DIFFERENTIAL GEOMETRY FOR PHYSICISTS AND MATHEMATICIANS: MOVING FRAMES AND DIFFERENTIAL FORMS: FROM EUCLID PAST RIEMANN

by Vargas Jose G

This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative — almost like a story being told — that does not impede sophistication and deep results.It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter.An appendix on the classic theory of curves and surfaces slashes is included. It does not only contain the traditional approach that uses vector calculus, but also the treatments of the subject by those who have already used differential forms for the same purpose.

Genres: Science

312 pages, Hardcover

First published January 1, 2014

Reviews

[Woflmao · Rating 2 out of 5]

This is a book for physics students who believe that mathematics is out to get them. It covers the differential geometry of vector bundles and connections on manifolds, including but not limited to affine and Levi-Civita connections. The exposition relies exclusively on computation in coordinates, abstract definitions are avoided or supplied only after the results are on them have been derived. There are barely any theorems or definitions throughout the text, most results are presented in a "narrative" style. Although well-intended, the author's handwaving approach to explaining concepts is often more confusing than helpful. Moreover, Vargas identifies a list of what he perceives to be problems in the mathematics education of pysicists, and goes to great lengths to explain his "right" way of doing mathematics. It leads me to suspect that Vargas is not very well-acquainted with the state-of-the-art literature in mathematics (rather than physics, and his bibliography supports this impression), for otherwise he would have realised that many of his "problems" are non-issues in contemporary mathematics.