基础操作
- 加法
$$ \begin{aligned} \mathbf{A} + \mathbf{B} &= \mathbf{B} + \mathbf{A}\\ (\mathbf{A} + \mathbf{B}) + \mathbf{C} &= \mathbf{A} + (\mathbf{B} + \mathbf{C})\\ \mathbf{A} -\mathbf{B} &= \mathbf{A} +(-\mathbf{B}) \end{aligned} $$
- 标乘(线性性)
$$ a(\mathbf{A} + \mathbf{B}) = a\mathbf{A} + a\mathbf{B} $$
- 点积
$$ \begin{aligned} \mathbf{A}\cdot\mathbf{B} &= AB\cos{\theta}\\ \mathbf{A}\cdot(\mathbf{B} + \mathbf{C}) &= \mathbf{A}\cdot\mathbf{B} + \mathbf{A}\cdot\mathbf{C} \end{aligned} $$
- 叉积
$$ \begin{aligned} \mathbf{A}\times\mathbf{B} & = AB\sin{\theta}\hat{\mathbf{n}}\\ \mathbf{A}\times(\mathbf{B} + \mathbf{C}) & = \mathbf{A}\times\mathbf{B} + \mathbf{A}\times\mathbf{C}\\ \mathbf{A}\times\mathbf{B} &= -\mathbf{B}\times\mathbf{A} \end{aligned} $$
代数方法:
$$ \begin{aligned} \mathbf{A}\times\mathbf{B} &= (A_{x}\hat{x} + A_{y}\hat{y} + A_{z}\hat{z})\times(B_{x}\hat{x} + B_{y}\hat{y} + B_{z}\hat{z})\\ &= (A_{y}B_{z} - A_{z}B_{y})\hat{x} + (A_{z}B_{x} - A_{x}B_{z})\hat{y} + (A_{x}B_{y} - A_{y}B_{x})\hat{z}\\ & =\left|\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z}\\ A_{x} & A_{y} & A_{z}\\ B_{x} & B_{y} & B_{z} \end{array}\right| \end{aligned} $$
- 三重积
$$ \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = \mathbf{B}\cdot(\mathbf{C}\times\mathbf{A}) = \mathbf{C}\cdot(\mathbf{A}\times\mathbf{B})\\ \mathbf{A}\cdot(\mathbf{C}\times\mathbf{B}) = \mathbf{B}\cdot(\mathbf{A}\times\mathbf{C}) = \mathbf{C}\cdot(\mathbf{B}\times\mathbf{A}) $$
行列式形式:
$$ \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) =\left|\begin{array}{ccc} A_{x} & A_{y} & A_{z}\\ B_{x} & B_{y} & B_{z}\\ C_{x} & C_{y} & C_{z} \end{array}\right| $$
所以可交换性:
$$ \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = (\mathbf{A}\times\mathbf{B})\cdot\mathbf{C} $$
BAC-CAB 规则:
$$ \mathbf{A}\times(\mathbf{B}\times\mathbf{C}) = \mathbf{B}(\mathbf{A}\cdot\mathbf{C}) - \mathbf{C}(\mathbf{A}\cdot\mathbf{B}) $$
矢量三重积不具有交换性: $$ (\mathbf{A}\times\mathbf{B})\times\mathbf{C} = -\mathbf{C}\times(\mathbf{A}\times\mathbf{B}) = -\mathbf{A}(\mathbf{B}\cdot\mathbf{C}) + \mathbf{B}(\mathbf{A}\cdot\mathbf{C}) $$
间隔矢量:
$$ \mathfrak{s} = \mathbf{r} - \mathbf{r}’ = (x - x’)\hat{x} + (y - y’)\hat{y} + (z - z’)\hat{z}\\ \mathfrak{s} = \sqrt{(x - x’)^{2} + (y - y’)^{2} + (z - z’)^{2}}\\ \hat{\mathbf{\mathfrak{s}}} = \frac{(x - x’)\hat{x} + (y - y’)\hat{y} + (z - z’)\hat{z}}{\sqrt{(x - x’)^{2} + (y - y’)^{2} + (z - z’)^{2}}} $$
$\mathbf{r}$ 指向场点, $\mathbf{r}’$ 指向源点.
