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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010332350 | QC20 V37 2014 | Open Access Book | Book | Searching... |
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Summary
Summary
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.In an appendix on the classical theory of curves and surfaces, the author slashes not only the main proofs of the traditional approach, which uses vector calculus, but even existing treatments that also use differential forms for the same purpose.
Reviews 1
Choice Review
Differential geometry courses for mathematics majors generally emphasize low-dimensional objects, curves, and surfaces-rich material certainly deserving of careful development. Unfortunately, this does not constitute sufficient grounding even for undergraduate-level general relatively. Higher-dimensional differential geometry represents a conceptual and pedagogical minefield though. Basic concepts (e.g., curvature) have highly nontrivial generalizations, diagrams cannot foster adequate intuition, notation often turns nightmarish, and even the experts have not achieved consensus concerning the "right" set-up. The current volume reads as differential geometry primer and also pedagogical/methodological diatribe. In 1924, famed geometer E. Cartan introduced an appealing formalism built around differential forms and what Vargas (PST Associates) here calls "teleparallel connections." Cartan's approach subsequently disappeared from the literature, with mathematicians doubting the possibility of making it rigorous. Vargas now revives Cartan's theory of connections, argues for its timeliness, and, following presumably previously unpublished suggestions of Y. Clifton, supplies the crucial definition and theorems that give it firm footing. Time will tell whether this book delivers the full-fledged paradigm shift that it virtually promises. Regardless, it makes very stimulating reading, much more than yet another introduction to a popular subject. Summing Up: Recommended. Upper-division undergraduates and above. --David V. Feldman, University of New Hampshire
