The following is the review I published in The UMAP Journal (Summer, 2009, Vol 30, no. 2) pp. 175-178.
My second review for this journal [1986] was of Gilbert Strang's Introduction to Applied Mathematics (hereafter IAM). I have never been too happy with that review, where I said that it is a "wonderful book." True enough; but more appropriately, it is an important book, as is the book reviewed here, Computational Science and Engineering (hereafter CSE).
CSE is--and is not--a second edition of IAM. Apparently, it is the result
of more than 20 years of Strang teaching his favorite course at MIT,
presumably out of IAM. Since CSE does not contain everything in IAM
and also contains topics not in IAM, it is a different text. CSE contains
Strang's further ruminations on the nature of applied mathematics, and
I view it as the superior text, but some individuals might prefer IAM. To
some extent, either book represents Strang's philosophy of teaching applied mathematics--that we need a new approach--but this conviction is much more explicit in CSE.
In particular, Strang believes that we should focus on both modeling and
computation. Many books are about one or the other, and he feels that applied mathematics is both. Furthermore, Strang believes that applied problems tend to have a common structure, and Chapter 2 is devoted to illustrating this principle through a wide variety of problems.
In my review of IAM, I tried to give an idea of the range of topics without enumerating the contents. CSE has the same difficulty: Enumerating the topics is tedious, but the titles of the chapters are informative (though listing them does not do justice to the sheer range of content):
1. Applied Linear Algebra
2. A Framework for Applied Mathematics
3. Boundary Value Problems
4. Fourier Series and Integrals
5. Analytic Functions
6. Initial Value Problems
7. Solving Large Systems
8. Optimization and Minimum Principles
Strang suggests that a course designed out of this text might follow the
structure that he uses (p. v):
* Applied linear algebra
* Applied differential equations
* Fourier series
I have long been a champion of Strang's books. I have reviewed different
editions of two texts on linear algebra, making clear that that I think he is the most influential author in linear algebra in the last 50 years. I have heaped high praise on his calculus text in my recent editorial on calculus [Cargal 2008]. I have done this for the exact reason that I have championed John Stillwell's books on geometry and algebra. These two authors, as well as a handful of others, write with authority leavened with the great enthusiasm of the born teacher. They are superb pedagogues.
What makes IAM and CSE so important is that they cover a great deal
of applied mathematics, and there is nothing in the literature that compares to them. Pedagogical works, as opposed to dry tomes, are simply
rarer in applied mathematics than they are in, say, calculus, linear algebra,geometry, and number theory. There are pedagogical works in differential equations and probability. But there is nothing that covers so much applied mathematics as these with comparative pedagogical skill and acumen.
Like IAM, CSE has a long first chapter that is a summary of applied linear
algebra (86 pp in IAM, 97 pp in CSE). Linear algebra is a key to applied
mathematics; it is the most important tool after calculus (this apparently is Strang's view). However, the first chapter is definitely a review. The reader needs to have had a course in linear algebra as well as the usual course in differential equations. These things are minimal. Courses in probability,numerical analysis, and so on certainly help. Knowledge of physics is a definite plus. These days, there are students of applied mathematics(computer science, statistics, operations research) who are physics-phobic. They would have problems with parts of the book. This necessity of a modicum of prior knowledge of applied mathematics means that the level of the book is for seniors and graduate students. The online comments about IAM are striking in their simplicity: Students who are not prepared despise the book, the others are enamored with it; there is no middle ground. The reader who is prepared should love this book. In particular, engineers and physicists should love this book.
People in industry, too, should love this book. Mathematicians and
engineers in industry benefit particularly froma book such as this for a very simple reason. Mathematicians in academia tend to specialize because of the need to publish. However, mathematicians in industry are motivated
to generalize. They don't have tenure; often they depend on contracts, so
that specializing can limit opportunities to get work. If a book like CSE (or AIM) had been available when I went into industry more than 30 years ago,it would have changed my life; it certainly would have made those first years easier. In fact, one topic that Strang covers very nicely in both books is the Kalman filter, a topic that is very big in industry and that occupied me in my first job.
The most important thing I tell my students is the need to study if they go into industry. This is particularly true if the student has stopped at the bachelor's degree, since a bachelor's degree is essentially a learner's permit. Few students go to work for national labs (those who do, do not need my advice--I need theirs), which means that on-the-job training is unlikely or superficial. Of people who have technical degrees, only a small portion maintain their technical skills; most simply travel along and forget much of what they learned. People tend to learn or they forget; nobody remains in stasis. In industry, you should take some of your time on the job to study.
Is spending work time studying material that is not clearly work related
to the work unethical? Typically, doing so does not create a problem (as
long as one gets one's tasks done). However, if your supervisor sees you
reading a newspaper, that could create a problem. On the other hand, if
you are studying number theory, there is no problem; that number theory
has nothing to do with your current job tasks will almost certainly not
register. Moreover, the worker who studies number theory will tend to
retain competence in differential equations far better than a worker who
just lets technical skills dissipate. In fact, those few workers who develop good technical reputations almost always study widely while on the job. Their ability to quickly respond to new problems on the job is a result of having used work time not to do company tasks. I view this behavior as a survival skill. The fact is, if one "steals" company time to study mathematics and engineering--even topics that have nothing to do with the job--one is far more likely to be promoted because of it than to be reprimanded.
However, the young worker almost always would benefit not only from
learning more number theory but--more urgently--needs to learn a lot
more applied mathematics. The undergraduate curriculum can't cover it
all. Key core areas are not just physics and differential equations, but probability,numerical analysis, and programming. For a worker in industry, CSE would be invaluable, and yet experienced engineers and mathematicians will also be impressed by this book.
Computational Science and Engineering should be in the library of every
applied mathematician, not to mention engineers. As a textbook, it is well suited for a senior or graduate course in applied mathematics.
References
Cargal, J.M. 1986. Review of Strang [1986]. The UMAP Journal 7 (4): 364-
365.
Cargal, J.M. 2008. Calculus: Textbooks, aids, and infinitesimals. The UMAP
Journal 29 (4): 399-416.
Strang, Gilbert. 1986. Introduction to Applied Mathematics. Wellesley, MA:
Wellesley-Cambridge Press.
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