微分算符
$\nabla$ 算子
$$ \begin{aligned} \mathrm{d}f &= \left(\frac{\mathrm{d}f}{\mathrm{d}x}\right)\mathrm{d}x\\ \mathrm{d}T &= \left(\frac{\partial T}{\partial x}\right)\mathrm{d}x + \left(\frac{\partial T}{\partial y}\right)\mathrm{d}y + \left(\frac{\partial T}{\partial z}\right)\mathrm{d}z\\ &=\left(\frac{\partial T}{\partial x} + \frac{\partial T}{\partial y} + \frac{\partial T}{\partial z}\right)\cdot\left(\mathrm{d}x\hat{x} + \mathrm{d}y\hat{y} + \mathrm{d}z\hat{z}\right)\\ & = (\nabla T)\cdot(\mathrm{d}\mathbf{l}) \end{aligned} $$
$$ \nabla T = \hat{x}\frac{\partial T}{\partial x} + \hat{y}\frac{\partial T}{\partial y} + \hat{z}\frac{\partial T}{\partial z} $$
散度
$$ \begin{aligned} \nabla\cdot\mathbf{v} &= \left(\hat{x}\frac{\partial}{\partial x} + \hat{y}\frac{\partial}{\partial y} + \hat{z}\frac{\partial}{\partial z}\right)\cdot\left(v_{x}\hat{x} + v_{y}\hat{y} + v_{z}\hat{z}\right)\\ &=\frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z} \end{aligned} $$
旋度
$$ \begin{aligned} \nabla\times\mathbf{v} = \left|\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ v_{x} & v_{y} & v_{z} \end{array}\right| = \hat{x}\left(\frac{\partial v_{z}}{\partial y} - \frac{\partial v_{y}}{\partial z}\right) + \hat{y}\left(\frac{\partial v_{x}}{\partial z} - \frac{\partial v_{z}}{\partial x}\right) + \hat{z}\left(\frac{\partial v_{y}}{\partial x} - \frac{\partial v_{x}}{\partial y}\right) \end{aligned} $$
积规则
$$ \begin{aligned} \nabla(fg) &= f\nabla g + g\nabla f\\ \nabla(\mathbf{A}\cdot\mathbf{B}) &= \mathbf{A}\times(\nabla\times\mathbf{B}) + \mathbf{B}\times(\nabla\times\mathbf{A}) + (\mathbf{A}\cdot\nabla)\mathbf{B} + (\mathbf{B}\cdot\nabla)\mathbf{A}\\ \nabla\cdot(f\mathbf{A}) &= f(\nabla\cdot\mathbf{A}) + \mathbf{A}\cdot(\nabla f)\\ \nabla\cdot(\mathbf{A}\times\mathbf{B}) & = \mathbf{B}\cdot(\nabla\times\mathbf{A}) - \mathbf{A}\cdot(\nabla\times\mathbf{B})\\ \nabla\times(f\mathbf{A}) &= f(\nabla\times\mathbf{A}) - \mathbf{A}\times(\nabla f)\\ \nabla\times(\mathbf{A}\times\mathbf{B}) &= (\mathbf{B}\cdot\nabla)\mathbf{A} - (\nabla\cdot\nabla)\mathbf{B} + \mathbf{A}(\nabla\cdot\mathbf{B}) - \mathbf{B}(\nabla\cdot\mathbf{A}) \end{aligned} $$
二阶微分
- 梯度的散度(Laplasin):
$$ \begin{aligned} \nabla\cdot(\nabla T) &= \left(\hat{x}\frac{\partial}{\partial x} + \hat{y}\frac{\partial}{\partial y} + \hat{z}\frac{\partial}{\partial z}\right)\cdot\left(\frac{\partial T}{\partial x}\hat{x} + \frac{\partial T}{\partial y}\hat{y} + \frac{\partial T}{\partial z}\hat{z}\right)\\ &= \frac{\partial ^{2}T}{\partial x^{2}} + \frac{\partial^{2}T}{\partial y^{2}} + \frac{\partial^{2}T}{\partial z^{2}} = \nabla^{2} T\\ \nabla^{2}\mathbf{v} &= (\nabla^{2} v_{x})\hat{x} + (\nabla^{2}v_{y})\hat{y} + (\nabla^{2} v_{z})\hat{z} \end{aligned} $$
- 梯度的旋度
