612
10 Variational Problems of Functionals with Vector, Tensor …
Taking the variation of the tensor A, we give
δ A n
· C
T
= C n
· δ A
T
= δ A
T n
· C = C
T n
· δ A
(10.1.5)
Let C =
∂ F
∂ A
, the variational expression of transposed tensor can be obtained
δ A n
·
∂ F
∂ A
T
=
∂ F
∂ A
n
· δ A
T
= δ A
T n
·
∂ F
∂ A
=
∂ F
∂ A
T
n
· δ A
(10.1.6)
Let A be a tensor of order m, A = a 1 a 2 · · · a m , B is a tensor of order n, B =
b 1 b 2 · · · b n , C is a tensor of order m + n, C = c 1 c 2 · · · c m+n , then the m + n time
parallel type inner product of the juxtaposition tensor AB of A and B and C has the
following operation properties
AB m + n
· C = a 1 a 2 · · · a m b 1 b 2 · · · b n m + n
· c 1 c 2 · · · c m+n = A m
· C n
· B
= C m + n
· AB = C T m + n
· B T A T = B T A T m + n
· C T
= A m
· (C n
· B) = (C n
· B) m
· A = A T m
· (B T n
· C T ) = (B T n
· C T ) m
· A T
= B n
· ( A m
· C) = ( A m
· C) n
· B = B T n
· (C T m
· A T ) = (C T m
· A T ) n
· B T
(10.1.7)
Let A be a tensor of order m, A = a 1 a 2 · · · a m , B is a tensor of order n, B =
b 1 b 2 · · · b n , C is a tensor of order m + n, C = c 1 c 2 · · · c m+n , then the m + n time
series type inner product of the juxtaposition tensor AB of A and B and C has the
following operation properties
AB m + n
· C = a 1 a 2 · · · a m b 1 b 2 · · · b n m + n
· c 1 c 2 · · · c m+n = B n
· C m
· A
= C m + n
· AB = C T m + n
· B T A T = B T A T m + n
· C T
= A m
· (B n
· C) = (B n
· C) m
· A = A T m
· (C T n
· B T ) = (C T n
· B T ) m
· A T
= B n
· (C m
· A) = (C m
· A) n
· B = B T n
· ( A T m
· C T ) = ( A T m
· C T ) n
· B T
(10.1.8)
Let A, B and C be all the tensors of order n, A = a 1 a 2 · · · a n , B = b 1 b 2 · · · b n ,
C = c 1 c 2 · · · c n , then the n time parallel type cross product A n
× B of A and B then
the n time parallel type inner product of A n
× B and C has the following operation
properties
10 Variational Problems of Functionals with Vector, Tensor …
Taking the variation of the tensor A, we give
δ A n
· C
T
= C n
· δ A
T
= δ A
T n
· C = C
T n
· δ A
(10.1.5)
Let C =
∂ F
∂ A
, the variational expression of transposed tensor can be obtained
δ A n
·
∂ F
∂ A
T
=
∂ F
∂ A
n
· δ A
T
= δ A
T n
·
∂ F
∂ A
=
∂ F
∂ A
T
n
· δ A
(10.1.6)
Let A be a tensor of order m, A = a 1 a 2 · · · a m , B is a tensor of order n, B =
b 1 b 2 · · · b n , C is a tensor of order m + n, C = c 1 c 2 · · · c m+n , then the m + n time
parallel type inner product of the juxtaposition tensor AB of A and B and C has the
following operation properties
AB m + n
· C = a 1 a 2 · · · a m b 1 b 2 · · · b n m + n
· c 1 c 2 · · · c m+n = A m
· C n
· B
= C m + n
· AB = C T m + n
· B T A T = B T A T m + n
· C T
= A m
· (C n
· B) = (C n
· B) m
· A = A T m
· (B T n
· C T ) = (B T n
· C T ) m
· A T
= B n
· ( A m
· C) = ( A m
· C) n
· B = B T n
· (C T m
· A T ) = (C T m
· A T ) n
· B T
(10.1.7)
Let A be a tensor of order m, A = a 1 a 2 · · · a m , B is a tensor of order n, B =
b 1 b 2 · · · b n , C is a tensor of order m + n, C = c 1 c 2 · · · c m+n , then the m + n time
series type inner product of the juxtaposition tensor AB of A and B and C has the
following operation properties
AB m + n
· C = a 1 a 2 · · · a m b 1 b 2 · · · b n m + n
· c 1 c 2 · · · c m+n = B n
· C m
· A
= C m + n
· AB = C T m + n
· B T A T = B T A T m + n
· C T
= A m
· (B n
· C) = (B n
· C) m
· A = A T m
· (C T n
· B T ) = (C T n
· B T ) m
· A T
= B n
· (C m
· A) = (C m
· A) n
· B = B T n
· ( A T m
· C T ) = ( A T m
· C T ) n
· B T
(10.1.8)
Let A, B and C be all the tensors of order n, A = a 1 a 2 · · · a n , B = b 1 b 2 · · · b n ,
C = c 1 c 2 · · · c n , then the n time parallel type cross product A n
× B of A and B then
the n time parallel type inner product of A n
× B and C has the following operation
properties