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
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