Tensor Analysis on Manifolds

by Richard L. Bishop, Samuel I. Goldberg

"This is a first-rate book and deserves to be widely read." — American Mathematical Monthly
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.

Genres: Mathematics, Science, Physics, Textbooks

288 pages, Paperback

First published December 1, 1980

Reviews

[Robert]

Abandoned after two chapters, that is after chapter 1: when a book that starts at chapter 0 it acts as a warning sign to me and I was right to be wary. This is a book by a mathematician for mathematicians and it might very well be excellent for its intended audience. I am not a mathematician, however, and I found it completely useless for my needs. Hence, I haven't rated the book because it may not actually be bad, just inappropriate for me.

[William Schram · Rating 4 out of 5]

I know very little about this subject. A tensor is a math thing, analysis is a higher math thing, and a manifold is a math thing that refers to a surface. When we put these things together, what do we get?

Unfortunately, we got something too advanced for me to understand and review properly. I need more grounding in Advanced Calculus to fully appreciate the book.

Despite my lack of understanding, the book is well written. I need to revisit this one later. Thanks for reading my review, and see you next time.

[Akli Bouchekal · Rating 4 out of 5]

Honestamente es un buen primer approa.ch para la geometría en mi caso quería familiarizarme con las formas y variedades simplecticas. Su intro para formas está muy bien con ejemplos y demás tb tienen un apartado duro de aplicaciones físicas donde introduce el fibrado cotangente .