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I am really confused about the use of tensors in special relativity. I'm not sure if I understood what is a covariant and a contravariant tensor and vector. I am looking for a book of relativity that teach this.

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Schutz's A First Course in General Relativity does a good job about tensors and relativity in the first few chapters.

He has another book dedicated to specifically the mathematics, but its title slips my mind. If its as good as those briefer chapters, you won't go wrong.

A more thorough treatment of the mathematics can be found in Tensors, Differential Forms, and Variational Principles

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    $\begingroup$ Great recommendation. Lovelock and Rund have done a good job $\endgroup$ – Cryo May 10 '19 at 21:41
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Almost all the standard books on general relativity include a section on tensors and special relativity.

Only for tensor calculus:

  • Schaum's Outline of Tensor Calculus: A textbook by David Kay covers all fundamental concepts along with 300+ fully worked problems and solutions. This can be supplemented with standard textbooks.

  • Introduction to Tensor Calculus: Freely available on arXiv. This can be used as a short reference for an introductory course on tensor algebra and calculus.

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