$$ \nabla\times(\nabla T) = 0 $$
- 旋度的散度
$$ \nabla\cdot(\nabla\times\mathbf{v}) = 0 $$
- 旋度的旋度
$$ \nabla\times(\nabla\times\mathbf{v}) = \nabla(\nabla\cdot\mathbf{v}) - \nabla^{2}\mathbf{v} $$
高斯定理(格林定理, 散度定理)(Gauss Theorem)
$$ \int_{\mathcal{V}}(\nabla\cdot\mathbf{v})\mathrm{d}\tau = \oint_{\mathcal{S}}\mathbf{v}\cdot\mathrm{d}\mathbf{a} $$
斯托克斯定理(Stokes Theorem)
$$ \int_{\mathcal{S}}(\nabla\times\mathbf{v})\cdot\mathrm{d}\mathbf{a} = \oint_{\mathcal{P}}\mathbf{v}\cdot\mathrm{d}\mathbf{l} $$
分部积分
$$ \nabla\cdot(f\mathbf{A}) = f(\nabla\cdot\mathbf{A}) + \mathbf{A}\cdot(\nabla f)\\ \Downarrow\\ \int\nabla\cdot(f\mathbf{A})\mathrm{d}\tau = \int f(\nabla\cdot\mathbf{A})\mathrm{d}\tau + \int\mathbf{A}\cdot(\nabla f)\mathrm{d}\tau = \oint f\mathbf{A}\cdot\mathrm{d}\mathbf{a} $$
坐标系
球坐标系
$$ \begin{cases} x = r\sin{\theta}\cos{\phi}\\ y = r\sin{\theta}\sin{\phi}\\ z = r\cos{\theta} \end{cases}\\ \mathbf{A} = A_{r}\hat{r} + A_{\theta}\hat{\theta} + A_{\phi}\hat{\phi}\\ \begin{cases} \hat{r} = \sin{\theta}\cos{\phi}\hat{x} + \sin{\theta}\sin{\phi}\hat{y} + \cos{\theta}\hat{z}\\ \hat{\theta} = \cos{\theta}\cos{\phi}\hat{x} + \cos{\theta}\sin{\phi}\hat{y} - \sin{\theta}\hat{z}\\ \hat{\phi} = -\sin{\phi}\hat{x} + \cos{\phi}\hat{y} \end{cases} $$
无限小微移
$$ \mathrm{d}\mathbf{l} = \hat{r}\mathrm{d}r + \hat{\theta}r\mathrm{d}\theta + \hat{\phi}r\sin{\theta}\mathrm{d}\phi $$
无限小体积元
$$ \mathrm{d}\tau = \mathrm{d}l_{\theta}\mathrm{d}l_{\theta}\mathrm{d}l_{\phi} = r^{2}\sin\theta\mathrm{d}r\mathrm{d}\theta\mathrm{d}\phi $$
梯度:
$$ \nabla T = \frac{\partial T}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial T}{\partial\theta}\hat{\theta} + \frac{1}{r\sin{\theta}}\frac{\partial T}{\partial\phi}\hat{\phi} $$
散度:
$$ \nabla\cdot\mathbf{v} = \frac{1}{v^{2}}\frac{\partial}{\partial r}(v^{2}v_{r}) + \frac{1}{r\sin{\theta}}\frac{\partial}{\partial\theta}(\sin{\theta}v_{\theta}) + \frac{1}{r\sin{\theta}}\frac{\partial v_{\phi}}{\partial \phi} $$
旋度:
$$ \nabla\times\mathbf{v} = \frac{1}{r\sin{\theta}}\left[\frac{\partial}{\partial\theta}(\sin{\theta}v_{\phi}) - \frac{\partial v_{\theta}}{\partial\phi}\right]\hat{r} + \frac{1}{r}\left[\frac{1}{\sin{\theta}}\frac{\partial v_{r}}{\partial\phi} - \frac{\partial}{\partial r}(rv_{\phi})\right]\hat{\theta} + \frac{1}{r}\left[\frac{\partial}{\partial v}(rv_{\theta}) - \frac{\partial v_{r}}{\partial\theta}\right]\hat{\phi} $$
Laplasin 算子:
$$ \nabla^{2}T = \frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial T}{\partial r}\right) + \frac{1}{r^{2}\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial T}{\partial\theta}\right) + \frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}T}{\partial\phi^{2}} $$
柱坐标系
