10.2 The Euler Equations of Functionals …
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∇ · a = ∂ i e i · a j e j = a i,i , a · ∇ = a j e j · ∂ i e i = a i,i
(10.2.5)
It follows from Eq. (10.2.5) that the left divergence of a vector equals the right
divergence of the vector.
For the parallel-type double inner product operation of the tensors of second order,
there exist the following identities
e i e s : e j e k = (e i · e j )(e s · e k ) = (e i · e j )(e k · e s ) = e i e k : e j e s
= (e j · e i )(e s · e k ) = e j e s : e i e k = (e j · e i )(e k · e s ) = e j e k : e i e s
(10.2.6)
Assume that the double inner product of two tensors of second order conforms
to the rule of the parallel-type operation of the tensors, take the divergence to T · a,
and note that T = T jk e j e k , that is to say that the sequence of base vector e j and e k
in the tensor must remain unchanged, making use of Eq. (10.2.6), there is
∇ · (T · a) = (T · a) · ∇ = ∂ i e i · (T jk e j e k · a s e s )
= (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 ) : (∂ i e i a s e s )
= (∇ · T ) · a + T : ∇a = a · (∇ · T ) + ∇a : T
(10.2.7)
Equation (10.2.1) is proved.
Taking the divergence to a · T , and using Eq. (10.2.6), 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 )
= (a s e s ∂ i e i ) : 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
(10.2.8)
Equation (10.2.2) is proved.
For the series-type double inner product operation of the tensors of second order,
there exist the following identities
e s e i : e j e k = (e i · e j )(e s · e k ) = (e i · e j )(e k · e s ) = e j e k : e s e i
= (e j · e i )(e s · e k ) = e j e s : e k e i = (e j · e i )(e k · e s ) = e k e j : e i e s
(10.2.9)
Assume that the double inner products of two tensors of second order conform
to the rule of the series-type operation of the tensors, take the divergence to T · a,
make use of Eq. (10.2.9), we have
∇ · (T · a) = (T · a) · ∇ = ∂ i e i · (T jk e j e k · a s e s )
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∇ · a = ∂ i e i · a j e j = a i,i , a · ∇ = a j e j · ∂ i e i = a i,i
(10.2.5)
It follows from Eq. (10.2.5) that the left divergence of a vector equals the right
divergence of the vector.
For the parallel-type double inner product operation of the tensors of second order,
there exist the following identities
e i e s : e j e k = (e i · e j )(e s · e k ) = (e i · e j )(e k · e s ) = e i e k : e j e s
= (e j · e i )(e s · e k ) = e j e s : e i e k = (e j · e i )(e k · e s ) = e j e k : e i e s
(10.2.6)
Assume that the double inner product of two tensors of second order conforms
to the rule of the parallel-type operation of the tensors, take the divergence to T · a,
and note that T = T jk e j e k , that is to say that the sequence of base vector e j and e k
in the tensor must remain unchanged, making use of Eq. (10.2.6), there is
∇ · (T · a) = (T · a) · ∇ = ∂ i e i · (T jk e j e k · a s e s )
= (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 ) : (∂ i e i a s e s )
= (∇ · T ) · a + T : ∇a = a · (∇ · T ) + ∇a : T
(10.2.7)
Equation (10.2.1) is proved.
Taking the divergence to a · T , and using Eq. (10.2.6), 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 )
= (a s e s ∂ i e i ) : 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
(10.2.8)
Equation (10.2.2) is proved.
For the series-type double inner product operation of the tensors of second order,
there exist the following identities
e s e i : e j e k = (e i · e j )(e s · e k ) = (e i · e j )(e k · e s ) = e j e k : e s e i
= (e j · e i )(e s · e k ) = e j e s : e k e i = (e j · e i )(e k · e s ) = e k e j : e i e s
(10.2.9)
Assume that the double inner products of two tensors of second order conform
to the rule of the series-type operation of the tensors, take the divergence to T · a,
make use of Eq. (10.2.9), we have
∇ · (T · a) = (T · a) · ∇ = ∂ i e i · (T jk e j e k · a s e s )