Mathematical physics: a modern introduction to its foundations
Choice Reviews Online · 2014
Preface to Second Edition by introducing vector bundles and linear connections, the stage becomes ready for the introduction of curvature tensor and torsion, two major players in differential geometry.This approach to differential geometry via fiber bundles is, in my opinion, the most elegant and intuitive approach, which avoids the ad hoc introduction of covariant derivative.Continuing with differential geometry, Chap.37 incorporates the notion of inner product and metric into it, coming up with the metric connection, so essential in the general theory of relativity.All these changes and additions required certain omissions.I was careful not to break the continuity and rigor of the book when omitting topics.Since none of the discussions of numerical analysis was used anywhere else in the book, these were the first casualties.A few mathematical treatments that were too dry, technical, and not inspiring were also removed from the new edition.However, I provided references in which the reader can find these missing details.The only casualty of this kind of omission was the discussion leading to the spectral decomposition theorem for compact operators in Chap.17.Aside from the above changes, I have also altered the style of the book considerably.Now all mathematical statements-theorems, propositions, corollaries, definitions, remarks, etc.-and examples are numbered consecutively without regard to their types.This makes finding those statements or examples considerably easier.I have also placed important mathematical statements in boxes which are more visible as they have dark backgrounds.Additionally, I have increased the number of marginal notes, and added many more entries to the index.Many readers and adopters provided invaluable feedback, both in spotting typos and in clarifying vague and even erroneous statements of the book.