A Course in Mathematics for Students of Physics: Volume 1

There are a number of reviews for this series, mixed good and bad, and rather than adding another, I just want to register agreement with most of the points on both sides; on the positive end, it's as far as I can tell unique in using electronic circuits as the primary motivation for the math, and provides a broad enough range of material that it should be fairly easy to pick up just these two volumes and be able to get a decent handle on the material.On the negative side, while I haven't noticed the errors that were mentioned, the lack of solutions and occasional hand-wavery can leave one wondering -- but that's by no means a problem unique to this text. The standard point of comparison seems to be Frankel's Geometry of Physics, which is an excellent text, but I'd argue that it suffers from the same problem. This series doesn't go as deep into differential geometry as Frankel's, but offers more breadth.

I'm going to mention the second volume almost exclusively. This is where the action is located. It is hard to find any equivalent treatment of circuit theory using algebraic topology. You should have some mathematics background or some patience and a stack of books on algebra and algebraic topology so you can understand what these authors write so poorly. However, while the text (vol.2) loses a star for being horribly written, it gains four stars for the amazing content. If you are patient you will see a side of circuits you have never dreamt of and then you will be led into the generalized (continuous) version which is electromagnetics.If you are looking for some really accessible and really interesting mathematics on circuits and EM buy this book (or buy it used, I bought my hardcover for 10$) You might also find it useful to consult the appendix in Frankel's Geometry of Physics for comparison.Have fun and keep in mind that the book is written by sadists, clever and intelligent, but sadists all the way!

excellent

In Memory of Shlomo Sternberg (January 20, 1936 – August 23, 2024).The preface: "course is aimed at students with an interest in Physics who have a good grounding in one-variable Calculus" and "our pedagogical approach follows the spiral method, wherein we cover the same topic several times at increasing levels of sophistication and range of application." (always an excellent pedagogic strategy ! ). Let me proclaim that I love these two volumes. Regards volume one:(1) Answers for the exercises are not provided. There are "hints" supplied for some of the exercises (see: page 54, #20; page 75, #15; page 114, #7,13,17; page 269, #8 &10). I note that many exercises are not too difficult (see: page 48, #5, 7 and 13; page 73, #1, 4, 5, 6 and 7; page 112, #1, 6 and 7 ). Solutions can often be verified on one's own ! To verify the inverse of a 2X2 determinant, merely matrix multiplication. Solutions to problem #11, a through f (differential equations, page 116) can be verified on one's own.(2) Chapter Three--Linear Differential Equations in the plane--is as lucid as one is likely to find. Now, see page 128, it is easy to follow along to the explicit construction using the Gram-Schmidt process, this but one example of the interplay between the abstract and the concrete permeating the textbook.(3) Special relativity begins section #4.8 (pages 148-166), an interesting account. This encompasses 20 pages, where learn: "twin paradox is, of course, no paradox, just an immediate corollary of the reverse triangle inequality." The logic is a bit reversed from that of a physics course.(4) Chapter Five: calculus in the plane with section 5.6, pullback notation, is lucid (pages 209-215).(5) Chapter Six: theorems of differential calculus. Lagrange multipliers, a lucid exposition ! The Section #6.3, inverse function theorem, along with Newton's algorithmic method, are a beautiful discussion.(6) Line integrals and differential forms are next (page 246): This is a fantastic chapter ! Note that Problem 12 a & b is one which many physics students should already have mastered. Note: orientation, parameterization---understanding these concepts are absolutely central and essential to understanding.(7) Double integrals, next up: The chapter begins with exterior derivative, then two-forms, Once again, pages 284 and 285 render that which is abstract to that which is concrete--the mere computation of an Integral. I highlight: a nice discussion between differences of two-forms and densities. Green's theorem the highpoint concluding this great discussion.(8) Tenth chapter will place everything previously assimilated in the context of axioms for vector spaces. The usual things (commutative, associative, identity, inverse), dual spaces, quotient spaces (equivalence class) and linear transformations, concluding with adjoint transformations make this a most challenging and rewarding part of the book.(9) Finally, determinants. from axioms to concrete computation, a satisfying conclusion to volume one. Not forgetting that the exercises for this chapter (pages 399-400) are a mix of the routine ( a,b,c,d ) and the innovative (remaining exercises).(10) A unique textbook which can be perused by both mathematics and physics students. Unfortunate, that it is not more often utilized by students and instructors. This text does offer comparison to Burke (Applied Differential Geometry) and Bressoud (Second Year Calculus). While it has much in common with those textbooks, all three should be studied ! The text is sufficiently interesting and innovative as to be heartily recommended for further enrichment. I am a proponent of the so-called "spiral method" of teaching (revisiting earlier material at increasing levels of abstraction as the text progresses).Thus, highly recommended !

The book itself is five stars. The kindle edition gets three: many equations are rendered in tiny, unreadable fonts.

I don't understand why some of the reviews here are mediocre. Although I haven't read the book to any depth, I have a knack of knowing whether a book exudes quaity by browsing briefly through it to get a general feel; the breadth of the material presented is breathtaking and at least eye-opening. Here are some reviews from the back cover...American Journal Of Physics:- Not only is the mathematics clean, elegant, and modern blah, blah, blah... This is first rate!Times Higher Education Supplement:-...There is to my knowledge no comparable book, and it is hard to imagine a more inspiring one.Remember also that there are two volumes.

This book covers a lot of material and presents the underlying concepts in a particularly useful manner. Silly digressions into self-indulgent proofs are pleasingly absent.