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10 Variational Problems of Functionals with Vector, Tensor …
= (T jk,i a s + T jk a s,i )(e i · e j )(e k · e s )
= (∂ i e i · T jk e j e k ) · a s e s + (T jk e j e k ) : (a s e s ∂ i e i )
= (∇ · T ) · a + T : a∇ = a · (∇ · T ) + a∇ : T
(10.2.10)
Equation (10.2.3) is proved.
Take the divergence to a · T , and makeuse of Eq. (10.2.9), we have
∇ · (a · T ) = (a · T ) · ∇ = ∂ i e i · (a s e s · T jk e j e k )
= (a s,i T jk + a s T jk,i )(e i · e k )(e s · e j )
= (∂ i e i a s e s ) : T jk e j e k + a s e s · (T jk e j e k · ∂ i e i )
= ∇a : T + a · (T · ∇) = T : ∇a + (T · ∇) · a
(10.2.11)
Equation (10.2.4) is proved. Quod erat demonstrandum.
The functional and its Euler equations with the vector, modulus of vector, and
Hamiltonian operators can be expressed by the following theorem.
Theorem 10.2.1 Let V be a spatial domain, S is the boundary surface of V, the
independent variables x, y, z ∈ V , the scalar function u(x, y, z) ∈ C
2
(V ), and the
vector function a(x, y, z) ∈ C
2
(V ), if the functional with the vector, modulus of
vector, and Hamiltonian operators
J =
V
F(u, a, ∇u, |∇u|, ∇ a, a∇, ∇ · a, ∇ × a, |a|, |∇ × a|)dV (10.2.12)
takes an extremum, where the double inner product of the second order tensors
∇ a and a∇ and any the second order tensors conforms to the rule of the paralleltype operation of the tensors, then the corresponding Euler equations and the natural
boundary conditions are
F u − ∇ ·
∂ F
∂∇u
− ∇ ·
∂ F
∂|∇u|
∇u
|∇u|
= 0
(10.2.13)
∂ F
∂ a
− ∇ ·
∂ F
∂∇ a
−
∂ F
∂ a∇
· ∇ − ∇
∂ F
∂∇ · a
+ ∇ ×
∂ F
∂∇ × a
+
∂ F
∂|a|
a
|a|
+ ∇
×
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
= 0
(10.2.14)
∂ F
∂∇u
· n
S
= 0,
∂ F
∂|∇u|
∇u
|∇u|
· n
S
= 0
(10.2.15)
n ·
∂ F
∂∇ a
S
= 0,
∂ F
∂ a∇
· n
S
= 0,
∂ F
∂∇ · a
n
S
= 0,
∂ F
∂∇ × a
× n
S
= 0,
∂ F
∂|∇ × a|
∇ × a
|∇ × a|
× n
S
= 0
(10.2.16)
For the third boundary condition in Eq. (10.2.16), the vector n can be neglected
when it does not appear together with other boundary condition.