The obvious relationships between components can easily be deduced when we represent ourvectors in the contravariant basis,ai=Mijbj,ai=Mjibj,ai=Mijbj.(1.26)Many books termMija contravariant second-order tensor;Mika covariant second-order tensorandMjia mixed second-order tensor, but they are simply representations of thesamecoordinate-independent object in different bases. Another more modern notation is to say thatMijis a type(2,0) tensor,MiJis type (0,2) andMijis a type (1,1) tensor, which allows the distinction betweenmixed tensors or orders greater than two.1.2.1Invariance of second-order tensorsLet us now consider a general change of coordinates fromξitoξi. Given thatai=Mijbj,(1.27a)we wish to find an expression forMijsuch thatai=Mijbj.(1.27b)Using the transformation rules for the components of vectors (1.19) it follows that (1.27a) becomes∂ξi∂ξnan=Mij∂ξj∂ξnbn.We now multiply both sides by∂ξm/∂ξito obtain∂ξm∂ξi∂ξi∂ξnan=δmnan=am=∂ξm∂ξiMij∂ξj∂ξnbn.
Comparing this expression to equation (1.27b) it follows thatMij=∂ξi∂ξnMnm∂ξm∂ξj,and thus we see that covariant components must transform covariantly and contravariant compo-nents must transform contravariantly in order for the invariance properties to hold. Similarly, itcan be shown thatMij=∂ξi∂ξn∂ξj∂ξmMnm,andMij=∂ξn∂ξi∂ξm∂ξjMnm.(1.28)An alternative definition of tensors is to require that they are sets of index quantities (multi-dimensional arrays) that obey the appropriate transformation laws under a change of coordinates.1.2.2Cartesian tensorsIf we restrict attention to orthonormal coordinate systems, then the transformation between co-ordinate systems must be orthogonal9and we do not need to distinguish between covariant andcontravariant behaviour. Consider the transformation from our Cartesian basiseIto another or-thonormal basiseI. The transformation rules for components of a tensor of order two becomeMIJ=∂xN∂xI∂xM∂xJMNM.The transformation between components of two vectors in the different bases are given byaI=∂xI∂xKaK=∂xK∂xIaK,which can be written the formaI=QIKaK,whereQIK=∂xI∂xK=∂xK∂xI,and the componentsQIKform an orthogonal matrix.Hence the transformation property of a(Cartesian) tensor of order two can be written asMIJ=QINMNMQJMor in matrix formM=QMQT.(1.29)In many textbooks, equation (1.29) is defined to be the transformation rule satisfied by a (Cartesian)tensor of order two.1.2.3Tensors vs matricesThere is a natural relationship between tensors and matrices because, as we have seen, we can writethe components of a second-order tensor in a particular coordinate system as a matrix. It is oftenhelpful to think of a tensor as a matrix when working with it, but the two concepts are distinct.