which are proven as well as many of the standard examples. For the most part difficult results, like the extreme value theorem, are stated but not proved. ( A glaring exception is limitsthe authors give a precise definition and a good discussion motivated by numerical approximation. ) Still, the number of applications to the physical and social sciences is richer then usual and assisted greatly by the JAVA animations and graphs. The book also has many exercises with an interactive twist: the answers are embedded in links that can be accessed by students right on the same page. If you’re preparing to teach a standard calculus course and want something careful, but not too difficult and will grab the students’ interest, this is a good choice and worth a look. Recommended.
 Calculus WikiBook 2011 Version Of course, there’s one of these. If every calculus book on Earth suddenly burst into flames, you can always count on a Wiki Calculus. It’s just a fact of life now. Wikibooks are a bit like Schaums Outlinesthere’s a vast range of quality, some are excellent and some need to be deleted for the benefit of mankind. That being said, this particular book is surprisingly good for all its flaws. It covers all the basic topics of any single variable calculus course, either standard or honorsprecalculus review, limits, differentiation and integration of one variable and their applications, giving both rigorous proofs and intutitive discussions depending on which sections you study. It also contains a ton of solved problems and a lot of terrific graphics of diverse kinds. Also, since the book is a Wiki and open to additions by anyone, there are many unusual applications in the book you won’t find in standard calculus books, such as to thermochemical statistical mechanics and kinetics. Unfortunately, the strength of the book is also it’s weaknessthere are many either missing or skeletal sections that have yet to be filled in or developed in more detail by future authors, mostly in the later “chapters” on multivariable calculus and differential equations. Consider it a work in progress that still has to be filled in quite a bit before it can be used as a basic source. That being said, there’s a lot of good and useful sections here for students at many different levels. Recommended as a supplement, but you’ve been warned how problematic it’ll be if you try and use it at this writing (June 2015) as your primary source material.
 Book 3 Relationships, change { and Mathematical Analysis Roy McWeeny
 Book 3a Calculus and differential equations John Avery H. C. Ørsted Institute University of Copenhagen (Denmark) (G) These 2 books are part of the remarkable BASIC BOOKS IN SCIENCE online series of texts, which attempts to provide introductory books free of charge online and written by professionals in each field as true introductions to students at all levels who wish to educate themselves in these areas. The entire series can be found here: http://www.learndev.org/ScienceWorkBooks.html. I heartily recommend them to all students and educators who believe, as I do, higher that education should be a right and not a priviledge for the wealthy. They are pitched at a level suitable for anyone with a decent high school education and would make terrific beginning texts for anyone. These 2 specific volumes are about calculus for absolute beginners and it’s relationship to the physical worldthey are designed to complement each other as the first volume by McWeeny discusses the basic concepts and methods of calculus from a purely mathematical yet intuitive viewpoint for beginnersi.e. discussing the geometry of calculus and how it motivates the basic ideas of limits, derivatives and integrals. The second volume is longer, deeper and more closely resembles a standard textbook in calculuswith several important differences. First of all, the text begins with a lengthy, detailed digression on the history of the ideas of calculus and how they arose simultaneously with developments in physics and geometry from ancient Mesopotamia to the development of modern real analysis in the 18^{th} and 19^{th} centuries. This gives enormous insight into not only the intuitive framework of calculus, but the applications that arose in their creation. Secondly, there is a much greater emphasis on applications of calculus to the physical sciences and how those applications usually manifest as differential equations with boundary conditions and their solution methods. There are no precise theorems or definitions in either bookso they’re hardly what one would call serious mathematics textbooks. Indeed, I’m not sure if you could call them textbooks in the usual sense of the wordthey’re more like extended essays on calculus, its history and applications. Still, they are remarkable works and I can’t think of better books to either give a beginner just learning in combination with a standard calculus textbook or as an intuitive supplement to a completely rigorous treatment like Clark, Hwang or Chen. Indeed, any of these sources combined with these 2 texts would provide an outstanding free text for a freshman honors calculus course. In any event, anyone just learning or teaching calculus should read them and will find their understanding of it greatly enriched by the experience.Very highly recommended.
 Calculus Textbook Daniel Kleitman MIT 18.013A (G) This is a yet another textbook in progress for an innovative calculus course at MIT. When I first saw it, I thought it was yet another calculus book for a standard “pencil pushing”calculus course. But clearly it’s intended for a more sophisticated audience, one that’s already been exposed to calculus in high school. However, it’s by no means a hard core theoretical “full proofs with epsilons and deltas/screw applications” calculus course like Clark or Spivak, either. Indeed, the word “limit” appears nowhere in the entire manuscript! A lot of mathematicians will shout heresy here. Kleitman is a very respected applied mathematician/mathematicial physicist at MIT and to be honest, I found his approach confusing at first. But as I read the notes and enjoyed all his insights, applications and visual aids, I began to understand the method of his madness. Judging from the content and many innovations within, what he’s attempting to write here is an honors calculus course for beginning hard science majors outside the mathematics department i.e. physics, engineering and
 chemistry students. In other words, he wants a course for beginners that is deep in understanding, but avoids the level of abstraction that a standard honors course taught to and by mathematicians usually has. The author is attempting to bring some creative use of technologyin this case, JAVA applets to a calculus course which is careful without being rigorous in the modern sense of the word. The author describes his goals as follows in the Preface:
 Chapter 0 which introduces the spreadsheet is entirely new. The discussion of standard functions in Chapter 1 is new. The geometric definitions of the trigonometric functions by the illustration is new in Chapter 2, and the section on various metrics (3.8) is new in Chapter 3. (It probably does more harm than good there.) The main innovations in Chapters 4 and 5 are the applets, though introduction of the concepts of eigenvectors and eigenvalues at this stage is perhaps unusual. Chapter 6, which involves definition of differentiation in all dimensions is novel; but I think it helps make it possible for students to see why the rules of differentiation are what they are, and why calculus is useful Chapter 7 on numerical differentiation is new at least to me. There is not much new in Chapter 8 except for it applying in all dimensions. Probably the major innovations in this course apart from the applets, are the sections on numerical analysis, the study of single and multiple variable calculus together, introduction of integration in the complex plane, and the applications to physics, which while commonplace in physics, rarely are discussed in a calculus course. Our intent here is to cover enough about more advanced areas of mathematics to make students aware of them and to encourage their wanting to learn more about them.
 The author certainly achieves all these goals in my opinion and the amazing this is that he does it without a single mathematically rigorous proof. Actually, that’s not truehe simply doesn’t give analytic proofs of all results. Instead, he focuses on explaining why the objects of calculus behave this way by giving entirely geometric proofs and explainations, along with full computational proofs when they’re called for. Indeed, many such computations are given as exercises for the student. For example, he defines the derivativein both one and several dimensionsas the slope of a tangent line or plane to the graph of a function at a point. It’s clear that a geometric explanation is going to be of more help not only to absolute beginners, but physics and engineering majors as well. The number of applications here is astonishing for a beginning calculus course, including applications to mechanics, thermodynamics, electrostatics and electrodynamics, differential geometry and even an intuitive discussion of quantum mechanics! As a mathematician and teacher, I’m on the fence about this. The mathematician in me wants to shake him and tell him shame on him, but the teacher in me is much more sympathetic. A course like this would be much more helpful to beginners prior to a first exposure to rigorous analysis, especially for nonmathematics hard science majors. Then againI think courses authored by Smith, Chen, Hwang, Duzhin and Shurman demonstrate very effectively this doesn’t have to be an eitheror situation. So despite
 my mixed feelings about it, I will recommend this to beginners in calculus because of the wonderful attention to detail and considerable effort Klientman puts into this course. However, I would add the caveat that this course would be far better and more useful to a much larger range of audiences if he added optional sections rigorously defining and proving all basic results. Until he does that, Hwang will remain the source of choice for those of us who believe both theory and applications in calculus are of equal import. But I do like what he’s trying to do here and I do recommend careful reading through the notes. And I heartily recommend them as a supplement to all.
