622
10 Variational Problems of Functionals with Vector, Tensor …
∇ · (a × b) = −∇ · (b × a) = b · ∇ × a − a · ∇ × b
(10.2.21)
a · a = |a|
2
, ∇u · ∇u = |∇u|
2
, (∇ × a) · (∇ × a) = |∇ × a|
2
(10.2.22)
δ|a| =
a
|a|
· δa, δ|∇u| =
∇u
|∇u|
· δ∇u =
∇u
|∇u|
· ∇δu,
δ|∇ × a| =
∇ × a
|∇ × a|
· δ∇ × a =
∇ × a
|∇ × a|
· ∇ × δa
(10.2.23)
Making use of Eqs. (10.2.1), (10.2.2), and (10.2.20)–(10.2.23), we obtain
∂ F
∂∇u
· δ∇u =
∂ F
∂∇u
· ∇δu = ∇ ·
∂ F
∂∇u
δu
−
∇ ·
∂ F
∂∇u
δu
(10.2.24)
∂ F
∂|∇u|
δ|∇u| =
∂ F
∂|∇u|
∇u
|∇u|
· δ∇u = ∇ ·
∂ F
∂|∇u|
∇u
|∇u|
δu
−
∇ ·
∂ F
∂|∇u|
∇u
|∇u|
δu
(10.2.25)
∂ F
∂∇ a
: δ∇ a =
∂ F
∂∇ a
: ∇δa = ∇ ·
∂ F
∂∇ a
· δa
−
∇ ·
∂ F
∂∇ a
· δa (10.2.26)
∂ F
∂ a∇
: δa∇ = ∇ ·
δa ·
∂ F
∂ a∇
−
∂ F
∂ a∇
· ∇
· δa
(10.2.27)
∂ F
∂∇ · a
δ∇ · a =
∂ F
∂∇ · a
∇ · δa = ∇ ·
∂ F
∂∇ · a
δa
−
∇
∂ F
∂∇ · a
· δa (10.2.28)
∂ F
∂∇ × a
· δ∇ × a =
∂ F
∂∇ × a
· ∇ × δa = −∇ ·
∂ F
∂∇ × a
× δa
+
∇ ×
∂ F
∂∇ × a
· δa
(10.2.29)
∂ F
∂|a|
δ|a| =
∂ F
∂|a|
a
|a|
· δa
(10.2.30)
∂ F
∂|∇ × a|
δ|∇ × a| =
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
· δ∇ × a
= −∇ ·
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
× δa
+
∇ ×
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
· δa
(10.2.31)
Substituting Eqs. (10.2.24)–(10.2.31) into Eq. (10.2.19), it can be written as
δ J =
V
∂ F
∂u
− ∇ ·
∂ F
∂∇u
− ∇ ·
∂ F
∂|∇u|
∇u
|∇u|
δu+
∂ F
∂ a
− ∇ ·
∂ F
∂∇ a
−
∂ F
∂ a∇
· ∇ − ∇
∂ F
∂∇ · a
+ ∇ ×
∂ F
∂∇ × a
+
∂ F
∂|a|
a
|a|
+ ∇ ×
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
· δa
dV +
V
∇ ·
∂ F
∂∇u
δu +
∂ F
∂|∇u|
∇u
|∇u|
δu
dV
+
V
∇ ·
∂ F
∂∇ a
· δa + δa ·
∂ F
∂ a∇
+
∂ F
∂∇ · a
δa −
∂ F
∂∇ × a
× δa −
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
× δa
dV = 0
(10.2.32)