Customer Reviews

Tensors, Differential Forms, and Variational Principles (Dover Books on Mathematics)

5 star:		(9)
4 star:		(1)
3 star:		(1)
2 star:		(3)
1 star:		(0)

 

 

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.

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.

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.

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.

I have to side with reviewers who say this book is near perfect. The original hardcover edition was a favorite of mine when I was an undergrad and its paperback reissue by Dover is a blessing. Readers who complain about the notation may be too easily blinded. Indices are certainly used abundantly, especially in the derivation of tensor theory on affine connections, but this practice actually fits well with the authors' objectives. By retaining traditional notation through most of the book, L&R provide the student with familiar handles to grasp, and the typesetting is the best I've seen for this kind of notation.

A huge value of this book is how it clarifies mathematical details that many other books make confusing. The material is well-organized and guides the reader clearly through most of the derivations. L&R carefully develop (as the notes say, by successive abstraction) the theory of tensor calculus on vector spaces to that of differential forms on Riemannian manifolds. They compare the deployment of affine connections in the former and metric tensors in the latter, enabling the reader to see how curvature can be defined in either context with minimum confusion.

Once differential forms are introduced, the notation becomes more modern and compact, though the use of boldface type common to "geometric" notation is nowhere in the book. Useful integral theorems of Stokes and the invariance properties of Lagrangian systems are derived in the language of differential forms. An appendix summarizing the theory of exterior calculus on differentiable manifolds closes the book.

In conclusion, this book is excellent complementary reading for anyone attempting to master the mathematics essential for an introduction to General Relativity theory, and in good company with other Dover reprints by Bishop & Goldberg, Flanders and Pauli. Even if you have the dough for Misner Thorne & Wheeler and the spine to lug it around, you might want these other books in your library too.

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.

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.

I bought this book togheter with Kay - Tensor analysis - when I began to study tensors. Kay could be not a masterpiece, but you can use it to learn the subject with some efforts but without a teacher. Don't try to do the same thing with Lovelock. May be this book is very profound, but you have better avoiding it as a self study text.

I bought this book six years ago for the exercises on tensor analysis and differential forms and it has become one of my favorite texts on invariant variational principles. The authors develop their account of tensors in a clear and logical manner, with several diagrams in the first few chapters to highlight geometrical concepts. The authors provide a commentary in words to accompany the mathematical exposition. I appreciate their pointing out the various implications of the results they develop and I found the exercises helpful and a useful aid to the development of the theory.

Chapter 6 on the calculus of variations for invariance problems under coordinate transformations is one of the best I have studied. Their explanation of the theorems of Noether and reference to Caratheodory's work clearly explain some of the deeper concepts of invariance. The authors include worked examples which demonstrate the application of the ideas as they are developed. Variational principles are developed further in subsequent chapters. Chapter 8 describes the application variational principles to invariant field theories and the Einstein vacuum field equations.

I think I have a decent handle on the prerequisites for this subject, and I frankly found this to be cryptic. Perhaps the style is a bit archaic; but, I found some of the general relativity texts to give a better introduction to the sujbect of tensors. I would not recommend this as an introductory text.