The obvious relationships between components can

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The obvious relationships between components can easily be deduced when we represent our vectors in the contravariant basis, a i = M ij b j , a i = M j i b j , a i = M ij b j . (1.26) Many books term M ij a contravariant second-order tensor; M ik a covariant second-order tensor and M j i a mixed second-order tensor, but they are simply representations of the same coordinate- independent object in different bases. Another more modern notation is to say that M ij is a type (2 , 0) tensor, M iJ is type (0 , 2) and M i j is a type (1 , 1) tensor, which allows the distinction between mixed tensors or orders greater than two. 1.2.1 Invariance of second-order tensors Let us now consider a general change of coordinates from ξ i to ξ i . Given that a i = M i j b j , (1.27a) we wish to find an expression for M i j such that a i = M i j b j . (1.27b) Using the transformation rules for the components of vectors (1.19) it follows that (1.27a) becomes ∂ξ i ξ n a n = M i j ∂ξ j ξ n b n . We now multiply both sides by ξ m /∂ξ i to obtain ξ m ∂ξ i ∂ξ i ξ n a n = δ m n a n = a m = ξ m ∂ξ i M i j ∂ξ j ξ n b n .
Comparing this expression to equation (1.27b) it follows that M i j = ξ i ∂ξ n M n m ∂ξ m ξ j , 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, it can be shown that M ij = ξ i ∂ξ n ξ j ∂ξ m M nm , and M ij = ∂ξ n ξ i ∂ξ m ξ j M nm . (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. 1.2.2 Cartesian tensors If we restrict attention to orthonormal coordinate systems, then the transformation between co- ordinate systems must be orthogonal 9 and we do not need to distinguish between covariant and contravariant behaviour. Consider the transformation from our Cartesian basis e I to another or- thonormal basis e I . The transformation rules for components of a tensor of order two become M IJ = ∂x N x I ∂x M x J M NM . The transformation between components of two vectors in the different bases are given by a I = x I ∂x K a K = ∂x K x I a K , which can be written the form a I = Q IK a K , where Q IK = x I ∂x K = ∂x K x I , and the components Q IK form an orthogonal matrix. Hence the transformation property of a (Cartesian) tensor of order two can be written as M IJ = Q IN M NM Q JM or in matrix form M = QMQ T . (1.29) In many textbooks, equation (1.29) is defined to be the transformation rule satisfied by a (Cartesian) tensor of order two. 1.2.3 Tensors vs matrices There is a natural relationship between tensors and matrices because, as we have seen, we can write the components of a second-order tensor in a particular coordinate system as a matrix. It is often helpful to think of a tensor as a matrix when working with it, but the two concepts are distinct.

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