矢量运算和基础场论

矢量运算

• 点积 (scalar product) $\mathit{a}\cdot \mathit{b}=a_i{b}_i$

爱因斯坦求和约定

出现重复下标时默认对该下标遍历求和。

• 矢积 (vector product)

$$\mathit{a}\times \mathit{b}=\left|\begin{array}{ccc}{\hat{e}}_1& {\hat{e}}_2& {\hat{e}}_3\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|={\epsilon }_{ijk}{a}_i{b}_j{\hat{e}}_k$$

${\epsilon }_{ijk}$

${\epsilon }_{ijk}$ 称为列维-席维塔符号，其定义为

$${\epsilon }_{ijk}=\{\begin{array}{ll}+1& \text{若}ijk\text{为}123\text{的偶排列}\\ -1& \text{若}ijk\text{为}123\text{的奇排列}\\ 0& \text{其他}\end{array}$$

• 三矢积公式 (triple vector product)

$$(\mathit{a}\times \mathit{b})\times \mathit{c}=(\mathit{a}\cdot \mathit{c})\mathit{b}-(\mathit{b}\cdot \mathit{c})\mathit{a}$$$$\mathit{c}\times (\mathit{a}\times \mathit{b})=(\mathit{b}\cdot \mathit{c})\mathit{a}-(\mathit{a}\cdot \mathit{c})\mathit{b}$$

基础场论

• 梯度 (gradient)

$$\begin{array}{rl}\mathrm{\nabla }\psi & =\frac{\partial \psi }{\partial x}\hat{x}+\frac{\partial \psi }{\partial y}\hat{y}+\frac{\partial \psi }{\partial z}\hat{z}\\ & =\frac{\partial \psi }{\partial r}\hat{r}+\frac{1}{r}\frac{\partial \psi }{\partial \theta }\hat{\theta }+\frac{\partial \psi }{\partial z}\hat{z}\\ & =\frac{\partial \psi }{\partial r}\hat{r}+\frac{1}{r}\frac{\partial \psi }{\partial \theta }\hat{\theta }+\frac{1}{r\sin \theta }\frac{\partial \psi }{\partial \varphi }\hat{\varphi }\end{array}$$

• 散度 (divergence)

$$\begin{array}{rl}\mathrm{\nabla }\cdot \mathit{a}& =\frac{\partial a_x}{\partial x}+\frac{\partial a_y}{\partial y}+\frac{\partial a_z}{\partial z}\\ & =\frac{1}{r}\frac{\partial }{\partial r}(r{a}_r)+\frac{1}{r}\frac{\partial a_{\theta }}{\partial \theta }+\frac{\partial a_z}{\partial z}\\ & =\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{a}_r)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta a_{\theta })+\frac{1}{r\sin \theta }\frac{\partial a_{\varphi }}{\partial \varphi }\end{array}$$

• 旋度 (curl)

$$\begin{array}{rl}\mathrm{\nabla }\times \mathit{a}& =\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ {\displaystyle \frac{\partial }{\partial x}}& {\displaystyle \frac{\partial }{\partial y}}& {\displaystyle \frac{\partial }{\partial z}}\\ a_x& a_y& a_z\end{array}\right|\\ & =\frac{1}{r}\left|\begin{array}{ccc}\hat{r}& r\hat{\theta }& \hat{z}\\ {\displaystyle \frac{\partial }{\partial r}}& {\displaystyle \frac{\partial }{\partial \theta }}& {\displaystyle \frac{\partial }{\partial z}}\\ a_r& r{a}_{\theta }& r_z\end{array}\right|\\ & =\frac{1}{r^2\sin \theta }\left|\begin{array}{ccc}\hat{r}& r\hat{\theta }& r\sin \theta \hat{\varphi }\\ {\displaystyle \frac{\partial }{\partial r}}& {\displaystyle \frac{\partial }{\partial \theta }}& {\displaystyle \frac{\partial }{\partial \varphi }}\\ a_r& r{a}_{\theta }& r\sin \theta a_{\varphi }\end{array}\right|\end{array}$$

• Laplace 算符 ( ${\mathrm{\nabla }}^2=$ div grad )

$$\begin{array}{rl}{\mathrm{\nabla }}^2\psi & =\frac{{\partial }^2\psi }{\partial x^2}\hat{x}+\frac{{\partial }^2\psi }{\partial y^2}\hat{y}+\frac{{\partial }^2\psi }{\partial z^2}\hat{z}\\ & =\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \psi }{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2\psi }{\partial {\theta }^2}+\frac{{\partial }^2\psi }{\partial z^2}\\ & =\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \psi }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial \psi }{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2\psi }{\partial {\theta }^2}\end{array}$$

