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Review of Boas: Mathematical Methods in the Physical Sciences

Wiley, 864 pages, 3rd edition, 2005

This is a very popular and classic (the first edition is from 1966 !) textbook for (undergrad) physics students who want to get an intuitive introduction to the topics in mathematics that are most important for studying physics. It is also useful for students in engineering or chemistry (depending on the mathematical areas they have to cover, I have tried to point out the relevance for EE students, please see the list below). As a prerequisite, you should be familiar with differentiation and integration, at about the level of AP calculus, so you might want to take a look at some introductions to calculus first.

The main idea behind the book is, that the student should learn the important areas of mathematics before he has to apply them to physics or engineering. This way he can later focus exclusively on the application. For example, if you first learn about vector analysis, it will be a lot easier to understand maxwells equations than if you have to learn both the mathematics and the physics simultaneously. To cover all necessary topics on 864 pages, there have to be some restrictions. Boas does not try to cover every detail of each topic, but focusses on the basics. Detailed proofs are omitted, but the intuitive explanations that Boas provides are done very carefully. Even though there is limited room for each topic, there are examples and many problems (and answers for selected problems).

If you are an undergrad student of physics, you will probably like this book (at some later time you may want to have a look at a more advanced treatment like in Arfken). Some reviewers say they wish the book were expanded into several volumes, so that each topic could be introduced more slowly. It may indeed happen that for a topic which you find especially difficult, the introduction in Boas will not be sufficient. In this case you may want to consult a specialized book that is devoted to the topic and provides a more gentle introduction (though I think that, considering the amount of space available for each topic, Boas does an excellent job of explaining them as clearly and intuitively as possible). But even if you need additional specialized books, it will still be useful that you can find all topics together in Boas, especially if you need a quick review of all of them when preparing for an exam. The only scenario that I can imagine in which you will really dislike the book, is if you are a fan of the formal aspects of mathematics and you want a mathematical rigorous treatment of each topic with lots of lemmas, proofs etc.

If you are studying EE, you may wonder how much of the book is relevant for you (and how much is only interesting for physicists). Below is a list of the content of the book, and for some topics I have added the most important application in electronics that came to my mind. Please note that the applications are in no way complete, but they may give you an idea what you will gain from the book:

Infinite Series, Power Series (The most important of these is the Taylor series. Whenever you have to analyze a nonlinear device, it is a good idea to approximate the characteristic curve by a linear function as a first step. Taylor series provide the mathematical foundation for this.)
Complex Numbers (Indispensable for analysing AC circuits in their steady state)
Linear Algebra (Provides a systematic way to analyze linear circuits of arbitrary size and configuration)
Partial Differentiation (Propagation of measurement errors, among many other applications)
Multiple Integrals (Very basic, many applications in physics and mechanics)
Vector Analysis (The heart of electromagnetic theory. Once you really understand about div and curl, understanding Maxwells famous equations is straightforward)
Fourier Series and Transforms (Whenever you see a waveform that is not a sine function, you can analyze it with the Fourier transform)
Ordinary Differential Equations (For the transient analysis of RLC circuits, i.e. how will they behave when you switch them on / off or send a pulse through them)
Calculus of Variations
Tensor Analysis
Special Functions
Series Solutions of Differential Equations: Legendre, Bessel, Hermite, and Laguerre Functions
Partial Differential Equations (You can describe all kinds of fields using PDEs - an advanced application)
Functions of a Complex Variable (You can solve some complicated problems in electrostatics by using conformal mapping - a rather theoretical and advanced application)
Probability and Statistics (Extremely important for practical purposes: For example, when you produce any device in series, you may want to understand the behavior of the whole series by looking at a small sample of the devices. Probability and statistics provide the mathematical foundations. You also need them if you want to predict the reliability of any device).

A list of errata is maintained by Boas son, who is also a mathematician (Mary L. Boas died on February 17, 2010).

You can find similar topics as in Boas, but at a slightly more advanced level, in the free Mathematical Tools for Physics by James Nearing. Other than Boas, Nearing assumes that you have already heard about the topics that he covers. He provides some additional intuitive explanations and teaches you some tricks how you can apply them more effectively to physics problems.

You might also be interested in some other Mathematics for Physics and Engineering resources.

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