 Multivariable Calculus, Saul Stahl Universiity of Kansas (PG)These are Stahl’s online notes for an honors vector calculus class at the University of Kansas and I have to say, they’re one of the more innovative entries in the current crop. Stahl says in the preface his goal with this course was to restructure the traditional rigorous course of multivariable calculus so that the Big Three theorems of vector calculus ( Green’s, Guass’ and Stokes’) could be covered in depth in one semestersomething a lot of instructors generally complain they have problems doing without running out of time . The usual topics are covered and in the usual geometric “old fashioned” mannervectors and coordinate systems in R^{3 }, limits and continuity of vector valued maps, the total derivative and the differential in higher dimension, partial derivatives, multiple integrals and the aforementioned big theorems, etc.all proven with complete rigor, lots of graphs and physical applications to boot. There are also many complete computations, which is critical for understanding this material for a beginner. But on a careful reading, you’ll realize this presentation is quite a bit different from the run of the mill one while still remaining classical in spirit. First of all, it’s quite a bit more rigorous and careful then the usual presentationwhich really isn’t breaking news in and of itself, but it certainly makes it superior to most of the usual treatments. Second, Stahl makes good on his stated intention in the preface by organizing the material in a quite different manner then usualthe major theorems of vector integral calculus are given before divergence and curl. This results in a rather pedantic chapter, but it does have the huge advantage of stating and proving these results from first principles and in a more elementary fashion then usual. Physics majorswho really need to understand this material and usually aren’t skilled in proofwill probably find this easier to master, especially since Stahl spends the first part of the chapter giving physical motivation to these results before proving them. Lastly, Stahl focuses a good deal more on the geometry of surfaces in R^{3} then usual. This allows him to cover some interesting and important topics that usually aren’t present in these courses, such as a geometric discussion of the classification of compact surfaces, a very complete
 and clear discussion of the quadratic forms and their conic surfaces, as well as isometries in R^{3 }. An excellent work from a first rate teacher and anyone learning or considering teaching multivariable calculus or vector analysis could do a lot worse then use this as the text for their course or as a supplement. Highly recommended.
 Vector Calculus David Lerner University of Kansas 2012 (PG) A very complete, careful and readable set of scanned handwritten lecture notes from a standard vector analysis course. About the same level as Bandaxall and Liebieck with similar prerequisites. Freely uses linear algebra throughout. Unfortunately, there are some annoying holes in the notes where they just drop off into blank space, betraying their “not ready for prime time” handwritten rough notes status. Still, there’s lots of examples and insights and Lerner is mathematically quite careful. A nice supplement for such a course. Recommended.
 Introduction to differential forms Donu Arapura Purdue University May 27, 2012 (PG)A supplement to an undergraduate course in vector analysis requiring a good working knowledge of calculus, pitched at a much lower level then some of the other sources on differential forms and multilinear algebra here. As I'll say several times at this page, I'm still on the fence about whether introducing differential forms at this level is really helpful to students or more confusing then helpful. That being said, if you really wanted to introduce forms in a beginning calculus course, you couldn't do better then these notes. They're extremely clear, computational and simple.Rigor is kept to a minimum, which is appropriate for a course like thisforms are carefully defined, but other then that they're used as calculation devices. There are many applications to line and surface integrals as well as to basic physics, such as the thermodynamic dforms of work and Maxwell's Equations in higher dimensions. It clocks in at 37 pages, which makes it a perfect supplement to a standard vector analysis course, whether a rigorous yet classical like Bendaxall and Liebeck or a more standard nonrigorous one like Corral. I would have added 20 pages of just examples.Forms are such an alien concept when students first see them, lots of examples are really needed no matter how clearly they're explained. Contents 1 1forms 2 Exactness in R2 3 Parametric curves 4 Line integrals 5 Work 6 Green's theorem for a rectangle 7 2forms 8 Exactness in R3 and conservation of energy 9 \d" of a 2form and divergence 10 Parameterized Surfaces 11 Surface Integrals 12 Surface Integrals (continued) 13 Length and Area 14 Green's and Stokes' Theorems 15 Cauchy's theorem 16 Triple integrals and the divergence theorem 17 Gravitational Flux 18 Laplace's equation 19 Beyond 3 dimensions 20 Maxwell's equations in R4 21 Further reading A Essentials of multivariable calculus A.1 Differential Calculus A.6 Integral Calculus
 Calculus JAMES W. BURGMEIER University of Vermont Fall 2012 (G)Handwritten scanned notes for a pretty standard pencil pushing calculus course. Nothing new to see here, move on.
 Calculus II Vincent Koehler University of Vermont 2012 (G)Yet another batch of handwritten, very standard notes for a basic calculus course. Lot of solved problems and nothing special to waste room on your PC downloading.
 Calculus I Lecture Notes Isaac Fried Boston University (G)This is a relatively brief but interesting set of notes for the first semester of a basic calculus course. They focus on the geometry of the real line and the plane, indicating how calculus is the process of composing linear approximations to real valued functions. They’re not earth shaking and this terrain is covered elsewhere in greater depth, but they’re worth a look anyway. Recommended.
 Calculus I Richard Kent University of Wisconsin Madison Spring 2013 (G)A vector calculus course for physics and engineering majors sans proofs. As I’ve said, I personally don’t like unitasker courses like that that strip the guts from the subject, but I will make exceptions if they’re done well and bring something new to the table a reader can benefit from. Sadly, except for some nice computer generated pictures and computations, this is a pretty cookier cutter set of notes. You’re much better off using a more rigorous presentation that contains many physical applications, such as Stahl or Bendaxall/ Liebeck.
 So, You Want to Become a Calculus Wizard Patrick LaVictoire Fall 2012 (G) A nice little 4 page commentary giving solid advice to beginners .
 Vector and Complex Calculus for the Physical Sciences Fabian Wale University of Wisconsin Madison Spring 2013 (G)Another vector calculus course for physics and engineering majors sans proofs. Same commentary as 100 other such coursesyou're better off with Stahl or Bendaxall/ Liebeck.
 Things my students heard me say in Math 231 Calculus, Fall 2010 Richard S. Laugesen UWUC Some amusing recollections of a wiseass calculus teacher. If only my teachers had made it this much fun the first time around.....................
 Analytic Geometry and Calculus (Fall 2012) Kiumars Kaveh University of Pittsburgh Course Materials (G) Yet another alltootypical pencil pushing calculus course with nothing materially mathematical within. There are so many of these now, it’s really aggravating to those of us who care about mathematics and its education. Courses like this really aren’t helping anyone but the University’s coffer holders. Calculus Aravind Asok University of Southern California 2009 Course Material (G)Yet another one. Nothing more to add.
 Calculus I Course Materials And Lecture Notes Math 1431 Jiwen He University of Houston
 Calculus II Course Materials And Lecture Notes Jiwen He University of Houston (G) Ah, here we gonow here’s something worth a close look. These are supplementary notes for 2 semesters of the standard calculus courseat the University of Houston. They are intended to be used by the students in class and for review, so they’re in outline formvirtually all the theorems are stated without proof. That being said, all definitions and examples are given in very good details and are very insightful. This can be very useful working in tandem with a rigorous source like Chen or Clarkhopefully most students would work with pencil and paper in hand to try and fill in the proofs. They are as mathematically careful and insightful as notes for a standard calculus course should be. There is also an enormous number of very good exercises and quizzescalculus students should try as many as they can. There are lots of physical applications and applications to geometry as well here. They can’t be used by themselves, but they’ll sure make an excellent study resource used in tandem with a good calculus textbook or set of lecture notes. Recommended.
