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Review of Nearing: Mathematical Tools for Physics

Go to Mathematical Tools for Physics 2003, James Nearing, University of Miami (You can download the whole book in two different formats, optimized for different screen sizes. Depending on the format it is either a 4 MB pdf file with 592 pages, or a 3.8 MB pdf file with 448 pages. Or you can download the separate chapters.)

Before telling you about the book in general, I would like to point out two small sections that should be read by anybody who has to use mathematics as a tool for physics or engineering. Please have a look at them even if the rest of the book should not be of interest to you:

In section 1.8 "Sketching Graphs", Nearing teaches you how you can quickly determine the rough picture of a given function. You might think that this is no big deal when a simple program can plot any function with arbitrary precision instantenously. But the point is, that applying Nearings method will give you some insight into a function that you would not gain by simply looking at a computer screen.

Even more important is section 2.9 "Checking Results" (for some strange reason hidden in the "Infinite Series" chapter, it would deserve to be expanded into a separate chapter). The first quick check that you should apply is dimensional analysis (sadly Nearing does not explain it in detail). The next step is to "play" with the equation and convince yourself that the results make sense. This is the real fun part, it does not only build trust that your equation is actually correct, but gives a really deep and intuitive understanding of what is going on. Nearing gives a detailed description of this important technique in several examples.

In general the book assumes that the reader has already received a (short) introduction to the topics that are covered (see the paragraph below. A very good book that only assumes knowledge of AP calculus is Mathematical Methods in the Physical Sciences by Boas). So most chapters start with an example that reminds the reader of what he knows about the topic and then discusses those techniques that are most important for physics, providing several examples. At the end of each chapter are lots of excercises and problems, some of them with answers. The book contains a few proofs, but in general the focus is not on mathematical rigor but rather on intuitive explanations. Of course some of the topics are quite difficult, but Nearings relaxed writing style ensures that the text is not too dry. There are several links to animations, and other links that reveal some quite surprising (and sometimes funny) facts.

So here are the covered topics: Basic Stuff (some useful facts and techniques about trigonometry, special functions erf and gamma, integrals and polar coordinates), Infinite Series, Complex Algebra, Differential Equations (focussing on a small number of techniques that are most important for physics), Fourier Series, Vector Spaces, Operators and Matrices, Multivariable Calculus, Vector Calculus 1 (div and curl), Partial Differential Equations, Numerical Analysis (this is a very wide field, Nearing just tries to raise your interest with a few examples), Tensors, Vector Calculus 2 (integrals, Gauss and Stokes theorem), Complex Variables, Fourier Analysis, Calculus of Variations, Densities and Distributions

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

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