A.  Properties and Identities of First and Second Rank Tensors

Several operations and identities of first and second order tensors are defined below for fast consulting. The same notation of the main text is used. The capitalized letters (U  , S  , T  ) refer to tensors and the amount of bars above them identifies the tensor ranking. The bold lower case letters (u  , v  ) refer to vectors, while the not bold letters are real numbers, except for f  which refer to a real function.

Addition
U = S + T = T + S →  Uij = Sij + Tij
(A.1)

Product of a Tensor and a Vector
     
v = Su →  vi = Sijuj
(A.2)

v = uS→  v =  u S
           j    j ji
(A.3)

Product of two Tensors
     
U = S ⋅T →  U = SikTkj
(A.4)

Transpose
      ⌊             ⌋
 T    S11  S21  S31
S ≡  ⌈S12  S22  S32⌉
       S13  S23  S33
(A.5)

Trace
      
trace(S) = S11 + S22 + S33
(A.6)

Inner Product
         ∑
S: T =       SijTij
        1<i<n
        1<j<m
(A.7)

Tensor Product
              ( u )               ( u v   u v   u v )
          T   (  1) (         )   (  1 1   1 2   1 3)
u⊗  v ≡ uv  =   u2   v1  v2 v3  =   u2v1  u2v2  u2v3
                u3                  u3v1  u3v2  u3v3
(A.8)

       ∑n  m∑
u⊗ v =        aibjui ⊗ vj
       i=1 j=1
(A.9)

In A.9 the vectors ui  and vi  are basis of the space of u  and v  respectively.

Tensor Identities
         
∇  ⋅(∇  × U) = 0
(A.10)

∇ ⋅(Δ U ) = Δ (∇ ⋅ U )
(A.11)

∇ ⋅(fU) = f∇ ⋅ U + U⋅∇f
(A.12)

∇ ⋅(U ∧ V) = V ⋅∇ × U − U ⋅∇ × V
(A.13)

            
∇ ⋅(∇U ) = ΔU
(A.14)

       t         
∇ ⋅(∇ U) = ∇ (∇ ⋅U )
(A.15)

∇ ⋅(U⊗ V) = U∇ ⋅ V + ∇U ⋅ V
(A.16)

∇ ⋅(f T) = f∇ ⋅T+ T⋅∇f
(A.17)

∇ ⋅(T ⋅ U ) = (∇ ⋅T⋅U)t + T : ∇ U
(A.18)

      
∇ ⋅(f I) = ∇f
(A.19)

                          
∇  × (∇  × U) = ∇ (∇  ⋅U) − ΔU
(A.20)

∇ × (Δ U ) = Δ (∇ × U )
(A.21)

                         
∇ × (fU ) = f∇ × U + ∇f ∧ U
(A.22)

∇ × (U ∧ V) = ∇ U ⋅ V + U ∇ ⋅ V − V ∇ ⋅ U − ∇ V ⋅U
(A.23)

                     
∇ (U ⋅ V) = ∇ U ⋅ V + ∇ V ⋅ U + U ∧ ∇ × V + V ∧ ∇ × U
(A.24)

                       U-2
∇ U⋅U = ∇ × U ∧ U + ∇  2
(A.25)