Cheetsheet¶
微分 ¶
常用的麦克劳林展开式
- \(\displaystyle e^x = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + \cdots, \quad x \in (-\infty, +\infty)\)
- \(\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + \cdots, \quad x \in (-\infty, +\infty)\)
- \(\displaystyle \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (-1)^n \frac{x^{2n}}{(2n)!} + \cdots, \quad x \in (-\infty, +\infty)\)
- \(\displaystyle \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^n \frac{x^{n+1}}{n+1} + \cdots, \quad x \in (-1, 1]\)
- \(\displaystyle (1 + x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \cdots + \frac{a(a-1)\cdots(a-n+1)}{n!}x^n + \cdots, \quad x \in (-1, 1)\)
- \(\displaystyle \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n, \quad |x| < 1\)
- \(\displaystyle \frac{1}{1 + x} = \sum_{n=0}^{\infty} (-1)^n x^n, \quad |x| < 1\)
曲率
\[
K = \frac{|y''|}{\left(1 + y'^2\right)^{3/2}}
\]
\[
K = \frac{|x'(t)\,y''(t) - y'(t)\,x''(t)|}{\left[\,x'(t)^2 + y'(t)^2\,\right]^{3/2}}
\]
雅可比行列式
\(\(\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cccc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array}\right|\)\)
积分 ¶
分部积分
\[\int u\mathrm{d}v=uv-\int v\mathrm{d}u\]
多元函数 ¶
格林公式
若函数 \(P\)，\(Q\) 在有界闭区域 \(D \subset \mathbf{R}^2\) 上连续且具有一阶连续偏导数，则
\[
\iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathrm{d}x \mathrm{d}y = \oint_{\Gamma} P \, \mathrm{d}x + Q \, \mathrm{d}y
\]
这里 \(\Gamma\) 为区域 \(D\) 的边界曲线，并取正向
格林公式的行列式表示法：
\[
\iint_{D} \begin{vmatrix} \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} \\[2ex] P & Q \end{vmatrix} \mathrm{d}x \mathrm{d}y = \oint_{\Gamma} P \, \mathrm{d}x + Q \, \mathrm{d}y
\]
高斯公式
\[\oiint_S F\cdot \mathrm{d}S=\iiint_V \nabla \cdot F \mathrm{d}V\]
梯度公式
\[\oiint_S p dS=\iiint_V \nabla pdV\]
散度
\[\nabla \cdot A=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}, A=A_x i + A_y j + A_zk\]
梯度
\[\nabla p=\frac{\partial p}{\partial x}i+\frac{\partial p}{\partial y}\mathbf{j}+\frac{\partial p}{\partial z}k, p=p(x,y,z)\]
旋度
\[\nabla \times A=\left|\begin{array}{cccc}i & j & k \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
A_x & A_y & A_z\end{array}\right|\]
速度散度
\[\nabla\cdot V=\frac{1}{\partial V}\frac{D(\partial V)}{Dt}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\]
速度散度表示单位体积的运动流体微元的体积随时间的变化率
环量
\[\Gamma=-\oint_C V \cdot \mathcal{d}s\]
级数 ¶
傅里叶级数
\[
\frac{a_0}{2}+\sum_{n=1}^{+\infty}(a_n \cos \frac{n\pi x}{l}+b_n \sin \frac{n\pi x}{l})
\]
\[
\left\{
\begin{aligned}
a_n=\frac{1}{l}\int_{-l}^{l} f(x)\cos\frac{n\pi x}{l}\mathrm{d}x, \quad n=0,1,2... \\
b_n=\frac{1}{l}\int_{-l}^{l} f(x)\sin\frac{n\pi x}{l}\mathrm{d}x, \quad n=1,2,3...
\end{aligned}
\right.
\]
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