Tensors of different order

The double contracting or double inner product of general tensors results in a tensor with the added order of the multiplied tensors lowered by four. The employed symbol of two dots alludes to the two scalar products of the particular base vectors. In the case of two tensors of second order, the outcome is of zeroth order, leading to the denomination as a scalar product of dyads. The double contracting product is commutative, given here for the case of dyads  

When only a single scalar product of base vectors is involved, the result of such a product has the added order of the multiplied tensors lowered by two. This contracting or inner product of general tensors thus comprises the scalar product of vectors as a special case. Further on, the tensorial or outer product 

For the occurrence of a transposed dyad within consecutive contracting products, the following rearrangement is permissible  

Concerning the partial differentiation of tensors, the following abbreviations for gradient and divergence are introduced  

For the manipulation of equations with products containing these operators. Gauss s divergence theorem will be needed. It is given for the usual case with the product of a scalar and a vector in Eq. (3.6a) and for the contracting product of a transposed dyad and a vector in Eq. (3.6b)  

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In general, there can be no coupling between flow processes represented by tensors of different orders. 

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