Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier MalcataЧитать онлайн книгу.
parallel to Eq. (4.20).
The scalar (or dot) product of a vector u by a tensor τ is a vector defined by
(5.31)
inspired by Eq. (3.52) and taking advantage of Eq. (4.47) – or, in condensed form,
if the order of multiplication is reversed, one obtains
(5.33)
that may be rephrased as
– so inspection of Eq. (5.34) vis‐à‐vis with Eq. (5.32) indicates that
(5.35)
except if τ is symmetric (as τji = τij under such circumstances).
The scalar product of two tensors, σ and τ (also known as double dot product, :), is a scalar defined as
(5.36)
where straightforward algebraic rearrangement was used to advantage – or, in condensed form,
(5.37)
Here tensor σ abides to
(5.38)
while (1 × 3) row vectors σ_x, σ_y, and σ_z are defined as
(5.39)
(5.40)
and
(5.41)
respectively, whereas (3 × 1) column vectors τx_, τy_, and τz_ abide to
(5.42)
(5.43)
and
(5.44)
respectively.
Finally, the dot product of two tensors, σ and τ, may be considered, according to
thus generating another tensor; Eq. (5.45) is equivalent to writing
(5.46)
which mimics the classical product of two matrices – see Eq. (4.47).
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