 Calculus 1 (804231) t MATC, Fall 2008
 Calculus 2 (804232) at MATC, Spring 2010
 Calculus 3 (804233) at MATC, Summer 2009 (G)These are the 3 semesters worth of calculus notes online from the Madison Area Technical College. They’re formatted in Microsoft Word and are nice and readable with lots of nice graphs and examples. They also have some spreadsheet examples that are good. But to be honest, I didn’t find anything new here. It’s a standard calculus course, presented nicely with some good examplesbut nothing you can’t find elsewhere or in a old copy of Stewart’s book. Feel free to download and use them, but nothing really exciting here.
 Vector Analysis Notes Matthew Hutton Autumn 2006 Lectures University of Warwick parts 110 Vector Calculus notes Matheew Hutton University of Warwick Chapter 11 2007 (PG)This is yet another semirigorous vector analysis course in the Cambridge and Oxford tradition that focuses mainly on geometric intuition and only proves simple results, such as the divergence and Green’s theorem in 2 dimensions. Intrestingly, the second half of the course gives a good introduction to complex analysis in the plane with full proofs. These notes focus mainly on the integral calculus in R^{n} in the first half and the complex calculus in the secondthe differential calculus in R^{n} is glanced over briefly. Although it doesn’t say so explicitly, I strongly suspect the differential calculus course by Pusey and friends given immediately preceding this link is a prerequisite for it. In any event, since full proofs are not given, the notes will be of limited use except for review and drill. Despite that flaw, the notes are very readable and sophisticated with many examples and graphs, so they’re still worth a look despite their limitations. Strong students should try working through them and filling in the blanks.
 Diifferentiation Lectures Vassili Gelfeich et al Based on Lectures in Spring 2007, updated August 27, 2007 University of Warwick (PG) A brief, concise set of notes on the differential calculus of several variables “by committee” at the University of Warwick. Very similar to the notes by Loke from Oxford described above, but with somewhat more detail. Shows some unevenness from the multiple authors, but it contains all the basics of differential calculus on R^{n}. The notes are pitched at a high level and are completely rigorous with full proofs, assuming a good command of linear algebra and rigorous single variable calculus.
 Calcuius XiangSheng Wang Memorial University of Newfoundland (G) A concise outline of a standard calculus course. It’s nicely written and does have some nice insights, including comparing continuous and discontinous functions to completed and noncompleted bridges. There’s also some nice step by step summaries of some needed skills in calculus, like curve sketching. But frankly, it’s nothing specialespecially after wasting 14 pages giving new students the 411 on what all past students thought of the professor, the course expectations,etc. That really couldn’t be done in class or in a separate link at the course homepage? Seriously?
 Multivariable Calculus Arthur Mattuck Supplementary Notes and Problems MIT OpenCourseWare (G) This is essentially a polished and less concise version of the second half of the Calculus With Theory notes above Mattuck wrote to supplement the standard calculus textbook with more rigorous proofs and problems for the course. As with anything by Mattuck, they are beautifully written, clear and full of analytic and geometric insights. Full proofs are given of all theorems, including the Big Three of vector analysis, by limiting the proofs to the low dimensional cases over the smoothest boundary curves in R^{2} and R^{3 }where the proofs are essentially computations. Linear algebra is limited to what is essentially needed, such as determinants and matrices and the results are expressed in this notation. For example, the second derivative test for functions of several variables is expressed by the Hessian matrix. If our standard calculus course notes at all American colleges were this well written, America wouldn’t be 31^{st} in the world in mathematics training at our schools. Very highly recommended.
 Multivariable Calculus James V. Lambers The University of Southern Mississipi June 10, 2013 (G/PG) This is a complete online textbook for a multivariable calculus course that presumes a good knowledge of single variable calculus and linear algebra. I looked at the original homepage for the course last summer and was a bit confused as it says Stewart would be the required book for the course. It’s not hard to guess from there what happenedLamberts probably didn’t like the presentation in that book, which avoids linear algebra. And after that,he probably couldn’t find a replacement text that suited his idea of what the Math 280 course should look like. So he did what any good, responsible teacher would dohe wrote his own. And a good book it is, too. It’s not rigorous, sadlymost hard proofs are avoided, although he gives a good careful discussion of limits and continuity in higher dimensions. He does give a ton of good examples and computations, as well as good computer generated graphs in R^{2} and R^{3 }. To be honest, I’d rather use either Bandaxall and Liebeck or C.H. Edwards for such a course, as some mathematical rigor and the use of linear algebra really shouldn’t be completely skirted in such a course. That being said, it’s very well written with lots of nice examples, so it’s worth a look if you’re either teaching or learning this material. And a good instructor or student can always fill in the proofs.
 Calculus I Jim Wang Old Dominion University Fall 2006 (G)Yet another standard calculus course with yet another set of typical standard calculus supplementary lecture notes. These are even more plain vanilla then usual since there are no pictures.I'd pass.
 Vector Calculus Birne Binegar Oaklahoma State University Summer 1998 (G)A relatively brief but very well written set of lecture notes on multivariable calculus, developing the prequisite linear algebra in the beginning on abstract vector spaces. Unfortunately, due to the brevity of the notes, more then half the proofs are missing, although theorems are stated carefully. The notes have many examples and explicit computations. If a student or instructor can fill in the proofs, these notes can be the skeleton of a very good course indeed. Fortunately, virtually all of the missing proofs are readily available in Bendaxall/Liebeck (Yeah, I told you it was one of favorites and for good reason.) Better used for selfstudy students to review after working through B/L or a comparable source.
 Multiple Integrals and Vector Calculus F.W. Nijhoff Semester 1, 20078. Course Notes University of Leeds (G) Yet another applied vector analysis course in the Cambridge/ Oxford mold, emphasizing the integration of functions of several variables and their applications. This is even more lacking in details then the usual suspects in this bunchit’s basically a bunch of definitions and theorem statements with graphs to go along with it. There are relatively few solved examples. Don’t expect much help from these. .
 Calculus David R. Guichard Whitman College 2013 (G/PG) A very solid textbook in both single and multivariable calculus that focuses on problem solving, but doesn’t pretend calculus doesn’t involve proofs, like a lot of other texts both online and in print try to do. It’s similar to Stewart in it’s choice of subject matter and emphasis on problem solving, but Whitman doesn’t shy away from mathematical arguments if he feels a proof is needed rather then an intuitive argument.For example, all the limit theorems are proven using the εδ definition, which is explained and only used selectively at certain points in the text. However, the chain rule isn’t given a full proof, although a good intuitive discussion is given of an ”infinitesimal” argument of what happens when x and y go to 0. There are many graphs which are well done and instructive, as well as a multitude of fully solved examples, lots of the usual applications to the sciences and best of all, numerical computation examples. In other words, this is a pencil pushing text for people who think. A great free alternative to Stewart’s text if you want to have mercy on your students’ bank accounts and also terrific for self study.
 Calculus I R. Mayer Reed College (PG) This is the calculus textbook at Reed College that inspired Shurman’s texts above. It too begins with in depth discussions of area and the history of the quadrature problem and its role in the development of calculusbut Mayer’s discussion seems simpler, clearer and more historically grounded. They’re equally rigorous and deep, yet remain elementary due to the geometric emphasis. Unusual and wonderfully readable, highly recommended.
