微积分公式的总结

1.从线积分开始讲起:
牛顿-莱布尼茨公式:\(\begin{aligned} \int_a^b f(x) \mathrm{d} x =F(b)-F(a) \end{aligned} \)
或者是:\(\begin{aligned} \int_a^b f(x) \mathrm{d} x =F(x) \mid_a^b \end{aligned} \)
然后是对弧长的曲线积分,物理意义是求质量:\(\begin{aligned} \int_L \mathrm{d} m = \int_L \rho(x,y) \mathrm{d} s \end{aligned} \)
然后是对坐标的曲线积分,物理意义是求做功:\(\begin{aligned} W=\int_L P(x,y)\mathrm{d}x+Q(x,y)\mathrm{d}y \end{aligned} \)
顺便打一些比较重要且有意思的公式:
\(Wallis公式\)
\(\begin{aligned}\int_{0}^{\frac{\pi}{2}} \sin^n(x) \mathrm{d} x=\int_{0}^{\frac{\pi}{2}} \cos^n(x) \mathrm{d} x= \frac{2k-1}{2k} \cdot \frac{2k-3}{2k-2} \cdots \frac{1}{2} \cdot \frac{\pi}{2}(n=2k)\end{aligned}\)
\(\begin{aligned}\int_{0}^{\frac{\pi}{2}} \sin^n(x) \mathrm{d} x=\int_{0}^{\frac{\pi}{2}} \cos^n(x) \mathrm{d} x= \frac{2k}{2k+1} \cdot \frac{2k-2}{2k-1} \cdots \frac{2}{3} \cdot 1 (n=2k+1)\end{aligned}\)
\(Gauss分布的积分公式:\)
1.先给出这个积分:\(\begin{aligned} \int_0^{\infty} e^{-x^2} \mathrm {d} x=\frac{\sqrt{\pi}}{2} \end{aligned}\)
2.然后定义\(\begin{aligned} f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^2}{2\sigma^2}} \end{aligned} \)且\( \begin{aligned}\int_{-\infty}^{\infty} f(x) \mathrm{d} x=1 \end{aligned}\)
3.\(Gamma函数又名为阶乘函数\)
\(\begin{aligned} \Gamma(x)=\int_0^{+\infty} t^{x-1}e^{-t} \mathrm{d}t (x>0) \end{aligned} \)
神奇的性质:\(\begin{aligned} \Gamma(x+1)=x\Gamma(x) \quad \Gamma(n)=(n-1)! \quad \Gamma(\frac{1}{2})=\sqrt{\pi} \end{aligned}\)

2.然后是二重积分:\(\begin{aligned} \iint \limits_D f(x,y) \mathrm{d} x =\int_a^b \mathrm{d} x \int_{\phi(x_1)}^{\phi(x_2)} f(x,y) \mathrm{d}y \end{aligned} \)
二重积分的极坐标形式:\(\begin{aligned} \iint \limits_D f(\rho\cos\theta,\rho\sin\theta)\rho \mathrm{d} \rho \mathrm{d} \theta =\int_{\alpha}^{\beta} \mathrm{d} \theta \int_{\phi(\theta_1)}^{\phi(\theta_2)} f(\rho\cos\theta,\rho\sin\theta) \rho \mathrm{d} \rho \end{aligned} \)
提一下\(Green\)公式:\(\begin{aligned}\iint\limits_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}) \mathrm{d}x \mathrm{d} y=\oint_L P\mathrm{d}x+Q\mathrm{d}y \end{aligned} \)
格林公式将线积分与面积分联系起来,他的具体的几何意义我还没弄明白,但物理意义想通了,很神奇的一个公式,对一个封闭曲线外围绕一周做功,相当于在所包围的面积下对某个函数做二重积分,所以那句经典的名言很是值得借鉴:
\(In \; mathematics \; you\; don't \; understand \; things.\)
\(You \; just\; get\; used \;to\; them.---- Johann\; von\; Neumann\)
反正弄不明白,就先拿来用,用着用着可能就想明白了。

3提一下微分方程的解:一阶非齐次线性微分方程
\(\begin{aligned} \frac{d y}{d x}+P(x) y=Q(x)\end{aligned}\)

其通解公式为:\( \begin{aligned} Y= \left(e^{ -\int P(x) d x} \right) \left(\int Q(x) e^{P(x) d x} d x+C\right) \end{aligned}\)

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