Second Year Calculus: From Celestial Mechanics to Special Relativity (Undergraduate Texts in Mathematics)

With the recent revival of Advanced Calculus texts floating around looking for departments wanting to give those courses a retry (realizing Calculus III may not have been the bargain originally hoped for), Second-Year Calculus (SYC) may be the offering that math departments are looking for.Set up almost as a text readable on its own, a quality not always there with the offerings in the Readings in Mathematics series, SYC has humor, wit, and clarity drawn from the same talent behind the more controversial series on Riemann and Lebesgue Integration. In this earlier volume, Bressoud brings a no less radical approach to advanced calculus with SYC.In the contexts I have seen during my career as a student in mid-tier universities, SYC could easily join Spivak's masterpiece as a, well ... second-year Calculus text. There is little overlap between the two, and SYC picks up, traditionally, where Spivak's Calculus leaves off: multidimensional theory. And unlike other possible choices one has for following up on Spivak (Several Real Variables, Multivariate Calculus and Geometry), the linear algebra - if needed to be learned - is taught within the text, while the book manages to escape the watered-down Jordan notion of Lebesgue measure that is reappearing in some newer texts as "paving." There are also plenty of illustrations - lacked by some books at this level (Rudin, and, though not totally devoid, the tougher, author-recommended H.M. Edwards) - and inclusions of important connections with physics, something gravely missing from curriculum and which should be encouraged if time allows, perhaps in collaboration with "those labcoated gals and guys down the hall."A forgotten text that is a well-written polymath of a book, SYC covers potentially difficult introductory material well and is, most importantly, fun. It deserves a re-write and a renaissance, since it fits so well after Spivak in a proof-light honors calculus setup and even with a teacher's or four-year liberal art program's Analysis sequence, in the light-proof settings allowed by a Lay or Ross for the first semester. Also excellent for self-study for people like me, who missed differential forms and even an introduction to the beauty of differential forms altogether, even in graduate school.

As the title indicates, this work has an unusual emphasis on physics; reflecting the author's inspiration, the appendix includes an excerpt from Freeman Dyson's speech "Missed Opportunities", which contains the stunning claim that if mathematicians had been keeping up with contemporary physics, in the 19th century they would have discovered "Einstein's theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis." Bressoud backs up the first part of this claim through a textbook for multivariable calculus of all things, concluding with a fast-paced crash-course on the historical development of the theory of electricity and a formal derivation of special relativity based only on the scientific knowledge at the time of Maxwell. With aesthetic unity unusual for math textbooks, the book opens with F=ma and a presentation of Newton's Law of Gravitation, derives all the mathematical formalism from physical concepts, and concludes with a derivation of E=mc^2 from F=ma. Taking its pedagogical inspiration from Apostol's Calculus and Edwards's Advanced Calculus, the work is also mathematically sophisticated and modern. I finished my reading grateful that I now have a far more developed physical intuition of this branch of math and an understanding of theoretical details of celestial mechanics and electromagnetism, in just 360 pages of straightforward prose and mathematical calculations.

Why did I buy this book? Well, I have a master's degree in math, though I have't seen multivariable calculus much at all in years. I also got into math so I could understand general relativity, but the math was way more fun than the lower level physics courses, so I stuck with math. Now that I'm done with my studies, I figured I can spend some time to make some progress toward my original goal.My physics background is virtually non-existent, and my geometry skills are very rusty (most of us who've been brought up in the public school systems over the past 30+ years seem to lack in formal geometry anyway.) So this book was the perfect start! Bressoud starts off with deriving some of Kepler's and Newton's laws with basic geometry and conic sections. Without wasting too much time, he jumps right into a more economical and modern approach using calculus. He quickly makes his way through the basics of multivariable calculus and differential forms, and exposes the reader to the core ideas behind celestial mechanics, E&M and concludes with relativity.I am still reading through this book, and haven't quite made it to the E&M/SR portions, but I've enjoyed what I've read so far! While this is sort of a textbook with theorems, proofs and exercises, Bressoud wrote with a prose that most mathematicians seem to lack (Herstein was another great writer!) Ultimately, I highly recommend this book to anyone wanting to see physics done in a clean, mathematical way!

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This is a really nice book. It contains the most readable introduction to Differential Forms I've ever seen. Not useful for aspiring mathematicians but highly recommended for beginners from all other disciplines.