 Single Variable Calculus Fall 2006 David Jerison MIT Fall 2006(G) A very beautiful,wonderfully written and comprehensive set of notes for a first semester calculus course of one variable. Jerison based the course on the terrificbut sadly,very priceycalculus textbook by George Simmons and seems to have been inspired by it. Contents: Graphing,derivatives, slope, velocity, rate of change,limits, continuity,trigonometric limits,derivatives of products, quotients, sine, cosine,chain rule,higher derivatives,implicit differentiation, inverse functions,exponential and logarithm functions, logarithmic differentiation,hyperbolic functions,hyperbolic functions,applications of differentiation,linear and quadratic approximations,curve sketching, extrema problems,related rates,Newton's method and other applications,mean value theorem,differentials, antiderivatives, differential equations, separation of variables, integration,definite integrals,first fundamental theorem of calculus,second fundamental theorem,applications to logarithms and geometry,volumes by disks and shells, techniques of integration.trigonometric integrals and substitution,integration by inverse substitution,partial fractions,integration by parts, reduction formulae,parametric equations, arclength, surface area,polar coordinates,area in polar coordinates,indeterminate forms and L'Hôspital's rule,improper integrals,infinite series and convergence tests,and Taylor series. It's a course for rank beginners and as such, it really doesn't make any pretense of caring about rigor. It's unapologetic in it's focus on the geometric and the applied aspects of calculus. As much as professional mathematiciansparticularly analystscan't stand it, that's the proper approach for students who have never seen calculus before in any form. Many beautiful diagrams and observations by Jerison, as well as many equally wonderful problem sets. This will be a terrific study aid for any student just beginning calculus or any teacher teaching calculus for the first time. It also will make a great intuitive supplement for an honors calculus
 focusing on theory. Highly recommended for beginners in calculus Active Calculus Matt Boelkins Grand Valley State University (G) This original text is part of a movement online to make widely available free OpenSource textbooks for mathematics in response to the aristocratic price ranges of standard textbooks like Stewart. This is an important phenomenon I’ll be commenting at length on in my blog. The
 author’s goal here was to have a completely open source free calculus text that could be used by students and, more importantly, could be used actively by the students to learn calculusi.e. by self study after free download. It’s still in progress at this writing (late September 2013) , so there’ll probably be updates to this review in the future. As for the
 book itself, it’s highly inituitive and geometric and there are virtually no proofs in the usual sense. The presentation is very dynamic as concepts are given entirely through examples and graphics. For example, derivatives are discussed before limits as the “limit” of a sequence of secant lines that “converges” to the tangent line on a fixed
 point of the graph of a function. There is also a very large number of exercises, with applications to both geometry and physics. In short, this is a book about the ideas of calculus and how to build and internalize those ideas in a
 first course through an inquiry based approach. While the pure mathematicians may quail loudly at such an approach, it’s hard to argue that the active manner of presentation has remarkable clarity and insightand that such an approach would be very helpful to the absolute beginner, such as a talented high school student trying to learn it on his or her own. But even more so, I think an exceptional honors course in calculus could be built from it using it as a supplement to a rigorous treatment, such as Clark’s. In any event, a very interesting and inventive text and worth a look whether or not you agree with the authors’ approach or not. UPDATE:The first edition of the book is now (Spring 2015) complete and available not only for download, but print on demand at Boelkins' website and it's associated links.
Calculus II Micheal Hill University of Virginia Fall 2008 (G) Another run of the mill set of supplementary calculus notes for integral calculus, infinite series and the rest. Again, nothing special .The Calculus of Several Variables Robert C. Rogers September 29, 2011 (PG) This terrific textinprogress was something I stumbled across on the internet quite by accident when I was finishing up the hyperlink list  and wanted to double check if there was anything of interest I missed in the basic subjects. Good thing I checked or I would have missed this jewel. Rogers is probably best known to mathematics students as the coauthor of a truly wonderful introduction to partial differential equations at the beginning graduate level. This book, while pitched at an undergraduate audience, is written in the same masterly style and with the same balance between theory and applications. Contents: Introduction Precalculus of Several Variables Vectors, Points, Norm, and Dot Product Angles and Projections Matrix Algebra Systems of Linear Equations and Gaussian Elimination Determinants The Cross Product and Triple Product in R^{3} Lines and Planes Functions, Limits, and Continuity Functions from R
 to R^{n} Functions from R^{n} to R Functions from R^{n} to R^{m} Vector Fields Parameterized Surfaces Curvilinear Coordinates Polar Coordinates in R^{2} Cylindrical Coordinates in R^{3} Spherical Coordinates in R^{3} Differential Calculus of Several Variables Introduction to Differential Calculus Derivatives of Functions from R to R^{n} Derivatives of Functions from R^{n} to R Derivatives of Functions from R^{n} to R^{m} Gradient, Divergence, and Curl Differential Operators in Curvilinear Coordinates Differentiation Rules Eigenvalues Quadratic Approximation and Taylor's Theorem MaxMin Problems Nonlinear Systems of Equations The Inverse Function Theorem The Implicit Function Theorem Integral Calculus of Several Variables Introduction to Integral Calculus Riemann Volume in R^{n} The Change of Variables Formula Hausdorf Dimension and Measure Integrals over Curves Integrals Over Surfaces Fundamental Theorems of Vector Calculus Introduction to the Fundamental Theorem of Calculus
 Green's Theorem in the Plane Fundamental Theorem of Gradients Stokes' Theorem The Divergence Theorem Integration by Parts Conservative Vector Fields The book is pitched at a very unusual level for such a text. While it's not advanced or rigorous enough to serve as an advanced calculus text the way Shurman or Munkres can, the prerequisites are very extensive for an undergraduate text: a full 2 year course of "mechanical" single and multivariable variable calculus, a semester each of linear algebra and basic differential equations and basic naive set theory. This list also implies Rogers expects the student to have some experience with formal proofs.Like Nitecki's more advanced bookinprogress, Rogers chooses to focus largely on theanalytic,geometry and algebraic structure of R^{2 }and R^{3} and downplaying higher dimensions for most of the book. This allows Rogers to be more careful then the average multivariable calculus text, in addition to supplying an enormous amount of intuition and visuals. However, Rogers' book is somewhat different from similar online "careful but not quite rigorous" sources like Jones and Stahl in several ways. The first is the very extensive prerequisites for a relatively low level treatment of multivariable calculus, which allows the author to present the subject concisely without lacking essential detailsit also allows for many definitions, examples and techniques that wouldn't be possible without this assumption. For example, Rogers covers orientation on both curves and surfacessomething that's very awkward to discuss without vector space theory. It also allows him to develop many physical examples using differential equations, such as the Guass' law in electromagnetism and the Laplacian of conservative vector fields and their gradients.Lastly, it allows him to be completely rigorous and careful in the presentation of both the geometry of Euclidean space and differential calculus in R^{n}. The linear algebra requirement, in fact, is kind of questionable since Rogers' uses many chapters to give a fairly comprebensive review of linear algebra. Theorems are not proven (references are given instead and many proofs are asked for in the exercises) , but precise definitions and many excellent examples are given.It's very well done,but to be honest, if a good linear algebra course is assumed, this much of a review really seems like overkill. Then again, considering how weak most students are in linear algebra these days, maybe not. Another way is that Rogers' isn't afraid to use or refer to concepts from advanced analysis when he feels he can present it reasonably simply to students. The main example is that Rogers gives a nonrigorous presentation of measure theory in order to motivate his careful constructions of the multivariable Riemann integral as well as the main "vector calculus integral theorems" He also gives an interesting intuitive development of the Hausdorff measure in the plane. Unfortunately, He defines what he calls the ndimensional volume on R^{n} which is essentially equivalent to the classical Jordan measure, which would allow for completely careful proofs of integration theorems in low dimensions with relatively smooth functions. The problem here is that since Rogers can't really give a precise formulation of the Jordan measure, he can't give exactly precise proofs the important results, either. Since a background in advanced calculus is enshewed, several needed results to precisely formulate the Jordan measuresuch as the upper bound property of the realsisn't available. The result is an annoyingbut unfortunately necessaryamount of handwaving in these chapters. I understand what Rogers is trying to do heregive an intuitive formulation that students can use as a foundation for a precise formulation in followup courses. And I have to admit, it's better then most attempts at this. But after he was so careful in the other sections of the book, this really feels like a big let down. I suppose given the level of the book, it really wasn't avoidable. Hopefully, the outstanding introductory presentation will inspire students to go on to advanced calculus courses and the rigorous formulation. In any event, Rogers has produced a first rate work for beginners that I hope will become one of the truly standard texts for either a standard course or an honors course with supplemented. proofs. Very highly recommended to both students and teachers of calculus.
