My first reaction to Modern Applied Mathematics for Engineers was that I wouldn't find it very interesting. After all, books with titles like this appear all the time. Then I ran into this:
All those who deal with engineering education are well aware of the precarious state of basic mathematical skills and conceptual comprehension which is typical of modern students. Most beginning engineering graduate students haven't had a good experience with mathematics. (...) The mathematical methods encountered in each course do not evolve into unified patterns which the future engineer would be able to recognize and use universally.(...)
We strongly believe that this situation can and should be radically improved. We think that it is truly possible to offer a beginning graduate student a concise yet comprehensive course which summarizes, unifies, and completes his/her mathematical knowledge by constructing a comprehensive system of operating mathematics and setting up patterns that can be recalled for use in a wide range of engineering disciplines. This book is our response to this challenge.
What the authors are saying is that we mathematicians aren't doing a good job of teaching mathematics. Specifically, we're not getting across concepts that have wide applicability. To counter this problem, their book packs a great deal of material into a small space, attempting a unifying point of view. And concepts are given priority over proofs:
With this in mind we have mostly forgone the theorem-proof format for a more informal style. We must confess that it was done with a certain relish; in this sense, as in many others, this is a book written by engineers for engineers. However, we have not eliminated all the theorems and have not presented the applied mathematics as merely a bag of tricks. The important theorems we retained are used as pivotal points in the exposition of particular concepts. They play an important role in summarizing concepts and making the student consciously realize that any established method or technique has well-defined limitations.
Well, that's interesting. Here's a way to "sell" the importance of theorems: they tell us the conditions for something to be true. And the book is also interesting. It starts with set theory, then covers logic, algebraic structures, linear algebra, metrics and topology, Banach and Hilbert spaces, Fourier series, transforms, and partial differential equations. While most mathematicians know most of this stuff, I suspect they may find much to ponder in the approach and style chosen by the authors. This one's worth a look.
Fernando Q. Gouvêa (firstname.lastname@example.org) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.