nculwell.github.io

Calculus

Standard calculus textbooks

These books are those designed to be the required textbook for a standard high school or university calculus course. Many of them have had a large number of revisions, which makes it likely that you can find slightly older but still perfectly useful copies on the used market. They try to have everything – proofs, intuitive explanations, illustrations, problems – but they aren’t necessarily great at anything. These books are widely derided by math majors, though I suspect this to be in part motivated by elitism and by grad students’ dislike of teaching lower-level math courses. In any case, these are useful even if you just want a cheap source of problems to practice with.

The OpenStax series follows a very common sequencing of topics, which I list here as a representative of what you can expect to find in the books in this section. OpenStax splits these into three separate books, but most series combine them all into one huge book of about 1000 pages. It’s also common to see 1 & 2 combined, and a separate book for 3.

• Volume 1: functions, limits, derivatives, integration.
• Volume 2: [more] integration, differential equations, sequences and series, parametric equations, polar coordinates.
• Volume 3: parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, second-order differential equations.

Here are some books in this category, including the most popular ones that I know of. I’ve tried to link mainly to the best-priced editions so you can get a book for $5-10 (though prices change).

• OpenStax: Strang, Herman et al. Calculus. (FREE ONLINE: Vol 1; Vol 2; Vol 3)
• Strang, Calculus (FREE ONLINE: Website, Direct link to PDF)
• Simmons. Calculus with Analytic Geometry (2e)
• Stewart. Calculus: Early Transcendentals (5e) – Has huge adoption in both high schools and universities.
• Leithold. (TC7; TCWAG 6) – I’ve seen Louis Leithold credited as the creator of the modern “standard” calculus textbook.
• Larson, Hostetler, Edwards. Calculus (8e; 7e; 6e)
• Thomas et al. Thomas’ Calculus (12e; Upd 10e; 9e) – Thomas retired in 1978, and editions since then have different co-authors, principally Ross Finney.
• Lial, Greenwell et al. Calculus with Applications (9e)
• Anton, Bivens and Davis. Calculus (7e; 7e Student Resource Manual; 9e multivariable)
• Marsden and Weinstein. Calculus (Vol I: 2e; Vol II: 2e; Vol III: 2e)
• Rogawski. Calculus: Early Transcendentals (2e)
• Briggs and Cochran. Calculus: Early Transcendentals (1e)
• Bittinger and Ellenbogen. Calculus and Its Applications (9e)

Elite calculus textbooks

These are books that math majors get enthusiastic about. They are typically used in elite universities and honors courses. They emphasize theory from the ground up beginning with the real number system, but they have fewer calculation problems and applications problems (no applications, in some cases). Note that many, maybe most students who read these books do not actually start calculus with them: often they’ve seen calculus before, maybe they’ve even had a course or two.

• Spivak, Calculus (4e; 3e; 3e/4e answer book)

  Spivak is the most revered of this class of textbooks. It’s also the most current: its fourth edition was published in 2008. Spivak has a chatty, conversational style. The book begins by establishing the properties of the real number system, and covers limits, differential and integral calculus, and infinite sequences. Unlike the books that follow, Spivak does not deal with calculus of several variables, so you need another book for that. (He has written a book on that subject, Calculus on Manifolds, but it’s not considered an introductory book.) Spivak’s book has been called an introduction to analysis; you might also call it “calculus for mathematicians”.

• Apostol, Calculus, volumes I and II (Vol I, Vol I intl, Vol II, Vol II intl)

  Apostol is, according to my own rather cursory research, the most widely used of this class of textbooks in college classrooms. Volume I in particular is excellent. The language is much more typical “math-speak” than Spivak’s, but the approaches to the material are unusual. Apostol begins with integration before covering limits and differential calculus. This approach works, but it is incovenient if you’re trying to use it in combination with another textbook or an online course, since almost all other courses follow the order limits-derivatives-integrals. Apostol’s problem sets are excellent, perhaps the best of the bunch. Chapter 1 in particular is a crash course in proof by induction that will very quickly get you comfortable with manipulating sums.

  Volume II introduces linear algebra and calculus of several variables. It seems to be regarded as not as good as Volume I, particularly because its presentation of linear alegebra is dated. However, it does cover some interesting topics not often seen in a calculus textbook: systems of differential equations, probability, and numerical analysis (i.e. approximation methods).

