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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)

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