10.7 Variational Problems of Functionals with Series-Type …
719
Equations (10.7.9) and (10.7.10) are the precisely needed results.
In Eq. (10.7.2), change A into A∇, there is
∂ F
∂ A∇∇
n + 2
· δ A∇∇ = ∇ ·
∂ F
∂ A∇∇
n + 1
· δ A∇
− ∇ ·
∂ F
∂ A∇∇
n + 1
· δ A∇
(10.7.11)
For the second term on the right side of Eq. (10.7.11), taking ∇ ·
∂ F
∂ A∇∇
as
∂ F
∂ A∇
,
making use of Eq. (10.7.2), we obtain
∂ F
∂ A∇∇
n + 2
· δ A∇∇ = ∇ ·
∂ F
∂ A∇∇
n + 1
· δ A∇
− ∇ ·
∇ ·
∂ F
∂ A∇∇
n
· δ A
+ ∇ · ∇ ·
∂ F
∂ A∇∇
n
· δ A
(10.7.12)
Repeat the above process for the last term in Eq. (10.7.12), the recurrence formula
of the right gradient variation of the tensor of order n can be obtained
∂ F
∂ A(∇) m
n + m
· δ A(∇) m =
m
i=1
(−1) i−1 ∇ ·
(∇·) i−1 ∂ F
∂ A(∇) m
n + m − i
·
δ A(∇) m−i
+ (−1) m (∇·) m ∂ F
∂ A(∇) m
n
· δ A
(10.7.13)
In Eq. (10.7.6), change A into A · ∇, there is
∂ F
∂ A · ∇ · ∇
n − 2
· δ A · ∇ · ∇ = ∇ ·
∂ F
∂ A · ∇ · ∇
n − 2
· δ A · ∇
− ∇
∂ F
∂ A · ∇ · ∇
n − 1
· δ A · ∇
(10.7.14)
Making use of Eq. (10.7.6) for the second term on the right side of Eq. (10.7.14),
we obtain
∂ F
∂ A · ∇ · ∇
n − 2
· δ A · ∇ · ∇ = ∇ ·
∂ F
∂ A · ∇ · ∇
n − 2
· δ A · ∇
− ∇ ·
∇
∂ F
∂ A · ∇ · ∇
n − 1
· δ A
− +∇∇
∂ F
∂ A · ∇ · ∇
n
· δ A
(10.7.15)
Repeat the above process for the last term in Eq. (10.7.15), the recurrence formula
of the right divergence variation of the tensor of order n can be obtained
∂ F
∂ A(·∇) m
n − m
· δ A(·∇) m =
m
i
(−1) i−1 ∇ ·
(∇) i−1 ∂ F
∂ A(·∇) m
n − m + i − 1
·
δ A(∇) m−i