33 Tensor Multiplication This may also be called outer or exterior or direct or

33 tensor multiplication this may also be called

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3.3 Tensor Multiplication This may also be called outer or exterior or direct or dyadic multiplication , although some of these names may be reserved for operations on vectors. On multiplying each component of a tensor of rank r by each component of a tensor of rank k , both of dimension m , a tensor of rank ( r + k ) with m r + k components is obtained where the variance type of each index (covariant or contravariant) is preserved . For example, if A and B are covariant tensors of rank-1, then on multiplying A by B we obtain a covariant tensor C of rank-2 where the components of C are given by: C ij = A i B j (102) while on multiplying B by A we obtain a covariant tensor D of rank-2 where the compo- nents of D are given by: D ij = B i A j (103) Similarly, if A is a contravariant tensor of rank-2 and B is a covariant tensor of rank-2, then on multiplying A by B we obtain a mixed tensor C of rank-4 where the components of C are given by: C ij kl = A ij B kl (104) while on multiplying B by A we obtain a mixed tensor D of rank-4 where the components of D are given by: D kl ij = B ij A kl (105)
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3.3 Tensor Multiplication 86 In the outer product operation, it is generally understood that all the indices of the involved tensors have the same range although this may not always be the case. [45] In general, the outer product of tensors yields a tensor . The outer product of a tensor of type ( m, n ) by a tensor of type ( p, q ) results in a tensor of type ( m + p, n + q ). This means that the tensor rank in the outer product operation is additive and the operation conserves the variance type of each index of the tensors involved. The direct multiplication of tensors may be marked by the symbol , mostly when using symbolic notation for tensors, e.g. A B . However, in the present book no symbol is being used for the operation of direct multiplication and hence the operation is symbolized by putting the symbols of the tensors side by side, e.g. AB where A and B are non- scalar tensors. In this regard, the reader should be vigilant to avoid confusion with the operation of matrix multiplication which, according to the notation of matrix algebra, is also symbolized as AB where A and B are matrices of compatible dimensions, since matrix multiplication is an inner product, rather than an outer product, operation. The direct multiplication of tensors is not commutative in general as indicated above; however it is distributive with respect to the algebraic sum of tensors, that is: [46] AB 6 = BA (106) A ( B ± C ) = AB ± AC ( B ± C ) A = BA ± CA (107) As indicated before, the rank-2 tensor constructed by the direct multiplication of two vectors is commonly called dyad . Tensors may be expressed as an outer product of vectors where the rank of the resultant product is equal to the number of the vectors involved, e.g.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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