686
10 Variational Problems of Functionals with Vector, Tensor …
+ A j 1 j 2 ··· j n D k 1 k 2 ···k n−1 ,i (e j 1 · e i )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )
= (∂ i e i · A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n ) n − 1
· D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
+ A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n
n
· ∂ i e i D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
= ∇ · A n − 1
· D + A n
· ∇ D = D n − 1
· ∇ · A + ∇ D n
· A
(10.6.1)
In Eq. (10.6.1), change D into δ A, change A into
∂ F
∂∇ A
, then the variational formula
of the left gradient of A is obtained
∂ F
∂∇ A
n + 1
· ∇δ A = ∇ ·
∂ F
∂∇ A
n
· δ A
− ∇ ·
∂ F
∂∇ A
n
· δ A
(10.6.2)
In Eq. (10.6.1), change A into δ A, let D =
∂ F
∂∇· A
, then the variational formula of
the left divergence of A is obtained
∂ F
∂∇ · A
n − 1
· ∇ · δ A = ∇ ·
δ A n − 1
·
∂ F
∂∇ · A
− ∇
∂ F
∂∇ · A
n
· δ A
(10.6.3)
Equation (10.6.2) and Eq. (10.6.3) are the precisely needed results.
Now do n−1 time inner product of D and A, then take the divergence, there is
∇ · ( D n − 1
· A) = ( D n − 1
· A) · ∇ = ∂ i e i · (D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
n − 1
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn )
= (D k 1 k 2 ···k n−1 ,i A j 1 j 2 ··· jn + D k 1 k 2 ···k n−1 A j 1 j 2 ··· jn ,i )(e i · e jn )(e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )
= D k 1 k 2 ···k n−1 ,i A j 1 j 2 ··· jn (e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )(e i · e jn )
+ D k 1 k 2 ···k n−1 A j 1 j 2 ··· jn ,i (e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )(e jn · e i )
= D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 ∂ i e i
n
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn
+ D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
n − 1
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn · ∂ i e i
= D∇ n
· A + D n − 1
· A · ∇ = A n
· D∇ + A · ∇ n − 1
· D
(10.6.4)
In Eq. (10.6.4), change D into δ A, change A into
∂ F
∂ A∇
, then the variational formula
of the right gradient of A is obtained
∂ F
∂ A∇
n + 1
· δ A∇ = ∇ ·
δ A n
·
∂ F
∂ A∇
−
∂ F
∂ A∇
· ∇ n
· δ A
(10.6.5)
In Eq. (10.6.4), change A into δ A, let D =
∂ F
∂ A·∇
, then the variational formula of
the right divergence of A is obtained
10 Variational Problems of Functionals with Vector, Tensor …
+ A j 1 j 2 ··· j n D k 1 k 2 ···k n−1 ,i (e j 1 · e i )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )
= (∂ i e i · A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n ) n − 1
· D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
+ A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n
n
· ∂ i e i D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
= ∇ · A n − 1
· D + A n
· ∇ D = D n − 1
· ∇ · A + ∇ D n
· A
(10.6.1)
In Eq. (10.6.1), change D into δ A, change A into
∂ F
∂∇ A
, then the variational formula
of the left gradient of A is obtained
∂ F
∂∇ A
n + 1
· ∇δ A = ∇ ·
∂ F
∂∇ A
n
· δ A
− ∇ ·
∂ F
∂∇ A
n
· δ A
(10.6.2)
In Eq. (10.6.1), change A into δ A, let D =
∂ F
∂∇· A
, then the variational formula of
the left divergence of A is obtained
∂ F
∂∇ · A
n − 1
· ∇ · δ A = ∇ ·
δ A n − 1
·
∂ F
∂∇ · A
− ∇
∂ F
∂∇ · A
n
· δ A
(10.6.3)
Equation (10.6.2) and Eq. (10.6.3) are the precisely needed results.
Now do n−1 time inner product of D and A, then take the divergence, there is
∇ · ( D n − 1
· A) = ( D n − 1
· A) · ∇ = ∂ i e i · (D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
n − 1
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn )
= (D k 1 k 2 ···k n−1 ,i A j 1 j 2 ··· jn + D k 1 k 2 ···k n−1 A j 1 j 2 ··· jn ,i )(e i · e jn )(e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )
= D k 1 k 2 ···k n−1 ,i A j 1 j 2 ··· jn (e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )(e i · e jn )
+ D k 1 k 2 ···k n−1 A j 1 j 2 ··· jn ,i (e k 1 · e j 1 ) · · · (e k n−1 · e j n−1 )(e jn · e i )
= D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 ∂ i e i
n
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn
+ D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1
n − 1
· A j 1 j 2 ··· jn e j 1 e j 2 · · · e jn · ∂ i e i
= D∇ n
· A + D n − 1
· A · ∇ = A n
· D∇ + A · ∇ n − 1
· D
(10.6.4)
In Eq. (10.6.4), change D into δ A, change A into
∂ F
∂ A∇
, then the variational formula
of the right gradient of A is obtained
∂ F
∂ A∇
n + 1
· δ A∇ = ∇ ·
δ A n
·
∂ F
∂ A∇
−
∂ F
∂ A∇
· ∇ n
· δ A
(10.6.5)
In Eq. (10.6.4), change A into δ A, let D =
∂ F
∂ A·∇
, then the variational formula of
the right divergence of A is obtained