$\mathrm{\nabla }$ 的运算法则

关于向量微分算符 $\mathrm{\nabla }$ 的相关运算法则如下：

其中 $\varphi ,\psi$ 代表标量场，$\mathit{f},\mathit{g}$ 代表矢量场。

$$\mathrm{\nabla }(\varphi \psi )=\varphi \mathrm{\nabla }\psi +\psi \mathrm{\nabla }\varphi$$$$\mathrm{\nabla }\cdot (\varphi \mathit{f})=(\mathrm{\nabla }\varphi )\cdot \mathit{f}+\varphi \mathrm{\nabla }\cdot \mathit{f}$$$$\mathrm{\nabla }\times (\varphi \mathit{f})=(\mathrm{\nabla }\varphi )\times \mathit{f}+\varphi \mathrm{\nabla }\times \mathit{f}$$$$\mathrm{\nabla }\cdot (\mathit{f}\times \mathit{g})=(\mathrm{\nabla }\times \mathit{f})\cdot \mathit{g}-\mathit{f}\cdot (\mathrm{\nabla }\times \mathit{g})$$$$\mathrm{\nabla }\times (\mathit{f}\times \mathit{g})=(\mathit{g}\cdot \mathrm{\nabla })\mathit{f}+(\mathrm{\nabla }\cdot \mathit{g})\mathit{f}-(\mathit{f}\cdot \mathrm{\nabla })\mathit{g}-(\mathrm{\nabla }\cdot \mathit{f})\mathit{g}$$$$\mathrm{\nabla }(\mathit{f}\cdot \mathit{g})=\mathit{f}\times (\mathrm{\nabla }\times \mathit{g})+(\mathit{f}\cdot \mathrm{\nabla })\mathit{g}+\mathit{g}\times (\mathrm{\nabla }\times \mathit{f})+(\mathit{g}\cdot \mathrm{\nabla })\mathit{f}$$

• 其他性质

$$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathit{f})=0$$$$\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathit{f})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathit{f})-{\mathrm{\nabla }}^2\mathit{f}$$

流体力学应用

在流体力学中常用的公式 $(\mathit{u}\cdot \mathrm{\nabla })Q$ 和 $(\mathit{u}\cdot \mathrm{\nabla })\mathit{Q}$ 在三种坐标系中的表达式：

• 直角坐标系 (Cartesian coordinates)
• 球坐标 (Spherical polar coordinates)
• 柱坐标 (Cylindrical polar coordinates)

