722
10 Variational Problems of Functionals with Vector, Tensor …
+
V
∂ F
∂ A∇
n + 1
·
δ A∇ +
∂ F
∂ A · ∇
n − 1
·
δ A · ∇ +
∂ F
∂ A × ∇
n
·
δ A × ∇
dV
=
V
∂ F
∂ A
−
∂ F
∂∇ A
· ∇ −
∂ F
∂∇ · A
∇ −
∂ F
∂∇ × A
× ∇ − ∇ ·
∂ F
∂ A∇
− ∇
∂ F
∂ A · ∇
− ∇ ×
∂ F
∂ A × ∇
n
·
δ AdV
+
V
∇ ·
δ A n
·
∂ F
∂∇ A
+ δ A n − 1
·
∂ F
∂∇ · A
+ δ A ×
(n − 1)·
∂ F
∂∇ × A
+
∂ F
∂ A∇
n
·
δ A +
∂ F
∂ A · ∇
n − 1
·
δ A +
∂ F
∂ A × ∇
×
(n − 1)·
δ A
dV
=
V
∂ F
∂ A
−
∂ F
∂∇ A
· ∇ −
∂ F
∂∇ · A
∇ −
∂ F
∂∇ × A
× ∇ − ∇ ·
∂ F
∂ A∇
− ∇
∂ F
∂ A · ∇
− ∇ ×
∂ F
∂ A × ∇
n
·
δ AdV
+
S
δ A n
·
∂ F
∂∇ A
+ δ A n − 1
·
∂ F
∂∇ · A
+ δ A ×
(n − 1)·
∂ F
∂∇ × A
+
∂ F
∂ A∇
n
·
δ A +
∂ F
∂ A · ∇
n − 1
·
δ A +
∂ F
∂ A × ∇
×
(n − 1)·
δ A
· n dS
=
V
∂ F
∂ A
−
∂ F
∂∇ A
· ∇ −
∂ F
∂∇ · A
∇ −
∂ F
∂∇ × A
× ∇ − ∇ ·
∂ F
∂ A∇
− ∇
∂ F
∂ A · ∇
− ∇ ×
∂ F
∂ A × ∇
n
·
δ AdV
+
S
∂ F
∂∇ A
· n +
∂ F
∂∇ · A
n +
∂ F
∂∇ × A
× n + n ·
∂ F
∂ A∇
+ n
∂ F
∂ A · ∇
+ n ×
∂ F
∂ A × ∇
n
·
δ AdS = 0
(10.7.29)
where, the integral of the area term is zero.
Since the arbitrariness of δ A and the independence of the terms in surface integral,
according to the fundamental lemma of the calculus of variations, it follows that the
bracket terms of the last volume integral in Eq. (10.7.29) give the Euler Eq. (10.7.27),
and the bracket terms of the last surface integral in Eq. (10.7.29) give the natural
boundary conditions (10.7.28). Quod erat demonstrandum.
It can be seen from comparing the volume integrals and the surface integrals of
Eq. (10.7.29) that the unit outward normal vector n s on the boundary surface S still
correspond with the Hamiltonian operators ∇, they are only opposite in signs, but
their corresponding terms are the same respectively.
It can be seen from Eq. (10.7.29) that for the variation of the left gradient of A,
if let T = ∇, u = δ A, v =
∂ F
∂∇ A
, then there is T
∗
= − · ∇, namely the negative
right divergence operator is the adjoint operator of the left gradient operator. For the
variation of the left divergence of A, if let T = ∇·, u = δ A, v =
∂ F
∂∇· A
, then there is
T
∗
= −∇, namely the negative right gradient operator is the adjoint operator of the
left divergence operator. For the variation of the left rotation of A, if let T = ∇×,
u = δ A, v =
∂ F
∂∇× A
, then there is T
∗
= − × ∇, that is to say that the negative right
rotation operator is the adjoint operator of the left rotation operator. Similarly, for
the variation of the right gradient of A, if let T = ∇, u = δ A, v =
∂ F
∂∇ A
, then there
is T
∗
= −∇·, the negative left divergence operator is the adjoint operator of the
right gradient operator. For the variation of the right divergence of A, if let T = ·∇,
u = δ A, v =
∂ F
∂ A·∇
, then there is T
∗
= −∇, namely the negative left gradient operator
is the adjoint operator of the right divergence operator. For the variation of the right
rotation of A, if let T = ×∇, u = δ A, v =
∂ F
∂ A×∇
, then there is T
∗
= −∇×, that
is to say that the negative left rotation operator is the adjoint operator of the right
rotation operator.
Let u be a scalar, a is a vector, then there is ∇u = u∇, ∇ · a = a · ∇. Making the
inner product of ∇δu (or δu∇) and a, there is (∇δu, a) = (δu, −∇ · a), (δu∇, a) =
(δu, −a · ∇), (∇δu, a) = (δu, −a · ∇), (δu∇, a) = (δu, −∇ · a). Making the inner
product of ∇ · δa and u can also obtain the similar results. Let a and b be both the
vectors, then there is ∇×a = −a×∇, making the inner product of ∇×δa and b, there
is (∇×δa, b) = (δa, ∇×b), (δa×∇, b) = (δa, b×∇), (∇×δa, b) = (δa, −b×∇),
(δa × ∇, b) = (δa, −∇ × b), this shows that in specific case, for the inner product