# 矢量场的散度

$\newcommand{\dif}{\mathop{}\!\mathrm{d}} \newcommand{\p}{\partial}$

# 矢量场

$\vec{F} \times \dif l = \begin{vmatrix} \vec{a}_x & \vec{a}_y & \vec{a}_z \\ F_x & F_y & F_z \\ \dif x & \dif y & \dif z \end{vmatrix} = 0$

# 矢量的通量（标量）

$\dif \Phi = \vec{A} \cdot \dif \vec{S} = A \cos \theta \dif S$

$\Phi = \int_S \vec{A} \cdot \dif \vec{S} = \int_S \vec{A} \cdot \vec{n} \dif S = \int_S A\cos \theta \dif S$

1. $\Phi>0$，正源，$S$ 内有发出通量线的源
2. $\Phi<0$，负源，$S$ 内有吸收通量的汇
3. $\Phi=0$，无源，$S$ 内无源无汇或源汇相消

# 散度

$\lim_{\Delta V \rightarrow 0} \frac{\oint_S \vec{A} \cdot \dif \vec{S}}{\Delta V}$

计算

直角坐标系

\begin{align} \mathrm{div} \vec{A} &= \frac{\p A_x}{\p x} + \frac{\p A_y}{\p y} + \frac{\p A_z}{\p z}\\ &= \nabla \cdot \vec{A} \end{align}

柱坐标系

\begin{align} \mathrm{div} \vec{A} &= \nabla \cdot \vec{A}\\ &= \frac{1}{r}\frac{\p}{\p r}(r A_r) + \frac{1}{r}\frac{\p A_\varphi}{\p \varphi} + \frac{\p A_z}{\p z} \end{align}

球坐标系

\begin{align} \mathrm{div} \vec{A} &= \nabla \cdot \vec{A}\\ &= \frac{1}{r^2}\frac{\p}{\p r}(r^2 A_r) + \frac{1}{r \sin \theta}\frac{\p}{\p \theta}(\sin\theta A_\theta) + \frac{1}{r\sin\theta}\frac{\p A_\varphi}{\p \varphi} \end{align}

\begin{align} \nabla\cdot \vec{F}&=\frac{1}{h_{1}h_{2}h_{3}} \left(\frac{\p }{\p u_{1}} (h_{2}h_{3}F_{1})+\frac{\p }{\p u_{2}} (h_{3}h_{1}F_{2})+\frac{\p }{\p u_{3}} (h_{1}h_{2}F_{3})\right)\\ &=\frac{1}{h_{1}h_{2}h_{3}} \sum_{j=1}^{3} \frac{\p }{\p u_{i}} \left(F_{j}\frac{h_{1}h_{2}h_{3}}{h_{j}}\right) \end{align}

1. 对柱坐标系：$h_1=1, h_2=r, h_3=1$
2. 对球坐标系：$h_1=1, h_2=r, h_3=r\sin\theta$

$\nabla\cdot\vec{F}=\frac{1}{r} \cdot\Big( \frac{\p rF_r}{\p r}+\frac{\p F_\varphi}{\p \varphi}+\frac{\p rF_z}{\p z}\Big)$

\begin{align} \nabla\cdot\vec{F}&=\frac{1}{r^2\sin\theta} \cdot\Big( \frac{\p r^2\sin\theta F_r}{\p r}+\frac{\p r\sin\theta F_\theta}{\p \varphi}+\frac{\p rF_\varphi}{\p z}\Big)\\ &=\frac{1}{r^2\sin\theta} \cdot\Big( \sin\theta \frac{\p r^2F_r}{\p r}+r\sin\theta \frac{\p F_\theta}{\p \varphi}+r\frac{\p F_\varphi}{\p z}\Big) \end{align}

1. $\nabla\cdot c\vec{A}=c\nabla\vec{A}$（c是常数）
2. $\nabla\cdot(\vec{A}\pm\vec{B})=\nabla\vec{A}\pm\nabla\vec{B}$
3. $\nabla u\vec{A}=u\nabla\cdot\vec{A}+A\nabla u$

# 高斯散度定理

$\int_V \nabla\cdot \vec{A} \dif V = \oint_S \vec{A}\cdot\dif \vec{S}$

# 拉普拉斯算子（补充）

$\Delta = \nabla^2 = \nabla\cdot\nabla f$

直角坐标系

\begin{align} \nabla^2 f = \frac{\p^2 f}{\p x^2}+\frac{\p^2 f}{\p y^2}+\frac{\p^2 f}{\p z^2} \end{align}

柱坐标系

\begin{align} \nabla^2 f = \frac{1}{r}\frac{\p}{\p r} \left(r \frac{\p f}{\p r}\right)+\frac{1}{r^2} \frac{\p^2 f}{\p \theta^2} + \frac{\p^2 f}{\p z^2} \end{align}

球坐标系

\begin{align} \nabla^2 f = \frac{1}{r^2}\frac{\p}{\p r} \left(r^2\frac{\p f}{\p r}\right)+\frac{1}{r^2\sin\theta}\frac{\p}{\p \theta} \left(\sin\theta\frac{\p f}{\p \theta}\right)+\frac{1}{r^2\sin^2\theta} \frac{\p^2 f}{\p \varphi^2} \end{align}

$\nabla^2 = \frac{1}{h_1h_2h_3} \sum_{i=1}^3 \frac{\p}{\p u_i} \left( \frac{h_1h_2h_3}{h_i} \frac{\p}{h_i \p u_i} \right)$

$\nabla^2 \vec{E} = (\nabla^2 E_1) \hat{a}_1+(\nabla^2 E_2) \hat{a}_2+(\nabla^2 E_3) \hat{a}_3$

• 分配律：$\nabla^2(u+v)=\nabla^2 u + \nabla^2 v$
• $\nabla^2(uv)=u\nabla^2 v+2\nabla u \cdot \nabla v +v\nabla^2 u$
• $\nabla \times (\nabla\times \vec{F}) = \nabla(\nabla\cdot\vec{F}) - \nabla^2\vec{F}$