哈密顿算子与梯度、散度、旋度

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哈密顿算子与梯度、散度、旋度

引用

CSDN

https://blog.csdn.net/irober/article/details/106231611

哈密顿算子点乘叉乘

1、定义与性质

哈密顿算子：（数学符号：∇ \nabla∇（又称nabla，奈布拉算子）），读来作Hamilton。

向量微分算子：∇ = ∂ ∂ x i ⃗ + ∂ ∂ y j ⃗ + ∂ ∂ z k ⃗ \nabla=\frac{\partial }{\partial x}\vec{i}+\frac{\partial}{\partial y} \vec{j}+\frac{\partial }{\partial z}\vec{k}∇=∂x∂ i+∂y∂ j +∂z∂ k

性质：

矢量性
微分算子
只对算子∇ \nabla∇右边的量发生微分作用

麦克斯韦方程的微分形式：

∂ D x ∂ x + ∂ D y ∂ y + ∂ D z ∂ z = ρ \frac{\partial D_{x}}{\partial x}+\frac{\partial D_{y}}{\partial y}+\frac{\partial D_{z}}{\partial z}=\rho∂x∂Dx +∂y∂Dy +∂z∂Dz =ρ∂ B x ∂ x + ∂ B y ∂ y + ∂ B z ∂ z = 0 \frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}=0∂x∂Bx +∂y∂By +∂z∂Bz =0

∂ H z ∂ y − ∂ H y ∂ z = δ x + ∂ D x ∂ t \frac{\partial H_{z}}{\partial y} -\frac{\partial H_y}{\partial z} = \delta_{x}+\frac{\partial D_{x}}{\partial t}∂y∂Hz −∂z∂Hy =δx +∂t∂Dx ∂ H x ∂ z − ∂ H z ∂ x = δ y + ∂ D y ∂ t \frac{\partial H_{x}}{\partial z}-\frac{\partial H_{z}}{\partial x}=\delta_{y}+\frac{\partial D_{y}}{\partial t}∂z∂Hx −∂x∂Hz =δy +∂t∂Dy ∂ H y ∂ x − ∂ H x ∂ y = δ z + ∂ D z ∂ t \frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}=\delta_{z}+\frac{\partial D_{z}}{\partial t}∂x∂Hy −∂y∂Hx =δz +∂t∂Dz

∂ E z ∂ y − ∂ E y ∂ z = − ∂ B x ∂ t \frac{\partial E_{z}}{\partial y}-\frac{\partial E_{y}}{\partial z}=-\frac{\partial B_{x}}{\partial t}∂y∂Ez −∂z∂Ey =−∂t∂Bx

∂ E x ∂ z − ∂ E z ∂ x = − ∂ B y ∂ t \frac{\partial E_{x}}{\partial z}-\frac{\partial E_{z}}{\partial x}=-\frac{\partial B_{y}}{\partial t}∂z∂Ex −∂x∂Ez =−∂t∂By

∂ E y ∂ x − ∂ E x ∂ y = − ∂ B z ∂ t \frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}=-\frac{\partial B_{z}}{\partial t}∂x∂Ey −∂y∂Ex =−∂t∂Bz

引进哈密顿算子，上式简化为：

{ ∇ ⋅ D ⃗ = ρ ∇ ⋅ B ⃗ = 0 ∇ × H ⃗ = δ ⃗ + ∂ D ⃗ ∂ t ∇ × E ⃗ = − ∂ B ⃗ ∂ t \begin{cases} \nabla \cdot \vec{D}=\rho \ \nabla \cdot \vec{B}=0 \ \nabla \times \vec{H}=\vec{\delta}+\frac{\partial \vec{D}}{\partial t} \ \nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t} \end{cases}⎩⎪⎪⎪⎨⎪⎪⎪⎧ ∇⋅D=ρ∇⋅B=0∇×H=δ+∂t∂D ∇×E=−∂t∂B

2、标量场的梯度

笛卡尔坐标系下的梯度：

∇ P = ∂ P ∂ x i ⃗ + ∂ P ∂ y j ⃗ + ∂ P ∂ z k ⃗ = g r a d P \nabla{P}=\frac{\partial P}{\partial x} \vec{i}+\frac{\partial P}{\partial y} \vec{j}+\frac{\partial P}{\partial z} \vec{k}=gradP∇P=∂x∂P i+∂y∂P j +∂z∂P k=gradP

（结果为矢量）

3、矢量场的散度

∇ ∙ V ⃗ = ∂ V x ∂ x + ∂ V y ∂ y + ∂ V z ∂ z = d i v V ⃗ \nabla \bullet \vec{V}=\frac{\partial V_{x}}{\partial x}+\frac{\partial V_{y}}{\partial y}+\frac{\partial V_{z}}{\partial z}=div \vec{V}∇∙V=∂x∂Vx +∂y∂Vy +∂z∂Vz =divV

（结果为标量）

4、矢量场的旋度

笛卡尔坐标系下旋度定义：

∇ × V ⃗ = ∣ i ⃗ j ⃗ k ⃗ ∂ ∂ x ∂ ∂ y ∂ ∂ z V x V y V z ∣ = ( ∂ V z ∂ y − ∂ V y ∂ z ) i ⃗ + ( ∂ V x ∂ z − ∂ V z ∂ x ) j ⃗ + ( ∂ V y ∂ x − ∂ V x ∂ y ) k ⃗ = r o t V ⃗ \nabla \times \vec{V}=\left|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ V_{x} & V_{y} & V_{z} \end{array}\right| = \left(\frac{\partial V_z}{\partial y} -\frac{\partial V_{y}}{\partial z}\right) \vec{i}+\left(\frac{\partial V_{x}}{\partial z}-\frac{\partial V_{z}}{\partial x}\right) \vec{j}+\left(\frac{\partial V_{y}}{\partial x}-\frac{\partial V_{x}}{\partial y}\right) \vec{k} = rot \vec{V}∇×V=∣∣∣∣∣∣ i∂x∂ Vx j ∂y∂ Vy k∂z∂ Vz ∣∣∣∣∣∣ =(∂y∂Vz −∂z∂Vy )i+(∂z∂Vx −∂x∂Vz )j +(∂x∂Vy −∂y∂Vx )k=rotV

（结果为矢量）

5、哈密顿算子重要运算性质

∇ ⋅ ( A ⃗ × B ⃗ ) = B ⃗ ⋅ ∇ × A ⃗ − A ⃗ ⋅ ∇ × B ⃗ \nabla \cdot(\vec{A} \times \vec{B})=\vec{B} \cdot \nabla \times \vec{A}-\vec{A} \cdot \nabla \times \vec{B}∇⋅(A×B)=B⋅∇×A−A⋅∇×B

证明