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| 当初学多元微积分没学好，散度旋度啥的都忘光了。现在复习一下。 | ||
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| # 数学基础 | ||
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| ## 几种坐标系 | ||
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| 二维平面上的： | ||
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| - 极坐标变换：$x=r\cos\theta,y=r\sin\theta$ | ||
| - 雅可比行列式为 $r$。 | ||
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| 三维空间中的： | ||
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| - 球坐标变换：$x=r\sin\theta\cos\phi,y=r\sin\theta\sin\phi,z=r\cos\theta$ | ||
| - 两个角分别是与 $x$ 轴和 $z$ 轴的夹角。 | ||
| - 雅可比行列式为 $r^2\sin\theta$，与 $z$ 轴夹角。 | ||
| - 柱坐标变换：$x=r\cos\theta,y=r\sin\theta,z=z$ | ||
| - 雅可比行列式为 $r$。 | ||
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| 一些概念 | ||
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| - 立体角：以观测点为球心，构造一个单位球面；任意物体投影到该单位球面上的投影面积，即为该物体相对于该观测点的立体角。$\Omega = \frac{S}{r^2}$，其中 $S$ 为球面上的面积，$r$ 为球心到球面的距离。 | ||
| - 球面坐标系中，任意球面的极小面积为：$\mathrm{d}A = (r\sin\theta\,\mathrm{d}\varphi)(r \mathrm{d}\theta)=r^2(\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi)$，极小立体角为 $\mathrm{d}\Omega = \frac{\mathrm{d}A}{r^2} = \sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi$ | ||
| - 对极小立体角做曲面积分即可得立体角：$\Omega= \iint_S \mathrm{d}\Omega = \iint_S \sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi$ | ||
| - 定向曲面：$\Omega= \iint_S \frac{dA}{r^{2}} =\iint_S \frac { \vec{r} \cdot \textrm{d}\vec{S}}{\left| \vec{r} \right|\, r^2}= \iint_S \frac { \vec{r} \cdot \textrm{d}\vec{S}}{ r^3}$。 | ||
| - 计算例：顶角为 $2\theta$ 的圆锥的立体角为 $\int_0^{2\pi} \int_0^{\theta} \sin \theta' \ d \theta' \ d \phi = 2\pi\int_0^{\theta} \sin \theta' \ d \theta' = 2\pi\left[ -\cos \theta' \right]_0^{\theta} \ = 2\pi\left(1 -\cos \theta \right)$. | ||
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| ## 其他 | ||
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| ### 幂次前缀 | ||
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| | Prefix | Symbol | Factor | | ||
| | ------ | ------ | ------ | | ||
| | yotta | Y | $10^{24}$ | | ||
| | zetta | Z | $10^{21}$ | | ||
| | exa | E | $10^{18}$ | | ||
| | peta | P | $10^{15}$ | | ||
| | tera | T | $10^{12}$ | | ||
| | giga | G | $10^{9}$ | | ||
| | mega | M | $10^{6}$ | | ||
| | kilo | k | $10^{3}$ | | ||
| | hecto | h | $10^{2}$ | | ||
| | deca | da | $10^{1}$ | | ||
| | deci | d | $10^{-1}$ | | ||
| | centi | c | $10^{-2}$ | | ||
| | milli | m | $10^{-3}$ | | ||
| | micro | μ | $10^{-6}$ | | ||
| | nano | n | $10^{-9}$ | | ||
| | pico | p | $10^{-12}$ | | ||
| | femto | f | $10^{-15}$ | | ||
| | atto | a | $10^{-18}$ | | ||
| | zepto | z | $10^{-21}$ | | ||
| | yocto | y | $10^{-24}$ | |
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| # 电学 | ||
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| ## 静止电荷的电场 | ||
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| ### 电荷 库仑定律 | ||
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| - 电荷是相对论不变量，与运动无关。 | ||
| - $e=1.6\times10^{-19}\mathrm{C}$ | ||
| - 库仑定律 | ||
| - $\varepsilon_0$ 真空电容率（permittivity of free space）或真空介电常数（dieletric constant of vacuum）。 | ||
| - 库仑定律只适用于点电荷。与叠加原理结合能够求解静电学中所有问题。 | ||
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| $$ | ||
| \begin{array}{l} | ||
| \vec{F}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\hat{r} \\ | ||
| k = \frac{1}{4\pi\varepsilon_0}=9\times10^9\mathrm{N\cdot m^2/C^2} \\ | ||
| \varepsilon_0=8.85\times10^{-12}\mathrm{C^2/N\cdot m^2} | ||
| \end{array} | ||
| $$ | ||
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| ### 静电场 电场强度 | ||
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| - 没有物质作传递介质的超距作用是不存在的。 | ||
| - 电磁场是物质存在的一种形态，分布在一定范围的空间里，具有能量、动量等属性，并通过交换场量子来实现相互作用的传递。电磁场的媒介子是光子，电荷间相互作用的传递速度也是电磁场的运动速度，光速。 | ||
| - 静电场是**相对于观察者静止的电荷**在其周围激发的电场。 | ||
| - 电场强度：$\vec{E}=\frac{\vec{F}}{q}$ | ||
| - 电场强度形成矢量场 $\vec{E}(\vec{r})$。 | ||
| - 电场强度叠加原理。 | ||
| - 连续分布电荷电场强度：$\vec{E}=\int \mathrm{d}\vec{E}=k\int \frac{\mathrm{d}q}{r^2}\hat{r}$ | ||
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| <!-- prettier-ignore-start --> | ||
| !!! example "电偶极子" | ||
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| 两个大小相等符号相反的电荷，间距为 $l$，比所考虑的场点到它们的距离小得多时，称为电偶极子（electric dipole）。电偶极子的电偶极矩（electric dipole moment）为 $\vec{p}=q\vec{l}$，方向由负电荷指向正电荷。 | ||
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| - 在轴线延长线上计算电场强度：$\vec{E_A}=\frac{1}{4\pi\varepsilon_0}\frac{2\vec{p}}{x^3}$ | ||
| - 在中垂线上计算电场强度：$\vec{E_B}=-\frac{1}{4\pi\varepsilon_0}\frac{\vec{p}}{y^3}$ | ||
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| 远离电偶极子处场强与距离三次方成反比，与电偶极矩成正比。 | ||
| <!-- prettier-ignore-end --> | ||
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| <!-- prettier-ignore-start --> | ||
| !!! example "均匀带电直棒" | ||
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| 无限长带点直棒附近某点电场强度 $E = \frac{1}{4\pi\varepsilon_0}\frac{2\lambda}{r}$，其中 $\lambda$ 为电荷线密度。以上结论对靠近有限长直棒中部区域也近似成立。 | ||
| <!-- prettier-ignore-end --> |
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