Mathematical Methods in the Physical Sciences

I can't understand why anyone would rank this book less than 5 stars.I am self studying physics in the hope of understanding particle physics one day. I have been through the standard calculus books, Strang's Linear Algebra and a some of Saff's Complex Analysis. Then I read Taylor's excellent Classical Mechanics book and then started Griffith's Electrodynamics. Griffith's math is more complex and even though he does a good job of teaching the math needed, I find it difficult to learn the math and the physics at the same time. I first got Byron and Fuller's book knowing that it might be advanced, but wanted to try anyway. It is way too advanced for my stage. I couldn't understand any equations on any page I opened to. I passed on Boas first time around as so many people said it was light on proofs. After the Byron and Fuller debacle, I thought I would try this book.IT IS FANTASTIC!!!I am about to finish the second to last chapter (Functions of a Complex Variable). I won't do the last chapter (Probability) as Probability is critical in thermodynamics and Quantum Mechanics. I will read a separate book on that. I have read every other chapter. With the exception of the Tensor chapter (more on this later), every chapter was outstanding. This book lays an extraordinary math foundation for an undergraduate program of study. I can open to any page in Byron and Fuller now and understand what they are trying to do. It is still over my head as you need to know quantum and advanced classical mechanics to understand their examples, but I know what aspects of math they are using and what they are trying to do because of Boas.It surprises me to see how many people have given this book negative reviews because it lacks rigorous proofs. All good books are written for a purpose and stay true to that purpose. Boas' purpose is laid out clearly in the preface. This book is for undergraduates who have completed at least a calculus series and probably also ordinary differential equations. It is meant to be a one year course to teach all the basic math needed for undergraduate study. It is meant for people like me who want a more complete understanding of math before starting books like Griffiths and don't like learning the physics and math at the same time. THAT IS HER PURPOSE AND SHE STAYS TRUE TO IT. This book is not meant for math majors. It is not meant for people who want Analysis level proofs of everything. If that is what you want, there are a million books out there for you. Use them. Why did you pick up this book?That said, there is nothing superficial about this book. If you are the intended audience, you will learn new material on every page. If you think you are going to skim through chapters to get their main points and then move on to bigger and better, you are in for a rude awakening. You will learn nothing if you don't read carefully and do the exercises. If you do that you will set a very solid foundation on which to build further math skills (which, of course, is the point of this book. It is not a be all, end all. It is the beginning of a deeper understanding of the advanced math skills you will need for graduate physical science study.) There are plenty of proofs if they are appropriate for this level. She leaves out extensive proofs that are very involved. Physical science students don't want those at the undergraduate level. Saying that this book is terrible because it leaves out extensive proofs is like saying Dr. Seuss books are terrible because they lack mathematical proofs. The book isn't intended to have those. Look elsewhere if that is what you want.That said, the chapter on Tensors is poor. Don't bother reading it. This is not her fault. Tensors are complex and simply cannot be taught well in the limited space they can be given in a book like this. The first half is okay where she discusses Kronecker delta and Levi-Civita permutation tensors, but then she just plops down the mathematical definition of covariant and contravariant tensors without giving any insight into the equations. It was incomprehensible to me after that. I put this book down and read Daniel Fleish's book on vectors and tensors (excellent book) and the Taha Sochi's Tensors Made Simple (pretty good book. Teaches you how to manipulate tensor symbols well). That gave me a better understanding, but I still don't feel like I have a good grasp of tensors. For my level, that is good enough. I'll learn more later when I need it. The Tensor chapter is probably good for someone who knows tensors and wants a refresher. It is not a good first introduction.I used to be intimdated of partial differential equations. ODEs are hard enough, then you add more variables. This book did such an excellent job with PDEs, I am now looking forward to reading a complete PDE book before I move on to graduate level studies.If you still have any doubts about this book, look at the number of places it is referenced. Taylor and Griffith (the standards for their respective subjects at the undergraduate level) both reference this book if you want a deeper understanding of the math they are using. Look at the number of graduate physics professors in the amazon comments who have ranked this book. They tell their GRADUATE students to start with Boas if they need some math they don't understand and then move to other books if they don't find the depth they need in Boas. That is saying something.Bottom line: If you are an undergraduate who has completed at least ODEs and want to do well in your advanced undergraduate physical science studies, you need to read this book. Every undergraduate physics program should teach a year long course based on this book.

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 EquationsSee chapter 12 summary, they go hand in hand.14. Functions of a Complex VariableI 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 StatisticsArguably 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.