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Advanced Calculus

David V. Widder
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

The term “advanced calculus” seems not to be used much any more, and even when it was widely used it did not have a fixed meaning, other than “the course you take after calculus”. It was usually some combination of (1) topics we didn’t get to in the first year, and (2) a more rigorous look at what we did get to. The present book follows this pattern, with a heavy emphasis on (1). It has a lot about multi-variable calculus, vector calculus (mostly line and surface integrals), and some topics not commonly found at this level, such as differential geometry and the Stieltjes integral. It also has shorter treatments of infinite series, the gamma function, Fourier series, and the Laplace transform. The present volume is a Dover 1989 corrected reprint of the 1961 second edition from Prentice-Hall, which was itself a lightly-revised version of the 1947 first edition.

Everything is proved. The proofs are kept simple by stating the theorems only for the most agreeable conditions, such as for functions that have several orders of continuous derivatives. Happily these conditions are met in nearly all cases that arise in practice, so you get the most useful parts of the subject although not the most general. As another example, the implicit function theorem is stated and proved only for real-valued functions of one variable. The book as a whole is slanted toward results that you would use on particular functions, rather than toward general properties of functions, and there are a lot of concrete examples given.

The multi-variable calculus part has an old-style treatment, where we extend the mechanics of single-variable calculus to cover functions of two or three variables; it is not the differential forms and manifolds approach found in (for example) Loomis & Sternberg’s Advanced Calculus. Similarly the vector part is ordinary 3-space vectors, and there are no Euclidean or linear spaces. The only use of topology is the Heine-Borel theorem for the line.

The Stieltjes integral allows handling continuous, discrete, and mixed problems, and some use is made of this, but not much. The development is straightforward and really no harder than a rigorous development of the Riemann integral. There are strong chapters on Fourier series (including convergence theorems and summability) and the Laplace transform (with many of its properties; not limited to its use for solving differential equations).

There are a reasonable number of exercises, of reasonable difficulty. They are split about half-and-half between drill and proof. A few have answers in the back.

The book is skimpy on applications, although it handles a few traditional subjects from physics, such as moment of inertia and the vibrating string. It has a fair amount about solution methods for differential equations.

Bottom line: still a valuable book, especially for physicists and engineers. More modern treatments cover the same topics in more generality and abstraction, but what’s here is fine for most uses. These more general treatments are usually called Mathematical Analysis rather than Advanced Calculus; some well-known ones are Apostol’s Mathematical Analysis and Rudin’s Principles of Mathematical Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

1. Partial Differentiation
2. Vectors
3. Differential Geometry
4. Applications of Partial Differentiation
5. Stieltjes Integral
6. Multiple Integrals
7. Line and Surface Integrals
8. Limits and Indeterminate Forms
9. Infinite Series
10. Convergence of Improper Integrals
11. The Gamma Function. Evaluation of Definite Integrals
12. Fourier Series
13. The Laplace Transform
14. Applications of the Laplace Transform