Universit¨at Stuttgart            Institut fu¨r Mechanik
                                                            Prof. Dr.-Ing. W. Ehlers
                                                            www.mechbau.uni-stuttgart.de
                               Vector and Tensor Calculus
                                       An Introduction
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                                     Last Change: 10 April 2018
                Chair of Continuum Mechanics, Pfaffenwaldring 7, D-70569 Stuttgart, Tel.: (0711) 685-66346
                  Contents
                  1 Mathematical Prerequisites                                                                   1
                     1.1   Basics of vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . .     1
                  2 Fundamentals of tensor calculus                                                              9
                     2.1   Introduction of the tensor concept . . . . . . . . . . . . . . . . . . . . . . .      9
                     2.2   Basic rules of tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . .    10
                     2.3   Specific tensors and operations . . . . . . . . . . . . . . . . . . . . . . . . .     13
                     2.4   Change of the basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    16
                     2.5   Higher order tensors    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25
                     2.6   Fundamental tensor of 3rd order (Ricci permutation tensor) . . . . . . . .           28
                     2.7   The axial vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   28
                     2.8   The outer tensor product of tensors . . . . . . . . . . . . . . . . . . . . . .      33
                     2.9   The eigenvalue problem and the invariants of tensors . . . . . . . . . . . .         34
                  3 Fundamentals of vector and tensor analysis                                                 36
                     3.1   Introduction of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .    36
                     3.2   Functions of scalar variables . . . . . . . . . . . . . . . . . . . . . . . . . .    36
                     3.3   Functions of vector and tensor variables . . . . . . . . . . . . . . . . . . . .     37
                     3.4   Integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    41
                     3.5   Transformations between actual and reference configurations . . . . . . . .           46