618
10 Variational Problems of Functionals with Vector, Tensor …
example, the elliptic equation problem in mathematical physics, the self-adaptive
mesh generation problem in numerical analysis, the fluid flow problem in fluid
mechanics, the expansive wave problem in gas dynamics, the heat conduction
problem in heat transfer, the superconductive problem in cryophysics, the electromagnetic field problem in electromagnetic field theory and electrodynamics, the wave
function problem in quantum mechanics, the liquid crystals phase transition problem
in condensed matter physics, the image processing problem in information science,
the sound transmission problem in acoustics, the vessels segmentation problem in
medical science, etc., all of these problems are the main contents researched in the
natural sciences, engineering, and technology, although they have different forms
of mathematical expression, the related functionals therein generally contain vector,
modulus of vector, and Hamiltonian operators. The variational problems of the functionals with vector, modulus of vector, and Hamiltonian operators namely those
with gradient, divergence, and rotation were investigated in this section, to derive the
corresponding Euler equations and natural boundary conditions, on the basis of these,
some famous equations will be presented, such as the Poisson equation, Helmholtz
equation, Maxwell equations, Schrödinger equation, Ginzburg-Landau equations and
image denoising equation, etc. In fact, as early as 1989, Chinese scholar professor
Su Jinghui had proposed the Euler equations and natural boundary conditions of the
functional with the Hamiltonian operators, but which did not cause enough attention.
Now four identities on the inner product operations of a vector and a tensor of
second order are proved.
For the double inner product parallel-type operation of the tensors of second order,
there are
∇ · (T · a) = (T · a) · ∇ = (∇ · T ) · a + T : ∇a = a · (∇ · T ) + ∇a : T
(10.2.1)
∇ · (a · T ) = (a · T ) · ∇ = a∇ : T + a · (T · ∇) = T : a∇ + (T · ∇) · a
(10.2.2)
For the double inner product series-type operation of the tensors of second order,
there are
∇ · (T · a) = (T · a) · ∇ = (∇ · T ) · a + T : a∇ = a · (∇ · T ) + a∇ : T
(10.2.3)
∇ · (a · T ) = (a · T ) · ∇ = ∇a : T + a · (T · ∇) = T : ∇a + (T · ∇) · a
(10.2.4)
where, T is a tensor of second order, a is a vector.
Proof Putting ∂ i =
∂
∂ x i
, and making use of the comma convention, Φ ,i =
∂Φ
∂ x i
= ∂ i Φ,
then the divergence of a vector a can be expressed as
10 Variational Problems of Functionals with Vector, Tensor …
example, the elliptic equation problem in mathematical physics, the self-adaptive
mesh generation problem in numerical analysis, the fluid flow problem in fluid
mechanics, the expansive wave problem in gas dynamics, the heat conduction
problem in heat transfer, the superconductive problem in cryophysics, the electromagnetic field problem in electromagnetic field theory and electrodynamics, the wave
function problem in quantum mechanics, the liquid crystals phase transition problem
in condensed matter physics, the image processing problem in information science,
the sound transmission problem in acoustics, the vessels segmentation problem in
medical science, etc., all of these problems are the main contents researched in the
natural sciences, engineering, and technology, although they have different forms
of mathematical expression, the related functionals therein generally contain vector,
modulus of vector, and Hamiltonian operators. The variational problems of the functionals with vector, modulus of vector, and Hamiltonian operators namely those
with gradient, divergence, and rotation were investigated in this section, to derive the
corresponding Euler equations and natural boundary conditions, on the basis of these,
some famous equations will be presented, such as the Poisson equation, Helmholtz
equation, Maxwell equations, Schrödinger equation, Ginzburg-Landau equations and
image denoising equation, etc. In fact, as early as 1989, Chinese scholar professor
Su Jinghui had proposed the Euler equations and natural boundary conditions of the
functional with the Hamiltonian operators, but which did not cause enough attention.
Now four identities on the inner product operations of a vector and a tensor of
second order are proved.
For the double inner product parallel-type operation of the tensors of second order,
there are
∇ · (T · a) = (T · a) · ∇ = (∇ · T ) · a + T : ∇a = a · (∇ · T ) + ∇a : T
(10.2.1)
∇ · (a · T ) = (a · T ) · ∇ = a∇ : T + a · (T · ∇) = T : a∇ + (T · ∇) · a
(10.2.2)
For the double inner product series-type operation of the tensors of second order,
there are
∇ · (T · a) = (T · a) · ∇ = (∇ · T ) · a + T : a∇ = a · (∇ · T ) + a∇ : T
(10.2.3)
∇ · (a · T ) = (a · T ) · ∇ = ∇a : T + a · (T · ∇) = T : ∇a + (T · ∇) · a
(10.2.4)
where, T is a tensor of second order, a is a vector.
Proof Putting ∂ i =
∂
∂ x i
, and making use of the comma convention, Φ ,i =
∂Φ
∂ x i
= ∂ i Φ,
then the divergence of a vector a can be expressed as