  The standard edition of these books is, unfortunately, exhorbitantly expensive. However, at least the physical quality of the books is extremely good (the one I have is, at least).

• Courant, Differential and Integral Calculus, volumes I and II

  • Courant (Wiley 2nd edition). Vol I Vol II
  • Courant and John (Springer). Vol I, Vol II/1, Vol II/2
  • Courant (original). Vol II
  • Courant (1937/1936 Blackie edition, printed 1961). At Archive.org: Vol I, Vol II

  Richard Courant, who died in 1972, was a highly respected German-American mathematician. With Herbert Robbins, he also wrote What is Mathematics?, a remarkable work that attempts to explain a large swathe of mathematics to a popular audience. This calculus textbook is particularly noted for its wealth of physics applications, making it by far the most “applied” of this group. However, its problems are also notoriously difficult, so much so that they’re not really appropriate for a learner at this level, at least not without a supplement of easier problems to practice with.

  Since Courant has been dead for several decades, a newer edition was published (by Springer) with Fritz John as co-author. That edition is also highly respected. The most notable difference is that Courant and John splits Volume II into two parts and adds several sections of new material.

• Shahriari, Approximately Calculus (1e)

  A more recent book of this style. It’s intended as a second book on calculus. It’s full of difficult problems that push your theoretical understanding of calculus, approaching the topic in unusual ways.

Other textbooks of note

These books don’t fit cleanly into the categories that I described above.

Intuitive approaches

These take the attitude that emphasis on rigorous proofs gets in the way of understanding calculus.

• Ash and Ash, The Calculus Tutoring Book (1e paperback 1e hardcover)

  This book just doesn’t bother with theorems or proofs unless they are necessary. It focuses on giving you an intuitive grasp of calculus. I like it a lot. I wouldn’t advocate it as your only book on calculus, but I think it would make a very good first book. It covers the standard calculus curriculum as I decribed in the “Standard calculus textbooks” section above. Although it focuses heavily on pedagogy, I think that this may not be a good book “for dummies”, so to speak. It may be that stronger math students are the ones who benefit the most from this approach, because they are used to learning by understanding rather than by memorizing rules. Though I mention that proofs are largely dispensed with here, they are not entirely absent: I found that, when they were included, it was most often in support of a statement that I wasn’t sure if I believed, which seems the perfect balance. Doesn’t cover a few standard topics, such as differential equations or parametric equations. The paperback seems a bit flimsy, so if you’re planning to use this you might want to get the hardcover edition. This book is out of print, so get a used copy while you still can!

• Thompson and Gardner, Calculus Made Easy (Revised)

  An intuitive introduction to calculus. This isn’t thorough enough to be a course textbook, but it makes a good companion to a textbook. It’s for someone who’s just beginning calculus. The original version (by just Thompson) is in the public domain. It’s shorter and some people feel that it does a better job of keeping things simple. Keep in mind that it’s old! It was published in 1914. Its casual style emphasizes its old-fashionedness. Some might find this charming, others annoying. You can get it online here: https://archive.org/details/CalculusMadeEasy

• Kline, Calculus: An Intuitive and Physical Approach (Dover)

  The title pretty much says what this is. It’s older (originally published in 1967, updated in 1977), and was written by Morris Kline, a prolific writer on mathematics (and critic of math education) who authored several books for a popular audience.

• Blåsjö. Intuitive Infinitesimal Calculus (FREE ONLINE)

  Focuses on supporting calculus with geometric and physical explanations, and motivating it with the historical problems that motivated the development of calculus. Problems focus on understanding. Early emphasis on differential equations.

• Zeldovich and Yaglom. Higher Math for Beginners (Archive.org)

Others

• Thomas, Calculus and Analytic Geometry (c) 3rd Alternate

  This entry refers to the original series of textbooks written by George B. Thomas himself, up to the Fourth Edition (1969). As of 2015 they can still be bought used (quite cheap!), and thanks to the durable hardback bindings they used in those days, you can probably still find them in good condition. (Just hope some jerk didn’t scribble all over your copy.) Depending on who you ask, the pinnacle of the series was either the 4th Edition or the 3rd Alternate Edition.

  As advertised, there is a strong emphasis here on analytic geometry and applications. Thomas taught at MIT, and this is clearly a “calculus for engineers” approach.

