Electromagnetic field learning notes - vector analysis and field theory foundation

Basis of vector analysis and field theory

---

---

1. Basis of vector analysis

1.1 Dot product of vectors

$$\mathbf{A}\ast \mathbf{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z=AB\cos \theta$$

1.2 The cross product of vectors

$$\mathbf{A}\times \mathbf{B}=\left|\begin{array}{ccc}{\mathbf{e}}_x& {\mathbf{e}}_y& {\mathbf{e}}_z\\ A_x& A_y& A_z\\ B_x& B_y& B_z\end{array}\right|=AB\sin \theta {\mathbf{e}}_n$$

${\mathbf{e}}_n$ Is and vector A,B Are vertical unit vectors , The three meet the right-hand spiral relationship .

2. Isolines and vector lines of the field

2.1 Basic concepts

Scalar and vector fields ：
$$\begin{gathered} u(M)=u(x,y,z) \\ \mathbf{A}(M)=\mathbf{A}(x,y,z)=A_x(x,y,z){\mathbf{e}}_x+A_x(x,y,z){\mathbf{e}}_x+A_x(x,y,z){\mathbf{e}}_x \end{gathered}$$
($\alpha ,\beta ,\gamma$ Respectively A The positive included angle with the three coordinate axes )

$A_x=A\cos \alpha A_y=A\cos \beta A_x=A\cos \gamma$

2.2 Scalar field isosurface

Electromagnetic field Potential field It's just one. Scalar fields . The equivalent surface composed of points with the same potential is the equipotential surface .
$$u(x,y,z)=C$$

2.3 Vector line of vector field

$$\mathrm{d}\mathbf{l}=\mathrm{d}x{\mathbf{e}}_x+\mathrm{d}y{\mathbf{e}}_y+\mathrm{d}z{\mathbf{e}}_z$$

Satisfy $\frac{dx}{A_x}=\frac{dy}{A_y}=\frac{dz}{A_z}$ The differential equation satisfied by the vector line , The solution is a family of vector lines .

3. Scalar field directional derivatives and gradients

Scalar fields ：$u(x,y,z)=C$

Directional derivative ： Scalar fields u stay $\mathbf{l}$ On the situation ：
$$\frac{\partial u}{\partial l}=\frac{\partial u}{\partial x}\cos \alpha +\frac{\partial u}{\partial y}\cos \beta +\frac{\partial u}{\partial z}\cos \gamma =(\frac{\partial u}{\partial x}{\mathbf{e}}_{\mathbf{x}}+\frac{\partial u}{\partial y}{\mathbf{e}}_{\mathbf{y}}+\frac{\partial u}{\partial z}{\mathbf{e}}_{\mathbf{z}})\ast (cos\alpha {\mathbf{e}}_{\mathbf{x}}+cos\beta {\mathbf{e}}_{\mathbf{y}}+cos\gamma {\mathbf{e}}_{\mathbf{z}})$$
​ $cos\alpha {\mathbf{e}}_{\mathbf{x}}+cos\beta {\mathbf{e}}_{\mathbf{y}}+cos\gamma {\mathbf{e}}_{\mathbf{z}}$ by ${\mathbf{e}}_{\mathbf{l}}$ by l Direction unit vector

gradient ：u The rate of change in all directions , That is, the fastest changing direction ,grad u = $\frac{\partial u}{\partial x}{\mathbf{e}}_{\mathbf{x}}+\frac{\partial u}{\partial y}{\mathbf{e}}_{\mathbf{y}}+\frac{\partial u}{\partial z}{\mathbf{e}}_{\mathbf{z}}$

Gradient along ${\mathbf{e}}_{\mathbf{n}}$:
$$\frac{\partial u}{\partial n}=|gradu|\ast {\mathbf{e}}_{\mathbf{n}}\ast {\mathbf{e}}_{\mathbf{n}}=|gradu|$$
Gradient along ${\mathbf{e}}_{\mathbf{l}}$:
$$\frac{\partial u}{\partial l}=|gradu|\ast {\mathbf{e}}_{\mathbf{n}}\ast {\mathbf{e}}_{\mathbf{l}}=|gradu|\cos \theta$$

4. Flux and divergence of vector field

4.1 Flux of vector field

vector $\vec{A}(M)$ The flux through the surface element is defined as ：
$$\begin{gathered} d\Phi =A_{n}dS=\vec{A}\cdot \overrightarrow{e_n}dS=\vec{A}d\vec{S} \\ \Phi ={\int }_S\vec{A}d\vec{S}\vec{S}\text{Is a directed surface S} \end{gathered}$$

4.2 Divergence of vector field

$$\begin{array}{rl}\mathrm{div}\vec{A}& =\underset{\Delta V\to 0}{\lim }\frac{{\oint }_s\vec{A}\cdot d\vec{S}}{\Delta V}\\ \iff \mathrm{div}\vec{A}& =\frac{\partial Ax}{\partial x}+\frac{\partial Ay}{\partial y}+\frac{\partial Az}{\partial z}\end{array}$$

4.3 Gauss divergence theorem

Flux emitted by any closed surface ：
$$\begin{array}{l}{\oint }_S\vec{A}\cdot d\vec{S}={\int }_V\mathrm{div}\vec{A}dV\\ {\oint }_S\left(A_{x}dydz+A_{y}dxdz+A_{z}dxdy\right)={\int }_V\left(\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\right)dxdydz\end{array}$$

1.5 Circulation and curl of vector field

1.5.1 Vector field circulation

$$\Gamma ={\oint }_l{A}_{t}dL={\oint }_{l}A\cos \theta dl=\oint \cdot \vec{A}\cdot d\vec{L}$$

1.5.2 Vector field curl

$$\begin{array}{rl}\mathrm{rot}\vec{A}=\vec{R}& \\ \Rightarrow & \mathrm{rot}\vec{A}=\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right){\vec{e}}_x+\left(\frac{\partial A_y}{\partial z}-\frac{\partial A_z}{\partial x}\right)\overrightarrow{e_y}+\left(\frac{\partial A_y}{\partial x}-\frac{\partial Ax}{\partial y}\right)\overrightarrow{e_z}\\ & =\left|\begin{array}{lll}\overrightarrow{e_x}& {\vec{e}}_y& {\vec{e}}_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ A_x& A_y& A_z\end{array}\right|\end{array}$$

1.5.3 Stokes The formula

Consider the circulation along any closed curve
$${\oint }_l\vec{A}\cdot d\vec{l}={\int }_S(\mathrm{rot}\vec{A})\cdot d\vec{S}$$

1.6 Common formula

hamilton operator $\nabla$ And Laplacian ${\nabla }^2$
$$\begin{gathered} \nabla =\frac{\partial }{\partial x}{\vec{e}}_x+\frac{\partial }{\partial y}{\vec{e}}_y+\frac{\partial }{\partial z}{\vec{e}}_z \\ \mathrm{grad}u=\nabla u \\ \mathrm{div}\vec{A}=\nabla \cdot \vec{A} \\ \mathrm{rot}\vec{A}=\nabla \times \vec{A} \\ {\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2} \end{gathered}$$