 Vector Analysis Ivan Avramidi New Mexico Institute of Mining and Technology Socorro May 19, 2004 (G) A surprisingly broad and comprehensive treatment of both classical and modern vector analysis for applications. Contents: Linear Algebra Vector and Tensor Algebra Geometry Vector Analysis Integration Potential Theory Basic Concepts of Differential Geometry Applications The notes are in "bullet point" form, where observations, definitions and theorems are presented in rapid fire procession and there are no pictures. Worst of all, there are no proofs. None. Nada. That being said, there's a lot of examples and topics not usually covered in more rigorous courses in a very lucid fashion, such as a terrific introduction to tensor notation and he Einstein convention. The final chapter is a treasure trove of physical applications to both classical and modern physics, such as mechanics, thermodynamics, fluid flow and elasticity. While I wouldn't recommend these notes as a sole source for either vector analysis or differential geometry courses at any level, they would certainly make a wonderful supplement and should be used that way. Highly recommended as a supplement, especially to students with an interest in the sciences.
 Calculus Lectures Richard Earl University of Oxford (G)Earl's notes are the current prototype for the "methods in calculus" courses at top British universities. This bears some explanation for those unfamiliar with the UK school system. These are semiapplied calculus courses that are designed to strengthen the basic knowledge of the mechanical aspects of calculus which the student was assumed to have learned in preuniversity studies, but the student may have a shaky grasp of while taking more advanced courses in mathematics or the sciences. They are intended to complement the purely theoretical "Analysis" sequence for pure majors and serve as the bulk of the analysis training for applied majorsalthough it's fairly common for the more serious applied majors to take the rigorous courses as well. These notes focus on the aspects of single and multivariable calculus that are critical in the solutions of ordinary and partial differential equations in R^{n}.Contents: Standard integrals, integration by parts,order of an ODE,separation of variables,general linear homogeneous ODEs,integrating factor for first order linear ODEs, second solution when one solution known for second order linear ODEs,first and second order linear ODEs with constant coefficients. General solution of linear inhomogeneous ODE as particular solution plus solution of homogeneous equation,examples of finding particular integrals by guesswork. Systems of linear coupled first order ODEs. The calculation of determinants,eigenvalues and eigenvectors,introduction to partial derivatives,chain rule, change of variable; examples to include plane polar coordinates. Examples of solving some simple partial differential equations,Jacobians for two variable systems, calculations of areas including basic examples of double integrals, Gradient vector; normal to surface, directional derivative,critical points and classification using directional derivatives (nondegenerate case only),Laplace’s and Poisson’s equation, including change of variable to plane polar coordinates and circularly symmetric solutions,the wave equation in two variables, including derivation of general solution. The notes are completely devoid of proofssomething I'm not crazy about. But the purpose of these notes are to strengthen computational skill and understanding. Appropriately, they are filled with many challenging computational examples and exercises, with many complete solutions.
 Calculus of Two or More Variables Eamonn Gafney University of Oxford 2010 Multivariable Calculus Eamonn Gaffney University of Oxford 2014 (G) 2 sets of notes for the calculus
 "methods" course at Oxford for applied majors and to supplement the pure mathematics sequence. They have considerable overlap with the complete notes of Earl above. but this one is more careful and proves many results carefully and with many pictures and calculations. I like practically all these notescombined with a more careful treatment like Cook or C.H. Edwards’ Dover book, they could very useful indeed for a serious mathematics student.
 Recommended.
 A Collection of Problems in Differential Calculus Problems 20002010 Veselin Jungic Petra Menz Randall Pyke Simon Fraser University December 6, 2011 (G) A very large collection of exercises for a standard calculus course on topics of differential calculus. Contents: Limits And Continuity, Differentiation Rules, Applications of Derivatives, Parametric Equations and Polar Coordinates. The problems are quite diverse and varied in nature; from limits and continuity, extrema problems, applications to mechanics and differential equations and more. And they all come with complete solutions. Unfortunately, many of these problems looked sadly familiar to me. Sure enough, many of them are drawn from James Stewart's ubiquitous text and the accompanying solution manual. Still, it's very handy to have all these exercises,complete with their solutions, collected into a nice free source for mathematics majors. Recommended as a supplement to a standard calculus course.
 Calculus A Modern Approach Volume II Multivariable Calculus Jeff Kenisly And Kevin Shirley East Tennesee State University Fall 2013 (G) A very impressive online bookinprogress on calculus. I was trying to think about what to say about the content and aims of this book, it's purpose,it's organization,etc. But Kinisly and Shirley have a homepage for the single variable part of the book where they do a far better and more detailed job of explaining their goals and motivations then I could. (There's a link to the multivariable partwhich is less developed and still in the drafting stagesat said homepage.) Basically, both authors were disappointed with the archaic style of how calculus was taught to beginners and wanted to create a calculus course that could serve as a foundation for all future studies by mathematics, physical and social science students that didn't require those students to completely relearn the subject as graduate students or professionals. They wanted to construct a course that accomplished basically 4 things: a) Use modern mathematical terminology without strict modern rigor; b) Present applications that are not only current and relevant to the actual future careers of the students, but are actualized as in the real world using computer algebra programs and their programming languages, c) Corrects many of the subtle misconceptions and distortions of calculus standard textbooks and mathematicians teaching it have introduced and perpetuated due to their inability to properly express calculus to beginners(for rather striking and alarming examples, see the brilliant article by the authors here). and most importantly, d) one that actually used educational and psychological research about the actual manner students learn calculus to organize and develop the material effectively. It's clear the authors have given the teaching of calculus an enormous amount of thought in writing this book and the result is remarkable. The full table of contents can be found here. What's more important then the table of contents, which really doesn't indicate how unique this book really is, is the author's description of what makes their book different from the other 11,798 books on the subject. My own impressions of the book? As I said, I was very impressednot only with the very original and deliberate construction of the book, but in the very determined manner the authors set about to achieve their critical ends. This is a book with a very specific philosophy and set of goals and this is clear on every page, problem and discussion. I was particularly impressed with how the authors sought to define everything precisely but were flexible on rigor, which they correctly realize will confuse absolute beginners. That is not to say the book is devoid of rigor, there are a number of optional rigorous sections that are very well done, such as the definition of the limit. But the authors use rigor carefully and sparingly and use it only when they believe it will clarify rather then obscure. I was also very impressed with the exercises and selection of applications, many of which are unusual for a beginning calculus text, such as applications of tangent lines in optics and the use of Pascal's Triangle in the solution of the differentiation of higher order polynomials. I really liked this book and would heartily recommend both the book and the wonderful websites by the author for not only students of basic calculus, but their teachers, who I think would benefit enormously from the author's commentaries on the crisis of current calculus courses. (And I'll give someone 10 bucks who can say that last sentence 5 times fast.) Highly recommended!
