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Theory Manual Version 3.6
 Section A.2: Higher Order Tensors Up Section A.2: Higher Order Tensors Subsection A.2.2: Fourth-Order Tensors 

A.2.1 Third-Order Tensors

A third-order tensor is a linear transformation that transforms any vector into a second-order tensor , In general, the dyadic product of three vectors is a third-order tensor which satisfies Any third-order tensor can be expressed in terms of its Cartesian components as and the Cartesian components of a third-order tensor may be evaluated from It follows from Eqs.(A.2.1-2) and (A.2.1-3) that so that the indicial form of eq.(A.2.1-1) is In particular,
The double dot product of a third-order tensor with a second-order tensor is defined by and Therefore, the double dot product of a third-order tensor with a second-order tensor is a vector given by Proof: Using , we find that . Similarly, , thus completing the proof.
For any second-order tensor , it also follows that
If we introduce the notation as the third-order (pseudo-)tensor of Cartesian components , the relation between an antisymmetric tensor and its dual vector can also be written as Similarly,
 Section A.2: Higher Order Tensors Up Section A.2: Higher Order Tensors Subsection A.2.2: Fourth-Order Tensors