Tensors, Differential Forms, and Variational Principles

Rating:
Summary: All in the indices
Review: As mentioned below, this book concentrates for 330+ pages on a "classical" index-based approach to tensors. Coordinate-free treatment is restricted to a brief appendix. There are only 10 illustrations in this very long book (none in the appendix). That means that you will need to concentrate very hard on the manipulation of many tiny indices in some long and hairy equations.

My hat's off to people who can learn this way, but I am in the camp that expects pictures for geometrical subjects. In this respect, Schutz is a much better book, as well as being coordinate-free from the get-go. (Ditto for Misner Thorne Wheeler and Carroll on general relativity, as well as various works by V. Arnol'd or J. Marsden that deal with this material in classical physics contexts.) I bought this book as a reference -- it might be useful also for people who want to come up with computational approaches to the material.

I might have given the book 3 stars but for the facts that (i) no solutions to exercises (only some of which are in the "show that this stuff = X" format) and (ii) text is filled with "clearly"s, "obviously"s and other superfluities.

Rating:
Summary: One of the best books ever
Review: I don't know how they did it but, this is the book you want to buy if you're trying to learn differential geometry, especially if you're learning general relativity. It takes you from the concepts you are already familiar with into differential geometry faster than any other book I've ever tried (and I've tried many!). Before you know it, you are comfortable with covariant derivatives and Lie derivatives and.. well the list could go on. Do not be turned off by the reputation of Dover books-- "cheap and not worth it!" This is a gem.

For those of you learning GR: Buy this book and Schutz's "Geometrical Methods of Mathematical Physics." Read Lovelock and Rund first and then dive into Schutz's book. This will provide you with the necessary mathematical background to handle Wald's "General Relativity" with (some amount of) ease. You might want to try Schutz's "A First Course in General Relativity" before Wald's more advanced book.

I've read many glowing reviews on Amazon about books that I "must have" and, quite frankly, they turned out to be poor choices. But in this case I have to say you "must have" this book! It is that good. And it's cheap, so if you do not agree with me, it's not much money out of your pocket.

Rating:
Summary: not so so rigorous
Review: Lots of material for the price, but this is one of those maths book with a "physical approach", IMO. Definitions for instances, like the definition of a Tensor, aren't always enounced clearly.

This just make things look more complex and different than what they are for no gain.

I believe that the book "Tensor calculus on manifold", same editor, Goldberg/Bishop does a better job: more rigorous and more concise.

Rating:
Summary: Rigorous, yet informal enough to be a lot of fun.
Review: Many years ago, this became the first book I had ever read about tensor calculus, differential geometry, or classical field theories, and I still have not found a better treatment of any of the three subjects anywhere else. I'm now not very far from a PhD in General Relativity theory, and I very rarely need to use any mathematics which I didn't first learn as a freshman undergraduate while reading this book independently. I owe a great debt to Lovelock and Rund, and could not recommend this book any more highly than I am right now.

Rating:
Summary: Amazing, it's only eleven dollars but worth HUNDREDS
Review: Many years ago, this became the first book I had ever read about tensor calculus, differential geometry, or classical field theories, and I still have not found a much better treatment of any of these subjects anywhere else.

The notation is often very classical, in the sense that there are a lot of indices, usually referring to coordinate bases, and there is a lot of talk of "transformation laws." While this style can be distressing to more advanced students, those familiar with the beautiful methods of avoiding such structures, I think it is useful to younger students, especially physicists, who yearn for concrete examples. Also, for the one section in which a more formal approach is advantageous, such a treatment is included as an appendix.

The book is also wonderful for its breadth. It is not a "tensor calculus" book, or a "differential geometry" book. It is really best described as a "geometrical methods" book "with applications to theoretical physics." Yet unlike most examples of this now-cliched subject, the breadth of material is matched by a cohesion of style.

Rating:
Summary: THe Mathematics of General Relativity
Review: The authors present a thorough development of TENSOR CALCULUS, from basic principals, such as ordinary three dimensional vector space. Tensors are generalizations of vectors to any number of dimensions (vectors are type (1,0) tensors, diff. forms are type (0,1) tensors). One of the key principles of General Relativity is that if physical laws are expressed in tensor form, then they are independent of local coordinate systems, and valid everywhere.

Chap. 1: Preliminary Obs.-- Chap. 2: Affine Tensor Algebra in Euclidean Geometry-- Chap. 3: Tensor Analysis on Manifolds -- Chap. 4: Additional Topics from the Tensor Calculus -- Chap. 5: The Calculus of Differential Forms -- Chap. 6: Invariant Problems in the Calculus of Variations -- Chap. 7: Riemannian Geometry -- Chap. 8: Invariant Var. Principles and Phys. Field Theories -

Chap. 8 covers a good deal of General Relativity. This book is a worthy addition to any mathematical library.

Rating:
Summary: Challenging, thorough introduction
Review: This book covers thoroughly the basics of tensor analysis, differential forms and variational calculus. The problems at the end of each chapter are a good blend of straight mechanical computation through to challenging, abstract exercises.

Rating:
Summary: Amazing, it's only eleven dollars but worth HUNDREDS
Review: This is a book from which you can learn about tensors and really KNOW WHAT'S GOING ON!!! Sure, so you can complain about it's slight lack of rigor--big deal!!!!! Once you're done reading this book, move on to the books that DO have rigor (if you're a mathematician-type rather than a physicist-type), but if you want an introduction to the theory of tensors which provides true intuitive understanding of what tensors are, why they are useful, and how the idea of tensors arose, then buy this book. This book requires almost no prerequisites except for a good background in vectors, matrices, and certain aspects of multivariable calculus. Buy it NOW, and thank God that it is published by Dover.