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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,

(5.32)

      if the order of multiplication is reversed, one obtains

      (5.33)

      that may be rephrased as

(5.34)

– so inspection of Eq. (5.34) vis‐à‐vis with Eq. (5.32) indicates that

      (5.35)

      except if τ is symmetric (as τ_ji = τ_ij 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 τ_{x_}, τ_{y_}, and τ_{z_} 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

(5.45)

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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