David M. Bressoud

As the title indicates, this work is 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 the book 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 uncluttered mathematical writing.… (mais)