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Introduction to Tensor Calculus KeesDullemond&KasperPeeters c 1991-2023 This booklet contains an explanation about tensor calculus for students of physics andengineeringwithabasicknowledgeoflinearalgebra. Thefocusliesmainlyon acquiring an understanding of the principles and ideas underlying the concept of ‘tensor’. We have not pursued mathematical strictness and pureness, but instead ´ emphasisepracticaluse(foramoremathematicallypureresume,pleaseseethebib- liography). Althoughtensorsareappliedinaverybroadrangeofphysicsandmath- ematics, this booklet focuses on the application in special and general relativity. Weareindebtedtoallpeoplewhoreadearlierversionsofthismanuscriptandgave ¨ useful comments, in particular G. Bauerle (University of Amsterdam) and C. Dulle- mondSr. (University of Nijmegen). The original version of this booklet, in Dutch, appeared on October 28th, 1991. A majorupdatefollowedonSeptember26th,1995. Thisversionisare-typesetEnglish translation made in 2008/2010. c Copyright 1991-2010KeesDullemond&KasperPeeters. 1 Theindexnotation 5 2 Bases, co- and contravariant vectors 9 2.1 Intuitive approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Mathematicalapproach . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Introduction to tensors 15 3.1 Thenewinnerproductandthefirsttensor . . . . . . . . . . . . . . . 15 3.2 Creating tensors from vectors . . . . . . . . . . . . . . . . . . . . . . . 17 4 Tensors, definitions and properties 21 4.1 Definition of a tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Symmetryandantisymmetry . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Contraction of indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Tensors as geometrical objects . . . . . . . . . . . . . . . . . . . . . . . 22 4.5 Tensors as operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Themetrictensorandthenewinnerproduct 25 5.1 Themetricasameasuringrod . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Properties of the metric tensor . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Coversuscontra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Tensorcalculus 29 6.1 The‘covariance’ of equations . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 Additionoftensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.3 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.4 First order derivatives: non-covariant version . . . . . . . . . . . . . . 31 6.5 Rot, cross-products and the permutation symbol . . . . . . . . . . . . 32 7 Covariant derivatives 35 7.1 Vectors in curved coordinates . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 Thecovariantderivative of a vector/tensor field . . . . . . . . . . . . 36 A Tensorsinspecialrelativity 39 B Geometricalrepresentation 41 C Exercises 47 C.1 Indexnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 C.2 Co-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 C.3 Introduction to tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 C.4 Tensors, general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 C.5 Metrictensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 C.6 Tensorcalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 4
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