• The calculus book from The Art of Problem Solving series.

• Lang, A First Course in Calculus (5e, Short Calculus, 4e @AbeBooks)

  Serge Lang is famous for his difficult graduate-level texts in a dozen different subjects, but he has also put his hand to writing a handful of textbooks for lower levels of math. Of those, probably the most successful is this, his calculus book.

  This book is really distinct from all the others here. It is relatively short. It ignores applications to focus on equations, but it is not really a rigorous textbook either. It has proofs, but only the ones that Lang thought were essential. It simply covers the topics of calculus in a clear and straightforward manner. There have been five editions, and each added more material (getting longer), until Lang actually brought back the original edition under the title Short Calculus. I have the Fourth Edition.

  Because of its focus on the standard topics without a bunch of fuss over applications, this is currently my preferred reference for anything that it covers.

• Körner, Calculus for the Ambitious (1e)

  CUP’s promotion says, “In a lively and easy-to-read style, Professor Körner uses approximation and estimates in a way that will easily merge into the standard development of analysis. By using Taylor’s theorem with error bounds he is able to discuss topics that are rarely covered at this introductory level.” I haven’t seen this book in the flesh yet, but it’s inexpensive.

• Stein. Calculus in the First Three Dimensions (Dover; Solutions manual (PDF))

• Kiesler. Elementary Calculus: An Infinitesimal Approach (FREE ONLINE; Dover)

• Lax and Terrell. Calculus With Applications

Free books

• https://openstax.org/subjects
• http://www.bodysoulmath.org/books/

Online calculus resources

MIT OCW has a series of online lectures for calculus. The lecturing isn’t outstanding, but the material covered is good for the most part. The lectures can be watched on YouTube here:

• https://www.youtube.com/playlist?list=PL590CCC2BC5AF3BC1
• https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38

Other online lecture offerings include ones from Ohio State and Khan Academy. Both seemed a bit slow and basic for my taste, whereas MIT’s was more my speed, but then again I’d already taken some calculus in high school so I wasn’t starting from scratch.

The University of Wisconsin-Madison also has free calculus texts that you can get online; they don’t look at all adequate as stand-alone texts for self study, but they do have problems with some solutions included. Whitman College also has free calculus texts.

• UW-Madison calculus students’ page, with links to PDFs of the latest texts
• Source code for the UW-Madison texts
• Whitman College’s Community Calculus page
• Fremlin’s problems with solutions (several subjects, Latex files)
• http://brianveitch.com/calculus/

Vector calculus

For multivariate calculus, there seems to be a general consensus that the texts that teach “all” of calculus don’t do a great job with it. (Leithold, for example, stops giving so many proofs at that point in the curriculum and states that they belong in a book dedicated to the subject.) The following texts are dedicated to this part of the calculus curriculum in particular (arranged in descending order according to my impression of their usefulness):

• Hubbard and Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach (5e at Matrix Editions, 5e at Amazon, 4e at Amazon)
• Schey, Div, Grad, Curl and All That (4e, 3e, 2e, 1e)
• Baxandall and Liebeck, Vector Calculus (Dover)
• Friedman, Advanced Calculus (Dover)
• C. E. Edwards. Advanced Calculus of Several Variables (1e)
• H. M. Edwards. Advanced Calculus: A Differential Forms Approach (3e, 2014 reissue, 3e, 1994)
• Shifrin, Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds (1e)
• Corral, Vector Calculus (FREE ONLINE)
• Marsden and Tromba. Vector Calculus (5e, 4e)
• Colley. Vector Calculus (2e, 1e, 2e)
• Matthews. Vector Calculus (1e)
• Bressoud. Second Year Calculus: From Celestial Mechanics to Special Relativity (1e)
• Flanigan and Kazdan. Calculus Two: Linear and Nonlinear Functions (2e)
• Bray. Multivariable Calculus (1e at CreateSpace, 1e) Author’s page: https://services.math.duke.edu/~cbray/mv/
• Shurman. Calculus and Analysis in Euclidean Space
• Lax and Terrell. Multivariable Calculus with Applications
• Callahan. Advanced Calculus: A Geometric View

The Feynman Lectures also discuss vector calculus in the context of electromagnetism, which is supposedly a great way to tie it in with your understanding of the physical, if you already have some background in physics. See Physics.

