Mathematical Methods in the Physical Sciences 3rd Edition

I used the first edition as a textbook in undergrad 35 years ago, I kept the book for a reference until it finally disintegrated at the binding. After searching, and seeing that I had two choices, the 2nd and 3rd editions, I opted for the 3rd edition and note that it is still the standard for Applied Mathematics. I have taught "Math Methods" on and off for quite a few years and still go back to this book as the go-to text for that course. I also use it as a supplemental text for my physical chemistry course, when I teach that. My students find that it is a pretty easy book to grasp, and especially useful, especially for those just starting second year calculus.

Mary L. Boas has passed away in 2010 and this is a must have book. I also have her first edition with a paper back dust jacket from 1967 or so. She improves what she has, and includes the definition of double factorial. It is her immense ability with the English language, that for me, leads to a new, shorter, clearer definition of the double factorial. One must also have the 1985 edition of "Mathematical Methods for Physicists" (uncertain of title accuracy) by just the author Arfken. He does have errors ; but they are easily picked up. The new stuff is also necessary, if you intend to become a physicist; however the 1985 edition far exceeds the new one. I do not know at this point if you can access it on Amazon.com.

Let me start off by saying, I have essentially covered every single chapter (with the exception of the multivariable calculus section, I took a separate course on that), and every single section in a 3-quarter mathematical methods course as part of my physics undergraduate requirements. And let me repeat this again, this book will make a man out of you. After you conquer this book, you will be on your way to conquering all undergraduate physics with ease; mathematics will no longer be a problem and the real learning of physics will begin.

1. Infinite Series, Power series:
Great coverage of series and series representations of functions. Introduces several methods of determining convergence or divergence and techniques to convert essentially any function into a series as well as determining accuracies in representations. These are invaluable tools to solve difficult and non-analytic functions that show up everywhere in physics.

2. Complex Numbers:
A great introduction to complex analysis, starts off slow and easy and picks up the tempo with powers and roots of complex functions. This chapter is missing a discussion on the argument of a function and its meaning and kind of sweeps under the rug a few more technical things that a real complex analysis course would cover but nevertheless well done.

3. Linear Algebra:
The linear algebra section is pretty solid as well and it went a bit further than my regular linear algebra course. The placement of planes and lines is a bit awkward and doesn't really deal with matrices in the sense that you don't need to write out matrices but still an appropriate spot. It is missing some discussion on abstract vector spaces and doesn't delve too deep into the theoretical side of things; a mild discussion of group theory ends the chapter.

4. Partial Differentiation:
(No comment - did not cover)

5. Multiple Integrals; Applications of Integration:
(No comment - did not cover)

6. Vector Analysis
(No comment - did not cover)

7. Fourier Series and Transform:
A great section to learn about fourier series, usually special series are left out of real analysis courses (or covered only slightly) but in physics we use these a lot. You learn how to represent oscillatory systems as a superposition of waves, that is a series, which is a really neat idea, at least to me. My only complaint is that the fourier transform is only limited to one section and I think it's a bit more important and deserves a more in depth discussion.

8. Ordinary Differential Equations:
The bread and butter of physics. No matter what you do in physics you'll always encounter ODE's. Even if you have never seen them you might be surprised to learn that a simple equation such as F = ma is, in fact, a differential equation. It gives you the tools you need to solve the problems you will encounter and gives you discussions on how to solve special cases that occur frequently in physics. It ends with Laplace transforms (related to Fourier transforms), convolution, dirac-delta functions (mathematicians cringe at our use of the term function here), and greens functions which are a bit more advanced topics but great introduction and are definitely worth looking at.

9. Calculus of Variations:
The most important principle you take out of variations is the principle of least action. Once you start doing big boy physics you'll be calculation Lagrangians and Hamiltonians to easily solve for systems. Definitely a good mathematical approach to variations and something that will be essential throughout physics.

10. Tensor Analysis:
I didn't really cover most of the chapter, and what I did cover was in such a short amount of time that I can't possibly write a review without being biased. All I have to say though, is that for those General Relativity lovers, this is going to be your best friend.

11. Special Functions:
As the chapter title itself says, these are just formulas and quick derivations for a variety of special functions that are everywhere in physics. You don't necessarily need to study these in great detail as they only help you solve integrals, but they are of some theoretical interest. Definitely a must read chapter.

12. Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Functions:
Solutions to partial differential equations everywhere, and I mean everywhere. Chapter 12 and 13 go hand in hand, first you learn the math stuff in chapter 12 without really knowing it's purpose and then jump into chapter 13 and find out these are solutions to partial differential equations. Just like ODE's, these are essential and found everywhere in physics. This chapter is very meaty and full of solutions to differential equations and chances are, if you ever run into a differential equation in your undergrad career the solutions are here.

13. Partial Differential Equations
See chapter 12 summary, they go hand in hand.

14. Functions of a Complex Variable
I still think this is an odd location for the second part of a complex analysis course, ideally I would have preferred right after chapter 2 or possibly 3 but nevertheless a good coverage and sum of complex analysis. You learn how to solve some really nasty integrals in a really trivial way using complex analysis.

15. Probability and Statistics
Arguably the worst of all chapters, at least in my opinion. The notation convention Boas uses isn't the most intuitive or the most frequently used and the explanation to some of the probability problems are not really helpful. Some are more naturally talented in probability, I however, am not thus found this chapter to be really annoying and confusing. Still, something worth knowing and if it works for you then let it be.

Overall this is a book I will be using for years and will keep coming back for years. It's not exactly mathematics and it's not exactly physics it fits that missing link between the two and helps clarify topics in advanced mathematics that will be useful in all undergraduate physics. I'm glad I went through this book and having seen these things at least once, even if I didn't understand it fully initially, definitely helped give me the courage to tackle my undergraduate physics courses. I recommend it to every physics student.

I've purchased plenty of books over the years to learn or re learn math concepts to apply to science and engineering. The majority of those books either wastes too much time over simplifying things, or they continue to get bogged down by mathematical proofs.

If you're a student of engineering or science and you've made it through single variable calculus you can benefit from this book. I think it's also safe to say that you're don't need things spelled out to you. You just need a clear and concise, cut to the chase, how will I use this, sort of text that doesn't kill you with proofs you don't care about. That's exactly what this classic book by Mary L. Boas provides. Easily the best study companion I've ever purchased'