32
1 Preliminaries
∇ × ∇ × a = ∇(∇ · a) − a
(1.3.100)
∇ × (a × b) = (b · ∇)a + (∇ · b)a − (a · ∇)b − (∇ · a)b
(1.3.101)
where, the operator = ∇ ·∇ = ∇
2 . The operator acting on the vector function is
called the vector Laplace operator or vector Laplacian, it is called the Laplacian
or Laplace operator for short. It should be pointed out that although both the scalar
operator and vector operator can be expresed with , but they are essentially different
two kinds of the second order differential operators.
Let ∇ × a = b in Eq. (1.3.97), then there is ∇ · b = 0, this expression shows that in
the domain of the field without source b, a vector function a can certainly be found,
such that ∇ × a = b holds. The vector function a is called the vector potential of
the field without source b. The vector potential of the field without source is not the
only one. If a is a vector potential of b, the vectot sum a
∗
= ∇ϕ + a of the gradient
of arbitrary scalar function ϕ that has a continuous second order partial derivative
and a is also the vector potential of b. The proof is as follows
∇ × a
∗
= ∇ × (∇ϕ + a) = ∇ × ∇ϕ + ∇ × a = ∇ × a = b
Quod erat demonstrandum.
In the vector field a, if both ∇ · a = 0 and ∇ × a = 0 hold, then a is called a
harmonic field. Or, the harmonic field refers to the both source-free and irrotational
vector field.
In a harmonic field, owing to ∇ × a = 0, there must be the potential function ϕ,
such that a = −∇ϕ, moreover owing to ∇ · a = 0, there is
∇ · a = ∇ · (−∇ϕ) = −ϕ = −
∂
2
ϕ
∂ x 2 +
∂
2
ϕ
∂ y 2 +
∂
2
ϕ
∂z 2
= 0
(1.3.102)
The above expression can be written as
ϕ =
∂
2
ϕ
∂ x 2 +
∂
2
ϕ
∂ y 2 +
∂
2
ϕ
∂z 2 = 0
(1.3.103)
This equation is a second order partial differential equation, it is called the threedimensional Laplace(’s) equation, it is called the Laplace(’s) equation for short,
ϕ is called the harmonic quantity. The potential function ϕ that satisfies the
Laplace equation and has the second order continuous partial derivative is called a
harmonic function.
Example 1.3.5 Verify ∇ × (ϕa) = ϕ∇ × a + ∇ϕ × a.
Proof According to the derivative rule of the two function multiplication and the
differential property of the operator ∇, there is
1 Preliminaries
∇ × ∇ × a = ∇(∇ · a) − a
(1.3.100)
∇ × (a × b) = (b · ∇)a + (∇ · b)a − (a · ∇)b − (∇ · a)b
(1.3.101)
where, the operator = ∇ ·∇ = ∇
2 . The operator acting on the vector function is
called the vector Laplace operator or vector Laplacian, it is called the Laplacian
or Laplace operator for short. It should be pointed out that although both the scalar
operator and vector operator can be expresed with , but they are essentially different
two kinds of the second order differential operators.
Let ∇ × a = b in Eq. (1.3.97), then there is ∇ · b = 0, this expression shows that in
the domain of the field without source b, a vector function a can certainly be found,
such that ∇ × a = b holds. The vector function a is called the vector potential of
the field without source b. The vector potential of the field without source is not the
only one. If a is a vector potential of b, the vectot sum a
∗
= ∇ϕ + a of the gradient
of arbitrary scalar function ϕ that has a continuous second order partial derivative
and a is also the vector potential of b. The proof is as follows
∇ × a
∗
= ∇ × (∇ϕ + a) = ∇ × ∇ϕ + ∇ × a = ∇ × a = b
Quod erat demonstrandum.
In the vector field a, if both ∇ · a = 0 and ∇ × a = 0 hold, then a is called a
harmonic field. Or, the harmonic field refers to the both source-free and irrotational
vector field.
In a harmonic field, owing to ∇ × a = 0, there must be the potential function ϕ,
such that a = −∇ϕ, moreover owing to ∇ · a = 0, there is
∇ · a = ∇ · (−∇ϕ) = −ϕ = −
∂
2
ϕ
∂ x 2 +
∂
2
ϕ
∂ y 2 +
∂
2
ϕ
∂z 2
= 0
(1.3.102)
The above expression can be written as
ϕ =
∂
2
ϕ
∂ x 2 +
∂
2
ϕ
∂ y 2 +
∂
2
ϕ
∂z 2 = 0
(1.3.103)
This equation is a second order partial differential equation, it is called the threedimensional Laplace(’s) equation, it is called the Laplace(’s) equation for short,
ϕ is called the harmonic quantity. The potential function ϕ that satisfies the
Laplace equation and has the second order continuous partial derivative is called a
harmonic function.
Example 1.3.5 Verify ∇ × (ϕa) = ϕ∇ × a + ∇ϕ × a.
Proof According to the derivative rule of the two function multiplication and the
differential property of the operator ∇, there is