$$ \begin{cases} x = s\cos{\phi}\\ y = s\sin{\phi}\\ z = z \end{cases}\\ \Downarrow\\ \begin{cases} \hat{s} = \cos{\phi}\hat{x} + \sin{\phi}\hat{y}\\ \hat{\phi} = -\sin{\phi}\hat{x} + \cos{\phi}\hat{y}\\ \hat{z} = \hat{z} \end{cases} $$
无限小微移:
$$ \mathrm{d}\mathbf{l} = \hat{s}\mathrm{d}s + \hat{\phi}s\mathrm{d}\phi + \hat{z}\mathrm{d}z $$
无限小体积元:
$$ \mathrm{d}\tau = s\mathrm{d}s\mathrm{d}\phi\mathrm{d}z $$
梯度:
$$ \nabla T = \frac{\partial T}{\partial s}\hat{s} + \frac{1}{s}\frac{\partial T}{\partial \phi}\hat{\phi} + \frac{\partial T}{\partial z}\hat{z} $$
散度:
$$ \nabla\cdot\mathbf{v} = \frac{1}{s}\frac{\partial}{\partial s}(sv_{s}) + \frac{1}{s}\frac{\partial v_{\phi}}{\partial\phi} + \frac{\partial v_{z}}{\partial z} $$
旋度:
$$ \nabla\times\mathbf{v} = \left(\frac{1}{s}\frac{\partial v_{z}}{\partial\phi} - \frac{\partial v_{\phi}}{\partial z}\right)\hat{s} + \left(\frac{\partial v_{s}}{\partial z} - \frac{\partial v_{z}}{\partial s}\right)\hat{\phi} + \frac{1}{s}\left[\frac{\partial}{\partial s}(sv_{\phi}) - \frac{\partial v_{s}}{\partial\phi}\right]\hat{z} $$
Laplasin 算子:
$$ \nabla^{2}T = \frac{1}{s}\frac{\partial}{\partial}\left(s\frac{\partial T}{\partial s}\right) + \frac{1}{s^{2}}\frac{\partial^{2}T}{\partial\phi^{2}} + \frac{\partial^{2}T}{\partial z^{2}} $$
狄拉克 $\delta$ 函数
1D
$$ \delta(x) = \begin{cases} 0 & x\neq 0\\ \infty & x = 0 \end{cases} $$
满足
$$ \int_{-\infty}^{+\infty}\delta(x)\mathrm{d}x = 1\\ \Downarrow\\ \int_{-\infty}^{+\infty}f(x)\delta(x-a)\mathrm{d}x = f(a) $$
3D
$$ \delta^{3}(\mathbf{r}) = \delta(x)\delta(y)\delta(z) $$
满足 $$ \iiint\delta^{3}(\mathbf{r})\mathrm{d}\tau = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\delta(x)\delta(y)\delta(z)\mathrm{d}x\mathrm{d}y\mathrm{d}z = 1 $$
$$ \int_{\text{All Space}}f(\mathbf{r})\delta^{3}(\mathbf{r} - \mathbf{a})\mathrm{d}\tau = f(\mathbf{a}) $$
$$ \nabla\cdot\left(\frac{\hat{\mathfrak{s}}}{\mathfrak{s}^{2}}\right) = 4\pi\delta^{3}(\mathbf{\mathfrak{s}}) $$
$$ \nabla\left(\frac{1}{\mathfrak{s}}\right) = -\left(\frac{\hat{\mathfrak{s}}}{\mathfrak{s}^{2}}\right)\\ \nabla^{2}\left(\frac{1}{\mathfrak{s}}\right) = -4\pi\delta^{3}(\mathfrak{s}) $$
矢量场
$$ \nabla\times\mathbf{F} = 0 \Longleftrightarrow \mathbf{F} = -\nabla V $$
无旋场
$$ \nabla\times\mathbf{F} = 0\\ \Updownarrow\\ \forall a\rightarrow b,\quad\int_{a}^{b}\mathbf{F}\cdot\mathrm{d}\mathbf{l} = \textit{Const.}\\ \Updownarrow\\ \forall S(a\rightarrow b\rightarrow a),\quad\oint_{S}\mathbf{F}\cdot\mathrm{d}\mathbf{l} = 0\\ \Updownarrow\\ \mathbf{F} = -\nabla V $$
无散度
$$ \nabla\cdot\mathbf{F} = 0\\ \Updownarrow\\ \forall \partial S,\quad\mathbf{F}\cdot\mathrm{d}\mathbf{a} = \textit{Const.}\\ \Updownarrow\\ \forall \partial V,\quad\oint\mathbf{F}\cdot\mathrm{d}\mathbf{a} = 0\\ \Updownarrow\\ \mathbf{F} = \nabla\times\mathbf{A} $$
两者结合起来, 即有 $$ \mathbf{F} = -\nabla V + \nabla\times\mathbf{A} $$