Elements of Linear Algebra and Direct Tensor Calculus
Since linear algebra and direct tensor calculus are widely used in what follows, some details of the appropriate techniques are given here for the convenience of the reader. While no attempt is made at completeness or mathematical rigor, the material present is nonetheless of value in that some of the relationships given here are not to be found in the accessible literature, at least not in the form used here.
KeywordsLinear Algebra Independent Vector Unit Tensor Orthogonal Tensor Tensor Calculus
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- A linear space is a set of elements termed vectors such that a sum of these, or the result of multiplying one by a number, also belongs to that set, these operations satisfying the axioms of commutation, association, and distribution (see  for details). Here we envisage only real linear spaces.Google Scholar
- These sets in general are hypercomplex numbers.Google Scholar
- A vector is taken as any quantity dependent on a single subscript, while a matrix is a quantity governed by two subscripts, all subscripts taking the same set of values.Google Scholar
- There are certain u that allow (10.17) and (10.18) to be obeyed although (10.15) and (10.16), respectively, are not.Google Scholar
- Conversely, a matrix is unit matrix if it leaves a vector unaltered.Google Scholar
- Note that u’ denotes the same vector as u, but with other components referred to a new frame of reference. The prime to u indicates precisely this feature.Google Scholar
- Transposition = permutation of two subscripts.Google Scholar
- The subscript to the quantity in bold type, n i, serves to distinguish the vectors; it should not be confused with the subscript to the same letter in ordinary type, which serves to distinguish components of a single vector n.Google Scholar
- We consider here for simplicity vectors and tensors that are zero, whereas it would be more accurate to speak of zero vectors (vectors all of whose components are zero) .Google Scholar
- Dyads are considered as independent if the left-hand vectors are linearly independent, as are the right-hand ones.Google Scholar