Mathematical Methods for Physics and Engineering

[Thinkaholic]

I’m thinking about getting this monster of a book because I heard it was incredibly awesome and useful, but I have one question. I want to be both a physicist and a mathematician in the future, so can aspiring pure mathematicians learn many useful things from this book, or does this book only focus on applications of the topics covered? Does the book go into excessive detail (and I say that as a good thing) about each topic?

 

[Orodruin]

First of all: Always mention not only the book title, but also the author(s). If you do not there may generally be degeneracies such as in this case, where my book has the same title as the book by Riley, Hobson, and Bence. (The title is one of the things you have the least control over as an author and, according to my publisher, similar and even the same title may not be as uncommon as you might think.)

That being said, this type of books (along the lines "mathematical methods (for/in/of) (physics/physicists) (and engineers/engineering)", i.e., RHB, my book, Arfken, Boas, etc) will generally be aimed at physicists and engineers, not budding mathematicians. As such, they typically focus more on the utility of the methods and their application in physics rather than on building the theory in a postulate-theorem-proof structure. Personally, my focus was on building the formalism needed and introducing its use in familiar physical situations in order to build intuition before applying it in more advanced settings.

Of course, this does not mean that this type of books does not have "excessive detail", it is just a question of what you mean by excessive detail. You might mean a book that has a very stringent formalism or you might mean a book that builds a detailed intuition for how the tools are used. The type of books we discuss here typically does the latter.

 

[vanhees71]

Well, when I studied physics, in some semesters, I listened more to mathematics than physics lectures (which was possible in the "good old times", where you could do at the university whatever you wanted without control over attendance etc. Of course, you had to pass your exams, but nobody cared how you got the knowledge to do so). There I was always a bit amused that the mathematicians could prove very complicated theorems in a very rigorous way, but were not able to do a simple basis transformation in linear algebra or evaluate a not-too-complicated integral. So I think also as a pure mathematician you can benefit a lot from books explaining how math is applied in the natural sciences and engineering. You do not only get a better education in "cranking out the numbers" but also some intuitive ideas about abstract mathematical concepts. E.g., knowing what surface integrals and the usual differential operators in vector calculus have to do with fluid mechanics gives you a great intuition about abstract ideas like de Rham cohomology and Stokes's theorem.

The same holds true also in the other way: It doesn't harm a theoretical physicist at all to have some idea about rigorous pure mathematics, because treating a distribution (in the sense of generalized function) as a function without thinking enough about it, can lead to big trouble and misunderstandings. So it's always good to have a broad overview over ones subject!

I'm often very thankful about the pretty abstract lectures by pure mathematicins I heard during my studies of physics. Although I've forgotten a lot already, because things you do not need regularly, you tend to forget unfortunately, I've an idea what might be going on when I encounter a problem in some applied use of mathematics.

 

[Demystifier]

There I was always a bit amused that the mathematicians could prove very complicated theorems in a very rigorous way, but were not able to do a simple basis transformation in linear algebra or evaluate a not-too-complicated integral.

I guess electric engineers are similarly amused how can theoretical physicists write the general solution of Maxwell equation in 10 dimensions, but cannot answer some basic practical questions about a dipole antenna.

 

[vanhees71]

True. The main difference, however, between engineers and physicists is that the former use the Laplace and the latter the Fourier transformation ;-).

 

[The Bill]

True. The main difference, however, between engineers and physicists is that the former use the Laplace and the latter the Fourier transformation ;-).

Engineers use plenty of FFTs these days.

 

[vanhees71]

It depends on what you want to calculate. For initial-value problems often the Laplace transformation is more convenient than the Fourier transformation.

 

[Demystifier]

The question is, are there non-applied physicists who use the Laplace transform?

 

[Orodruin]

Did somebody say Wick rotation?

I think this is drifting off-topic ...