Tensor Calculus [Paperback]

J. L. Synge (Author), A. Schild (Author), Mathematics (Author)

Book Description

Series: Dover Books on Mathematics | Publication Date: July 1, 1978

Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, special types of space, relative tensors, ideas of volume, more.

Product Details

• Paperback: 324 pages
• Publisher: Dover Publications (July 1, 1978)
• Language: English
• ISBN-10: 0486636127
• ISBN-13: 978-0486636122
• Product Dimensions: 8.2 x 5.6 x 0.6 inches
• Shipping Weight: 12.6 ounces (View shipping rates and policies)
• Average Customer Review: 2.8 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #337,611 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

76 of 80 people found the following review helpful:

5.0 out of 5 stars The best classical introduction to tensors, July 21, 1998

By 

henrique fleming - See all my reviews

This review is from: Tensor Calculus (Paperback)

This is probably the clearest classical treatment of tensors you can find. Tensors are objects whose components transform in some linear and homogeneous way. This is the original definition, by Ricci, the founder of the theory. Today one prefers to define them as the members of some vector space and avoid talking of components. However, most physicists adhere to the classical formulation. After all this was the tensor calculus known to Einstein! Anyway the job is extremely well done: you end up knowing about parallel transportation and covariant derivative, curvature tensor and several applications. You'll be able to write the Laplacian operator in any corrdinate system whatsoever, and so on. I think the chapter on Integration is much more difficult than the others, but, then, invariant integration is the realm of exterior differential forms, and building them from tensors is inevitably clumsy.

26 of 29 people found the following review helpful:

5.0 out of 5 stars This is really a good book, despite what some people are saying..., March 10, 2006

By 

Neal J. King (Munich, Germany) - See all my reviews
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This review is from: Tensor Calculus (Paperback)

I find it rather strange that several of the negative reviews of Synge & Schild are really negative opinions about the lack of elegance of tensors, compared to the new-fangled differential forms.

This is like blaming the author of a book on the grammar of a language, because you think the grammar is too complicated. Sorry, but the author of the book can only explain as well as he/she can the grammar that exists, it's not within his scope to improve upon it!

This book is a relatively easy-to-read and carefully motivated text on tensor calculus, a subject that does tend to lead to that eye-glazing-over effect because of the numerous indices. It does a very good job of keeping the focus on the concepts, without getting too bogged down in the equations - most of the time.

Does it need to be said that this subject is still useful, despite its comparative inelegance, because so many classic texts and articles on general relativity use this language? Will those who scorn to deal with indices demand that all these papers be properly translated into differential forms before they deign to read them?

64 of 87 people found the following review helpful:

1.0 out of 5 stars Antiquated, September 30, 2001

By 

Dougabug "dougabug" (Orangevale, CA United States) - See all my reviews

This review is from: Tensor Calculus (Paperback)

"The introduction of numbers as coordinates...is an act of violence..." -- H. Weyl.

If that's so, this is a very violent book. While it's true that physicists, particularly those working in General Relativity, were slow to abandon the coordinate approach, there can be little doubt that the sea of indicies form of Tensor Calculus runs counter to the modern approach to Differential Geometry, with its emphasis on abstract spaces, manifolds, bundles, exterior algebra, differential forms, diffeomorphisms, Lie groups, etc.

Physicists trained prior to the trend towards employing modern mathematics will likely be right at home with this book, which presents the tensor calculus in the form developed by Levi-Civita and Ricci in the late 19th/early 20th Century. On the other hand, classically trained Physicists tend to be hopelessly confused when confronted by modern Differential Geometry, which relies on so much more of the modern machinery from areas such as Topology, Global Analysis, and Group Theory/Representation Theory.

Students would be better served to pursue the subject framed in a more modern context. That means learning about manifolds and analysis on manifolds. The best introduction is probably Spivak's "Calculus on Manifolds", followed by Munkres "Analysis on Manifolds". Darling's "Differential Forms and Connections" and Sternberg's "Lectures on Differential Geometry" are well regarded, as is do Carmo's "Differential Geometry of Curves and Surfaces". A working knowledge of multivariable calculus, linear algebra, and elementary analysis are required for making heads or tails out of these books, even though they are introductory in nature. Having digested all that, one can now embark on the study of Riemannian geometry, say through do Carmo's "Riemannian Geometry", or Spivak's "A Comprehensive Course in Differential Geometry" (5 vols.). If you survived that then attentively study Kobayashi/Nomizu "Foundations of Differential Geometry" (2 vols., the diffeomorphism/bundle perspective) or Helgason "Differential Geometry, Lie Groups, and Symmetric Spaces" (from the perspective of Representation Theory) and go write your dissertation. Then come back and explain it all to me.