Untitled - Aerobib - Universidad Politécnica de Madrid

3.11. APPENDIX: NOTATION AND REPERTOIRE OF VECTORIAL AND TENSORIAL FORMULAE 85
Symmetric tensor: τ = ¯τ .
If τ is symmetric:
τ ij = τ ji (i, j = 1, 2, 3)
Product of a vector by a tensor:
ā · τ = ī 1 (a 1 τ 11 + a 2 τ 21 + a 3 τ 31 )
+ ī 2 (a 1 τ 12 + a 2 τ 22 + a 3 τ 32 )
+ ī 3 (a 1 τ 13 + a 2 τ 23 + a 3 τ 33 ) (vector)
Divergence of a tensor:
∇ · τ =
(
∂τ11
ī 1 + ∂τ 21
+ ∂τ )
31
∂x 1 ∂x 2 ∂x 3
(
∂τ12
+ ī 2 + ∂τ 22
+ ∂τ )
32
∂x 1 ∂x 2 ∂x 3
(
∂τ13
+ ī 3 + ∂τ 23
+ ∂τ )
33
∂x 1 ∂x 2 ∂x 3
(vector)
Gradient of a vector:
⎛
∇ ā =
⎜
⎝
∂a 1
∂x 1
∂a 2
∂x 1
∂a 3
∂x 1
∂a 1
∂x 2
∂a 2
∂x 2
∂a 3
∂x 2
∂a 1
∂x 3
∂a 2
∂x 3
∂a 3
∂x 3
⎞
⎟
⎠
(tensor)
Double product:
ā · (¯b · τ
)
(scalar)
If τ is symmetric, then:
ā · (¯b · τ
)
= ¯b ·
(ā
· τ
)
Contraction of two tensors:
3∑ 3∑
τ : τ ′ = τ ij τ ji
′
i=1 j=1
(scalar)
Therefore
and
τ : τ ′ = τ ′ : τ
∇ · (ā · τ) = ā · (∇ · τ) + (∇ā) : τ (scalar)