Tensor Calculus

Rating:
Summary: Antiquated
Review: "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.

Rating:
Summary: Hard to follow
Review: It's an old fashioned text, confusing and hard to follow.

Rating:
Summary: A good solid introduction
Review: Synge and Schild is a good solid introduction to tensor calculus, as it is used by most physicists, and was used throughout the 20th century.

Rating:
Summary: okay book
Review: this book dosen't take things from basics but goes to do high level calculus.

Rating:
Summary: The best classical introduction to tensors
Review: 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.