 Calculus Alexander Belton University of Lancaster MATH143 (G) Another very standard second semester applied calculus course covering the basics of single variable differentiation and integration. Contents: Introduction to differentiation using graphical methods. Differentiation from first principles (up to thirdorder polynomials only). Basic differentiation using formulae. Chain, product and quotient rules. Secondorder and higher derivatives. Stationary points and their classification. Parametric, implicit and logarithmic differentiation. Integration The definite integral (as signed area). Indefinite integration (as the process inverse to differentiation). Integration of trigonometric functions using product and doubleangle formulae. Integration of rational functions by completing the square and by partial fractions. Integration by parts and by substitution. Applications: arc length of a plane curve; area and centroid of a plane region; surface area, volume and centre of mass of a solid of revolution.Numerical methods The trapezium rule. Simpson’s rule (the derivation is not examinable). The NewtonRaphson method. Taylor and Maclaurin series; estimation of integrals. L’Hopital’s rule. Some nice examples and exercises, but this is all pretty typical and nothing you can't get from the more substantial and superior sources like Dawkins or Strang. Worth a look, but nothing major.
 Vector Calculus Carol Ash University of Illinois UrbanaChamplaign 2013(G) A very standard, geometric and nonrigorous course in vector analysis for engineering and physical science students. Contents: CHAPTER 1 VECTOR DERIVATIVES CHAPTER 2 COORDINATE SYSTEMS CHAPTER 3 COMPUTING LINE INTEGRALS AND SURFACE INTEGRALS CHAPTER 4 INTEGRAL THEORY CHAPTER 5 COORDINATE SYSTEMS CONTINUED FROM CHAPTER 2 Ash is the wife of the aforementioned Robert B. Ash, who also teaches at UrbanaChamplaign, for those who are wondering. She seems to have as much concern and talent for teaching as her husband, despite the very run of the mill nature of these notes. While not completely devoid of proofs, it's clear the overriding concern here is teaching students how to use classical vector analysis to solve equally classical problems in physics and engineering, such as fluid flow and path length. Still, as I said, Ash writes very lucidly and there are many excellent diagrams,exercises and solved problems. A good book for a basic Calculus III/ Vector Analysis course for undergraduates, but not exceptional. There are many better sources we cover on this list,particularly for mathematics rather then physics majors.
 Calculus I Keith K. C. Chow Hong Kong University of Science And Technology Fall 2013 (G)
 Calculus II Keith K. C. Chow Hong Kong University of Science And Technology Spring 2014 (G)A very unusual set of supplementary notes for the first 2 semesters of a calculus sequence. Contents (I)Functions 1.1 Fundamentals of Functions Linear Functions and Lines 1.3 Quadratic Functions and Parabolas 1.4 Exponential Functions 1.5 Logarithmic Functions 1.6 Trigonometric Functions 2 Limits and Continuity 2.1 Limits of Functions 2.2 Techniques for Evaluating Limits 2.3 Existence of Limits 2.4 Continuity 3 Differentiation 3.1 Derivatives 3.2 Tangent Lines and Differentials 3.3 Basic Rules for Differentiation 3.4 Chain Rule 3.5 Transcendental Functions 3.6 Higher Derivatives 3.7 Implicit Differentiation 4 Applications of Differentiation 4.1 Increasing and Decreasing Functions 4.2 Relative Extrema 4.3 Absolute Extrema on a Closed Interval 4.4 Curve Sketching 4.5 Mean Value Theorem 4.6 L’Hopital's Rule 4.7 High Order Approximation 4.8 Business and Economics Applications Integration 5.1 Indefinite Integrals 5.2 Some Elementary Examples 5.3 Definite Integrals 5.4 Properties of Definite Integrals 5.5 The Fundamental Theorem of Calculus 5.6 Techniques of Integration 5.7 Applications of Integration 5.8 Improper Integrals (II) 1 Applications of Integration 1.1 Applications in Geometry 1.2 Applications in Physics 1.3 Applications in Business and Economics 1.4 Improper Integrals 2 Methods of Integration 2.1 Integration by Substitution 2.2 Integration by Parts 2.3 Integration by Partial Fractions 2.4 Integrals of Various Special Forms 2.5 Miscellaneous Techniques 3 Differential Equations 3.1 What are Differential Equations? 3.2 Classification of Differential Equations 3.3 Solutions of Ordinary Differential Equations 3.4 FirstOrder Linear Differential Equations 3.5 Separable Equations 3.6 The Existence and Uniqueness Theorem 3.7 Modeling with FirstOrder Equations 4 Infinite Series 4.1 Sequences 4.2 Infinite Series 4.3 Tests for Convergence and Divergence 4.4 Power Series 4.5 Taylor Series These notes have very few proofs of theorems. That would indicate they're purely practical notes with little rigor, but you'd be wrong. The definitionssuch as the definition of a limit and neighborhood of a point on the real lineare done extremely carefully, with many equally careful and lucid examplesmore importantly and unusually for a basic course, there are many simple counterexamples to theorems, such as nondifferentiable functions and functions that lack absolute extrema. There are also many instructive diagrams and applications. Chow's notes aren't complete enough to serve as a calculus text, but I certainly recommend them very highly as a supplement to such a course.
 Elementary Calculus For Economics Christopher Cooper McQuarrie University (G) A set of notes on the applications of calculus to economics that are even more intuitive then the above Concepts in Calculus. They lack the substance of the other notes,thoughand really don't know how useful they'll be. I would just assign Concepts to such studentsthey'd get a lot more out of them.
 Concepts of Calculus Christopher Cooper McQuarrie University
Techniques of Calculus Christopher Cooper McQuarrie University (G) This is a strange pair of notes for 2 different versions of a single variable calculus course. Contents: (Concepts) CONTENTS 1. GRAPHS AND THE STORIES THEY TELL 2. THE xy PLANE 3. DIFFERENTIATION 4 ELEMENTARY FUNCTIONS 5. TANGENTS AND NORMALS 6. MAXIMA AND MINIMA 7. OPTIMIZATION PROBLEMS 8. NEWTON’S METHOD 9. INTEGRATION 10. AREAS BETWEEN CURVES 11. NUMERICAL INTEGRATION 11.APPENDIX: FORMULAE AND RULES (Techniques) 1. DIFFERENTIATION 2.INTEGRATION 3. TRIGONOMETRIC FUNCTIONS 4. TECHNIQUES OF INTEGRATION 5.SEQUENCES AND SERIES The first set of notes is a purely intuitive,informal course in calculus, as one would expect either good high school students or freshman college students in the US to take. What's interesting here is that Cooper works unusually hard to develop the intuitive part of the coursehe introduces many geometric and visual tools to present the conceptual ideas in as careful and complete but nonrigorous a manner as possible. For example, he introduces the derivative and the integral as geometric functions defined on graphsthe derivative is a slope function on a graph and the integral is the area function of a graph. He uses many careful geometric arguments, many of them embedded in specific examples, such as the slope change of a mountain trail, the zero slope at the top of a hill, the rate of change of a curve representing weekly savings and many many more.The notes also contain many excellent exercises that help develop the student's geometric intuition in both one and 2 dimensions. Not only is this is an extremely readable and insightful way to proceed with introducing calculus to beginners, it could be equally useful as a supplement to a purely theoretical treatment like Muldowney above or the book by Spivak. The second set of notes is a bit more standard and less inventive, but quite nice nevertheless. They are the notes for a more rigorous "standard" calculus course that presumes the material from the first course is known. Most of the basic results are carefully defined and provenwith the exception of the most sophisticated properties that rely on the completeness property of the reals, such as the Intermediate Value Theorem. Cooper also, as in the previous notes, includes many excellent examples and exercises. As a result, these notes can serve as the basis for a gentle advanced calculus or honors calculus course.Together with the previous notes, they can be used for an excellent and complete year long calculus course for strong students. Such a course would also be very versatile since the instructor can decide how many of the proofs in Techniques he or she wishes to cover in depth. However you choose to use them, Cooper has written a very readable and informative set of notes that I highly recommend to all students and teachers of calculus.  Calculus III Grant Lakeland University of Wisconsin Urbana Champlaign Fall 2013
 Calculus III Grant Lakefield UIUC Fall 2013 Class Diary and Homework Assignments, 2013 (G) These are the links to Lakefield's scanned handwritten notes to a typical plug and chug multivariable calculus course based on Stewart. They're nicely written and clear with good diagrams, but basically just rehash what's in the textbook without much in the way of new insights. You can take a look, but I really don't think there's much worth bookmarking here for a calculus student.