Calculus problem books

It may be that the problems in whatever textbook you’re using don’t give you as much practice as you want. One option is to get another textbook and do the problems from that, too. However, there are a number of books dedicated to providing problems. One advantage to these is that they generally tend to give you explanations of how to solve them as well.

• Demidovich, Problems in mathematical analysis [can be found online]
• Maron, Problems in Calculus of One Variable (c)
• Mendelson, Schaum’s 3,000 Solved Problems in Calculus (c) [reportedly has some errors]
• Kelley, The Humongous Book of Calculus Problems (c)
• Bluman, Problem Book for First Year Calculus
• Jones, Calculus: 1,001 Practice Problems For Dummies (1e) – By PatrickJMT of YouTube fame.
• REA. The Calculus Problem Solver (Unspecified edition)

John Erdman of Portland State University has posted a number of free problem books as PDFs on his website (answers to odd-numbered exercises given): http://web.pdx.edu/~erdman/

This site has links to lots of other sites with problem sets and practice exams: http://www.calculus.org

Differential Equations

Combined

• Simmons. Differential Equations with Applications and Historical Notes (3e, 2e)

• Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (9e, 8e)

• Edwards and Penney. Differential Equations and Linear Algebra (2e)

• Edwards, Penney, Calvis. Differential Equations and Boundary Value Problems: Computing and Modeling (5e, 4e (Edwards/Penney))

• Piskunov, Differential and Integral Calculus (Vol I: Mir/CBS; Vol II: Mir/CBS)

  From what I gather, this pair of volumes is legendary for its quality (though problems are too easy), but the physical quality of this edition is terrible.

ODEs

• Tenenbaum, Ordinary Differential Equations (Dover)
• Hurewicz, Lectures on Ordinary Differential Equations (Dover)
• Coddington, An Introduction to Ordinary Differential Equations (Dover)
• Brauer and Nohel. The Qualitative Theory of Ordinary Differential Equations: An Introduction (Dover)
• Arnold. Ordinary Differential Equations (MIT Press, tr. Silverman, Springer, tr. Cooke) - Springer is a later version.
• Coddington and Levinson, Theory of Ordinary Differential Equations (Krieger 1984)
• Imhoff, Differential Equations in 24 Hours: with Solutions and Historical Notes (1ed, 2015)
• Logan. A First Course in Differential Equations

PDEs

Basic

• Strauss. Partial Differential Equations: An Introduction (1e, 2e)
• Farlow. Partial Differential Equations for Scientists and Engineers (Dover)
• Logan. Applied Partial Differential Equations (3e, 2e)
• John. Partial Differential Equations (4e)
• Bleecker and Csordas. Basic Partial Differential Equations (1e)
• Zachmanoglou and Thoe. Introduction to Partial Differential Equations with Applications (Dover)
• Asmar. Partial Differential Equations with Fourier Series and Boundary Value Problems (3e Dover)
• Hillen, Leonard, van Roessel. Partial Differential Equations: Theory and Completely Solved Problems (1e)
• Olver. Introduction to Partial Differential Equations
• Logan. Applied Partial Differential Equations

Advanced (more prequisites)

• Vasy. *Partial Differential Equations: An Accessible Route Through Theory and Applications( (1e)
• Gustafson. Introduction to Partial Differential Equations and Hilbert Space Methods (Dover)
• Taylor. Partial Differential Equations (Vol I: Basic Theory 2e; Vol II: Qualitative Studies of Linear Equations 2e; Vol III: Nonlinear Equations 2e)
• Evans. Partial Differential Equations: Second Edition (2e)
• Folland. Introduction to Partial Differential Equations (2e)
• McOwen. Partial Differential Equations: Methods and Applications (2e)

Advanced Calculus

The term “advanced calculus” seems to be somewhat vague. Sometimes it means calculus of several variables, sometimes it means analysis, sometimes it means applications that become possible after you have a couple of years of calculus under your belt. The ones I list here straddle the line between calculus and analysis.

• Buck. Advanced Calculus (3e)
• Hildebrand. Advanced Calculus for Applications (Dover 2e)
• Widder. Advanced Calculus (Dover)

Calculus of variations

https://www.amazon.com/Calculus-Variations-Applications-Physics-Engineering/dp/0486630692 https://smile.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485 https://smile.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677