$$\begin{array}{rl}(\mathit{u}\cdot \mathrm{\nabla })Q& =u_x\frac{\partial Q}{\partial x}+u_y\frac{\partial Q}{\partial y}+u_z\frac{\partial Q}{\partial z}\\ & =u_r\frac{\partial Q}{\partial r}+\frac{u_{\theta }}{r}\frac{\partial Q}{\partial \theta }+\frac{u_{\varphi }}{r\sin \theta }\frac{\partial Q}{\partial \varphi }\\ & =u_R\frac{\partial Q}{\partial R}+\frac{u_{\varphi }}{R}\frac{\partial Q}{\partial \varphi }+u_z\frac{\partial Q}{\partial z}\end{array}$$$$\begin{array}{rl}(\mathit{u}\cdot \mathrm{\nabla })\mathit{Q}& ={\left[\begin{array}{c}{u}_x{\displaystyle \frac{\partial Q_x}{\partial x}}+u_y{\displaystyle \frac{\partial Q_x}{\partial y}}+u_z{\displaystyle \frac{\partial Q_x}{\partial z}}\\ u_x{\displaystyle \frac{\partial Q_y}{\partial x}}+u_y{\displaystyle \frac{\partial Q_y}{\partial y}}+u_z{\displaystyle \frac{\partial Q_y}{\partial z}}\\ u_x{\displaystyle \frac{\partial Q_z}{\partial x}}+u_y{\displaystyle \frac{\partial Q_z}{\partial y}}+u_z{\displaystyle \frac{\partial Q_z}{\partial z}}\end{array}\right]}_{x,y,z}\\ & ={\left[\begin{array}{c}{u}_r{\displaystyle \frac{\partial Q_r}{\partial r}}+{\displaystyle \frac{u_{\theta }}{r}}{\displaystyle \frac{\partial Q_r}{\partial \theta }}+{\displaystyle \frac{u_{\varphi }}{r\sin \theta }}{\displaystyle \frac{\partial Q_r}{\partial \varphi }}-{\displaystyle \frac{u_{\theta }{Q}_{\theta }+u_{\varphi }{Q}_{\varphi }}{r}}\\ u_r{\displaystyle \frac{\partial Q_{\theta }}{\partial r}}+{\displaystyle \frac{u_{\theta }}{r}}{\displaystyle \frac{\partial Q_{\theta }}{\partial \theta }}+{\displaystyle \frac{u_{\varphi }}{r\sin \theta }}{\displaystyle \frac{\partial Q_{\theta }}{\partial \varphi }}+{\displaystyle \frac{u_{\theta }{Q}_r}{r}}-{\displaystyle \frac{u_{\varphi }{Q}_{\varphi }\cot \theta }{r}}\\ u_r{\displaystyle \frac{\partial Q_{\varphi }}{\partial r}}+{\displaystyle \frac{u_{\theta }}{r}}{\displaystyle \frac{\partial Q_{\varphi }}{\partial \theta }}+{\displaystyle \frac{u_{\varphi }}{r\sin \theta }}{\displaystyle \frac{\partial Q_{\varphi }}{\partial \varphi }}+{\displaystyle \frac{u_{\varphi }{Q}_r}{r}}+{\displaystyle \frac{u_{\varphi }{Q}_{\theta }\cot \theta }{r}}\end{array}\right]}_{r,\theta ,\varphi }\\ & ={\left[\begin{array}{c}{u}_R{\displaystyle \frac{\partial Q_R}{\partial R}}+{\displaystyle \frac{u_{\varphi }}{R}}{\displaystyle \frac{\partial Q_R}{\partial \varphi }}+u_z{\displaystyle \frac{\partial Q_R}{\partial z}}-{\displaystyle \frac{u_{\varphi }{Q}_{\varphi }}{R}}\\ u_R{\displaystyle \frac{\partial Q_{\varphi }}{\partial R}}+{\displaystyle \frac{u_{\varphi }}{R}}{\displaystyle \frac{\partial Q_{\varphi }}{\partial \varphi }}+u_z{\displaystyle \frac{\partial Q_{\varphi }}{\partial z}}+{\displaystyle \frac{u_{\varphi }{Q}_R}{R}}\\ u_R{\displaystyle \frac{\partial Q_z}{\partial R}}+{\displaystyle \frac{u_{\varphi }}{R}}{\displaystyle \frac{\partial Q_z}{\partial \varphi }}+u_z{\displaystyle \frac{\partial Q_z}{\partial z}}\end{array}\right]}_{R,\varphi ,z}\end{array}$$

some useful results

• 球坐标下 Laplace 算符的 $r$ 分量

  $$\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^2\frac{\mathrm{d}R(r)}{\mathrm{d}r}\right)=\frac{1}{r}\frac{{\mathrm{d}}^2(rR(r))}{\mathrm{d}{r}^2}$$

  proof:

  $$\begin{array}{rl}\frac{{\mathrm{d}}^2(rR)}{\mathrm{d}{r}^2}& =\frac{\mathrm{d}}{\mathrm{d}r}(R+r\frac{\mathrm{d}R}{\mathrm{d}r})\\ & =\frac{\mathrm{d}R}{\mathrm{d}r}+\frac{\mathrm{d}R}{\mathrm{d}r}+r\frac{{\mathrm{d}}^{2}R}{\mathrm{d}{r}^2}\\ & =2\frac{\mathrm{d}R}{\mathrm{d}r}+r\frac{{\mathrm{d}}^{2}R}{\mathrm{d}{r}^2}\\ & =\frac{1}{r}[2r\frac{\mathrm{d}R}{\mathrm{d}r}+r^2\frac{{\mathrm{d}}^{2}R}{\mathrm{d}{r}^2}]\\ & =\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^2\frac{\mathrm{d}R(r)}{\mathrm{d}r}\right)\end{array}$$

• 经典势场及其力场

  $$\mathrm{\nabla }\left(\frac{1}{r}\right)=-\frac{\overrightarrow{r}}{r^3}$$
  $${\mathrm{\nabla }}_i\left(\frac{1}{r_{ij}}\right)={\mathrm{\nabla }}_i\left(\frac{1}{r_j-r_i}\right)=\frac{1}{(r_j-r_i{)}^2}\hat{r}=\frac{{\overrightarrow{r}}_{ij}}{r_{ij}^3}$$
  $$\mathrm{\nabla }\left(\frac{\overrightarrow{r}\cdot \overrightarrow{a}}{a^3}\right)=\frac{\overrightarrow{a}}{a^3}$$