 Calculus II Class Notes Second Semester Calculus Hang Huang (Amy) University of Wisconsin UrbanaChamplaign Spring 2013 (G)Very standard and brief set of handwritten notes to accompany a second semester calculus course.Nothing to see here, move on.
 Calculus for Biology I Joseph M. Mahaffy San Diego State University Fall 2012 Calculus for Biology II Joseph M. Mahaffy San Diego State University Spring 2013(G)A very lively and unusual applied calculus course for serious biology or chemistry majors needing to learn the bare bones of mathematics needed for the burgeoning field of mathematical biology. Mahaffy is a mathematical biologist, which means his approach to the material is rather different then that of either field. Contents: (I) Introduction Linear Models Least Squares Analysis Function Review and Quadratics Other Functions and Asymptotes Allometric Modeling Discrete Malthusian Growth Linear Discrete Models Introduction to the Derivative Velocity and Tangent Lines Limits, Continuity, and the Derivative Rules of Differentiation Application of the Derivative  Graphing The Derivative of e^{x} and ln(x) Product Rule Quotient Rule Chain Rule Optimization Logistic Growth and Dynamical Systems Applications of Dynamical Systems (II) Introduction Logistic Growth and Dynamical Systems Applications of Dynamical Systems Optimization Trigonometric Functions Differentiation of Trigonometric Functions Newton's Method Introduction to Differential Equations Linear Differential Equations Numerical Methods for Differential Equations Differential Equations and Integration Separable Differential Equations Integration by Substitution Riemann Sums Definite Integral Integration by Parts Qualitative Analysis for Differential Equations Competition Models The notes are fascinatingthey aren't rigorous in the classical analytic sense, but they aren't really plug and chug notes,either. Although the emphasis in the notes are, of course, on applications, Mahaffy doesn't completely avoid proofs either. For example, he develops limits in the context of providing a careful definition of the derivative as the convergence of a sequence of secant lines to a specific point on the graph of a function. Another good example is how he develops Riemann sums via the midpoint trapezoid rule. What these approaches have in common is that they emphasize the geometric aspects of calculus over the purely analytic ones. This makes complete sense as these are the aspects that will be of most importance in physical applications. The number of applications in these notes is truly extraordinaryand all of them from biological systems. Why this area is usually neglected in most standard calculus courses and texts is a mystery to me given the incredible growth in this area of applied mathematics over the last 30 years. Mahaffy rectifies this with a traincar full of applications to population growth, DNA and RNA replication rate models, disease modeling, tumor growth and Ricker's function and so much more. This is a treasure trove for serious students and teachers of both the biological and mathematical sciences. Moreover, it should be required reading for both pure and applied mathematicians to convince them that mathematical biology can and should be introduced to undergraduates. Very highly recommended!
 Calculus IIII by Alan Weinstein and Jerrold Mardsen 2nd edition Complete Texts And Study Guides (G) This classic trilogy of calculus textbooks by 2 eminent mathematicians at the University of California at Berkeley are a benchmark of calculus texts in recent decades.The fact the textbooks and thier study guideswith many complete solutions to the exercisesare available free for online download is a real gift to both calculus students and their teachers. Contents: Preface How to Use thls Book: A Note to the Student Orientation Quizzes Chapter R Review of Fundamentals Chapter I Derivatives and Limits Chapter 2 Rates of Change and the Chain Rule Chapter 3 Graphing and MaximumMinimum Problems Chapter 4 The Integral Chapter 5 Trigonometric Functions Chapter 6 Exponentials and Logarithms Chapter 7 Basic Methods of integration Chapter 8 Differential Equations Chapter 9 Applications of Integration Chapter 10 Further Techniques and Applications of lntegration Chapter 11 Limits, L'Hopital's Rule, and Numerical Methods Chapter 12 infinite Series Chapter 13 Vectors Chapter 14 Curves and Surfaces Chapter 15 Partial Derivatives Chapter 16 Gradients, Maxima and Minima Chapter 17 Multiple integration Chapter 18 Vector Analysis The syllabus is almost redundant in it's standardnessthere are no surprises in what's covered in any of these texts. The authors are very carefulalthough not entirely rigorous,which is appropriate at this leveland complete. It's an excellent representation of what a good,standard old school calculus course at a good US university looked like back in the daybefore Mathematica, the calculus reform movement and the general push to reinvent the wheel in calculus in all it's myriad manifestations. Highly recommended to both students and teachers of calculus.
 Calculus III Will Sutherland Queen Mary College of London 2013 (G) A multivariable calculus course focusing on vector integral calculus and has an interesting practical introduction to Fourier series and their applications. Contents: Arclength of plane curves: length of a parametric curve, length of a curve y=f(x) . Length of the circumference of a circle, ellipse. Area and length in polar coordinates.Vector fields, line, surface and volume integrals.Grad, div and curl operators in Cartesian coordinates. Grad, div, and curl of products etc. Vector and scalar forms of divergence and Stokes's theorems. Conservative fields: equivalence to curlfree and existence of scalar potential. Green's theorem in the plane.Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical and cylindrical polar coordinates as examples.Fourier series: full, half and arbitrary range series. Parseval's Theorem. Laplace's equation. Uniqueness under suitable boundary conditions. Separation of variables. Twodimensional solutions in Cartesian and polar coordinates. Axisymmetric spherical harmonic solutions.Notice partial derivatives and differentiation in R^{n} aren't covered, so it's not quite equivalent to a typical US "Calculus III" course. This course focuses largely on intuition and examples, proofs are sparse. That being said, everything is done clearly and readably, and the final section on Fourier analysis is quite illuminating and will make a nice introduction. There are also a lot of nice exercises. Nothing special, but a good study aid for basic calculus.
 Calculus A Modern Approach Volume I Single Variable Calculus Jeff Kenisly And Kevin Shirley East Tennesee State University Fall 2013
 MultiVariable Calculus and Linear Algebra Alexander Nagel Department of Mathematics University of WisconsinMadison Spring 2014 Course Materials (PG) Very terse and brief set of notes for an undergraduate seminar on linear algebra and calculus based on Apostol's classic honors calculus book. Nothing here you can't find elsewhere in linear algebra or vector calculus notes. Move on.
 Honors Calculus Willard Miller University of Minnesota Spring Semester 2007 (G) This is the website for an honors course based on Simmon's outstanding calculus book. The only reason this link is here is the section towards the bottom of the page called "Rocket Science"containing some beautiful and very informative notes on applications of calculus to the sciences, such as epicycles,conic sections in polar coordinates, Kepler's laws, and the gravitational force. Well written and explained and will make a very nice supplement to a theoretical treatment.
