Great Calculus Reference Texts for Students Who Want to Learn More
The standard college calculus textbooks (popular examples are Anton, Larson, and Stewart, although Simmons seems to be a superior text to me) are supposed to provide thorough calculus training for large bodies of students with diverse background and intentions. They also form an indispensible resource for the instructor by supplying a large collection of problems for practice. Mathematics cannot be learned properly without doing a large number of problems.
Yet, for the student who wants to go deeper, or who wants to see the “big picture”, these books are generally not sufficient. In the usual college calculus course sequence, there is simply not enough time to spend on the main mathematical concepts at the heart of calculus, such as linear approximation, differentiability, orders of contact, or rectifiability of curves. The deeper facts about limits and continuity and the properties of real numbers, sequences, and functions which follow from order-completeness (such as the maximum value theorem) are normally skipped. In multivariable calculus, the routine texts do not usually have an adequate coverage of certain fundamental concepts related to geometry and physics, such as differential forms, or a more in-depth look at fields and potentials, or a unified treatment of the higher dimensional versions of the fundamental theorem of calculus.
Here is a list of some great calculus reference texts for students who want to dig deeper.
Single Variable Calculus
The books listed below are timeless masterpieces which go much deeper into calculus than ordinary popular textbooks. They are meant for the student who cannot be content with remaining just a user of calculus, but who intends to become a master of the subject.
Introduction to Calculus and Analysis, Volume I, by Richard Courant and Fritz John (Springer).
This classic text combines emphasizing the roots of calculus in the physical sciences and applications, the intuition behind the concepts and their nuances, as well as mathemetical correctness. It acheives a perfect balance between thorough rigor and an informal, lucid, pleasant presentation style. Covers integration before differentiation, which is the order I prefer for presentation, although this order is no longer common, perhaps because it requires more algebra skills.
If I have to recommend one serious calculus book for lifetime use, this would be it.
If you master the material in Courant's book, you will learn a great deal about single-variable calculus, as well as several topics in advanced calculus (such as Fourier series), with equal emphasis on theory and applications.
Calculus, Volume 1, by Tom Apostol (Wiley).
Another classic, known among mathematicians for its thoroughness, rigor, and mathematical elegance. It is difficult to name a calculus reference text having the clarity and precision as that of Apostol's book.
Coverage is similar to Courant's book, but it has a more structured presentation. Integration is presented before derivatives (like Courant), in fact even before formal treatment of limits and continuity! The book (Volume 1) ends with linear algebra.
But with its focus on mathematical principles underlying calculus, Apostol's book probably appeals more to mathematicians than physicists and engineers.
On the other hand, Courant's book extensively covers the connections and applications of calculus to geometry and the physical sciences, as well as the underlying mathematical principles. That, combined with its more informal style of presentation, makes Courant's book suitable equally for physicists and engineers as well as mathematicians. This wider appeal is the main reason that if I have to choose a single Calculus book as a reference text, it would be Courant's.
Appropriate for the more mathematically inclined student, Spivak's book has superb and detailed exposition of the deeper ideas behind calculus. Like Courant and Apostol, Spivak straddles both calculus and analysis. In fact, it is a beautiful and masterly introduction to real analysis with full rigour, but is gentler, more motivated, and less terse than Rudin. It can serve as a great bridge between calculus and real analysis as done in Rudin.
A Course of Pure Mathematics, by G. H. Hardy (Cambridge, 10th edition).
A timeless classic written by one of the greatest mathematicians, this book is full of insights, instructive examples, and challenging problems. Intended for serious, truly mathematically inclined students.
Hardy at Amazon, with reviews: Centenary version for the 10th edition, and Previous version of the 10th edition.
What if I do not want to go deep?
Everyone can benefit from the short, enjoyable, and easy to read book below.
Calculus Made Easy, by Silvanus P. Thompson and Martin Gardner (St. Martin's Press).
This book presents the basic ideas and methods of calculus intuitively without the details and rigor found in advanced courses. Not a reference text, it is extremely useful for the novice or the lay person who is looking for an overview of the subject. But even for serious students of calculus, it is a highly valuable text, mainly because it provides the original intuition of “infinitesimals” missing in so many modern texts.
Other Related Web Pages
- The most enlightening Calculus books, by Antonio Cangiano
Multivariable and Vector Calculus
Some good books on Calculus of Several Variables
- Calculus of Several Variables, by Serge Lang (Springer).
- Second Year Calculus, by David M. Bressoud (Springer)
- Introduction to Calculus and Analysis, Volume II, by Richard Courant and Fritz John (Springer).
- Advanced Calculus: A Differential Forms Approach, by Harold M. Edwards (Birkhauser).
- Calculus, Volume 2, by Tom Apostol (Wiley).
- Advanced Calculus, by R. Creighton Buck (Waveland).
- Calculus on Manifolds, by Michael Spivak (HarperCollins)