 Honors Calculus II Nathan Dunfield Math 231 University of Wisconsin Urbana
Fall 2008 Course Materials Calculus III Honors section Nathan Dunfield University of Wisconsin Urbana Course Materials Spring 2008 (G) These are 2 very solid sets of lecture notes from Dunfield's honors calculus course at UrbanaChamplain.Contents: (II) Introduction; foundations of integration. foundations of integration and the Fundamental Theorem. Integration by parts Trigonometric integrals. Trig substitution. Improper integrals.Improper integrals II.Improper integrals III; Partial Fractions.Partial Fractions II; Integration in elementary terms.Midterm Review. Differential Equations.Limits of sequences. More on sequences.Convergence of monotone sequences; Intro to series Infinite series. Series with positive terms: the Integral Test.Comparison and Limit Tests; Alternating series.Alternating series; Absolute convergence.Absolute convergence.Conditional convergence; Intro to power series.Review for Miderm II. Power series.Differentiating and integrating power series.Taylor series.Taylor's Theorem.Applications of Taylor series.Applications of Taylor Series II.Fourier series.Fourier series II; intro to Chapter 9. Plane curves.Properties of plane curves.Review for Midterm III. Polar coordinates. Properties of curves in polar coordinates. Conic sections: ellipses.Conic sections: parabolas and hyperbolas.Complex numbers.Complex numbers II.Miscellaneous.Final review. (III) Introduction and outline of course.Vectors and the dot product. More on the dot product. Matrices and linear transformations.Linear transformations and matrix multiplication. Properties of matrix multiplication; the determinant.Cross product.Functions of several variables: graphs and level sets.Level sets in 3dimensions. Limits in several variables.Limits and continuity. Derivatives.More on derivatives.The chain rule.Derivative miscellanae.More on the gradient.Introduction to min/max. Review.Taylor series. Unconstrained min/max. Extreme Value Theorem; Intro to constrained min/max. Lagrange Multipliers; Partial Differential Equations. More on the sinking of the Sleipner A.Feb 25: Linear Programming. Interesting applications: Trucking, Biology. Curves and their lengths. Integration over paths.More on integration over paths.Conservative vector fields I.Conservative vector fields II.Midterm Review.Curl and conservativity; multivariable integration.More on multivariable integration.Change of variables I.Change of variables II.Triple integrals.Change of coordinates in 3dimensions.Surfaces in R3.Surface area and integration on surfaces.More on integrating over surfaces.Integrating vector fields over surfaces. Green's Theorem. Aside: Planimeters and Green's Theorem.The Divergence Theorem.Gauss's Law.More applications of the divergence theorem.Midterm review. Stokes Theorem I. Stokes Theorem II.Conservative vector fields and Stokes Theorem.Surfaces bounded by knots; Maxwell's equations. Differential Forms I.Differential Forms II.Differential Forms III Stokes' Theorem for manifolds.Why Stokes' Theorem works. Cohomology to cosmology.Additional resources on cosmology: Jeff Week's really excellent book and website, especially the "torus games" and "curved spaces". Final Review. Every university has it's own version of honors calculus; larger and more prominent ones have several versions of the course depending on the preparedness and talents of the entering freshman. The problem with such a course, as I've commented here, is to decide how much theory to cover i.e. how different to make the course is from the standard one. UrbanaChamplain has a very interesting solution to this problem: They have the same main section for both honors and standard calculus students which covers the standard calculus material and they add a special additional section the honors students register for where the more difficult material is covered. The notes here are Dunfield's notes for the standard section for parts II and III. The textbook for part II's main section was James Stewart's book and Dunfield's notes don't deviate too much from that book in it's presentation of the materialthere's not many proofs or rigorous definitions in this course. He does cover fairly well 2 additional topics that one usually sees in honors calculus coursesFourier series and complex numbers. They are done completely nonrigorously, but the exposure will do beginners good, especially those in the physical sciences. Overall, very standard presentation, but very visual and lucid with many examples.The Part III are much more original and detailed, giving a beautiful and careful course in vector analysis in R^{3 }, focusing on vectors as linear transformations. Dunfield relies much more heavily on his notes then the chosen textbook, the excellent Vector Calculus by Miroslav Lovric ,then he did in the previous course. Linear algebra isn't explicitly covered, but by limiting most of the presentation to the plane and 3space, Dunfield makes it unnecessary. The linear mapping approach to limits, continuity and differential calculus in higher dimensions is prominent, all examples and proofs are done carefully and Dunfield doesn't hesitate to state advanced results, like the Fixed Poiut Theorem, without proof. There are many applications as wellnot only the standard ones to mechanics,electromagnetism and fluid flow, but to biology, economics and finance. He draws examples directly from the literature, such as one modeling the collapse of Sleipner oil rig and a PDF of an entire paper modeling metabolic pathways via the vector form of flux partial differential equations. There are many wonderful hand drawn figures and differential forms are used to explain vector integration. In summary, while Dunfield has written fine lecture notes and both make for excellent lecture notes in a strong calculus course, it's the part III notes that really represent a superior source unto themselves. The Part III notes are highly recommended for all teachers and students of serious multivariable calculus courses.  Calculus With Vectors Jay Treiman Western Michigan University 2014 Version (G) I found this book draft in the very last stages of preparing the first online version of the Freemathversity. and was very pleasantly surprised by it. Contents: Chapter 1. Points and Vectors Chapter 2. Limits and Derivatives Chapter 3. More on Limits Chapter 4. Rules for Finding Derivatives Chapter 5. Applications of Limits and Derivatives Chapter 6. Integration Chapter 7. Th Cross Product Chapter 8. More Techniques of Integration Chapter 9. Applications of Integration Chapter 10. Series Index Appendix A. Mathematical Preliminaries A.1. Order of Operations A.2. Algebra and Functions A.3. Trigonometry I suggest all students and teachers considering using this book read carefully through Treiman's preface, which explains his motivations for writing yet another calculus text. Basically, his intent was to write a treatment of calculus that could be used as a common starting point in mathematics for either mathematics, physics or engineering students without any special considerations for any specific group. I was really impressed with this goal. The book clearly is intended for students with bare minimal backgrounds in high school mathematics and emphasizes intuition and practical methods, I found the book well written, very informative and surprisingly original. While Treiman isn't setting out to write a rigorous text, he also wants it to be careful enough that mathematics majors and strong high school students can begin their studies with it and he accomplishes this several ways. One of the ways he does this is to shift from the εδ functional approach to limits to a sequential approach to limits.I find it interesting many texts are now making this shift. One reason is the fact that the intuition given in most calculus texts coincides more closely with the sequence definition of limits of functions. A side benefit is that the explanation of convergence and sums of infinite series then becomes much easier. Another thing that's original is the derivation of the derivatives of sine and cosine, leaving explicit discussion about the connection between increasing and decreasing functions and derivatives until the Mean Value Theorem is available, rethinking the use of “tables” of integration and reordering the techniques of integration to allow the use of partial fractions for trigonometric integrals.Lastly, the early introduction of vectors and vector valued maps allows for a deeper presentation of basic physics examples, which the author thinks is critical for nonmathematics majors. There are many beautiful computer generated graphs, examples and various exercises of different levels. The result is a very versatile textbook in calculus that can and should be adopted for many different kinds of beginning calculus courses, from high school to university level and would be excellent for self study as well. Very highly recommended for the beginner in calculus! Update: The final version of the book has been published by SpringerVerlag in January 2015. I don't know if the author will continue to allow the earlier drafts to remain online. So download the free version while it's still available!
 Ready.Set.....Calculus Precalculus Notes prepared by Department of Mathematical Sciences
Rensselaer Polytechnic Institute Rensselaer Polytechnic Institute 2007  Multivariable Calculus for Engineers Masha Vlasenko University College Dublin Semester 1 2013