layout: post title: 伟大的公式 categories:
网上有一篇所谓的“世上最伟大的十个公式”, 先简要转载一下.
英国科学期刊《物理世界》曾让读者投票评选了“最伟大的公式”, 最终榜上有名的十个公式既有无人不知的1+1=2, 又有著名的 E=mc^2; 既有简单的圆周公式, 又有复杂的欧拉公式……
从什么时候起我们开始厌恶数学？这些东西原本如此美丽, 如此精妙. 这个地球上有多少伟大的智慧曾耗尽一生, 才最终写下一个等号. 每当你解不开方程的时候, 不妨换一个角度想, 暂且放下对理科的厌恶和对考试的痛恨. 因为你正在见证的, 是科学的美丽与人类的尊严.
No.10 圆周长公式（The Length of the Circumference of a Circle） $$c=2 \pi r$$
No.9 傅立叶变换（The Fourier Transform） $$\hat {F} (\xi)=\int_{-\infty}^{+\infty} f(x) e^{-2 \pi i \xi x} dx$$
No.8 德布罗意关系式（The de Broglie Relations） $$\boldsymbol{p} = \hbar \boldsymbol{k} \;\;\;\; E=\hbar \omega$$
No.7 1+1=2 $$1+1=2$$
No.6 薛定谔方程（The Schrödinger's Equation） $$\hat{H}\Psi = i\hbar {\partial{ \Psi} \over \partial t }$$
No.5 质能方程（Mass-energy Equivalence） $$E=mc^2$$
No.4 勾股定理/毕达哥拉斯定理（Pythagorean Theorem） $$a^2+b^2=c^2$$
No.3 牛顿第二定律（Newton's Second Law of Motion） $$\mathbf{F} = m \mathbf{a}$$
No.2 欧拉公式（Euler's Identity） $$e^{i\pi}+1=0$$
No.1 麦克斯韦方程组（The Maxwell's Equations）
微分形式： $$\nabla \cdot \mathbf{E} = {\rho \over \varepsilon} \ \nabla \cdot \mathbf{B} = 0 \ \nabla \times \mathbf{E} = -{ \partial \mathbf{B} \over \partial t} \ \nabla \times \mathbf{B} = \mu \mathbf{J} + \mu \varepsilon {\partial \mathbf{E} \over \partial t }$$
积分形式： $$\oint{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = {Q \over \varepsilon} \ \oint{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 \ \oint{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \iint\Sigma {\partial \mathbf{B} \over \partial t} \cdot d\mathbf{S} \ \oint{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu I + \mu\varepsilon \iint\Sigma {\partial \mathbf{E} \over \partial t} \cdot d\mathbf{S}$$
考究起来, 网上这个说法与真正的Top Ten Greatest Equations Ever存在差距:
Maxwell's four equations describing how an electromagnetic field varies in space and time.
Euler's equation of momentum-flow and force-density in fluid dynamics. $${\partial \boldsymbol{v} \over \partial t } + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} = -{\nabla P \over \rho} + \boldsymbol{f}$$
Newton's Second Law (F=ma) - 'The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction'.
Pythagoras's Theorum (a^2+b^2=c^2;) - In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
Schrödinger's equation describing the time-dependence of quantum mechanical systems.
Boltzmann equation describing the statistical distribution of particles in a fluid. $${\partial f \over \partial t} +\boldsymbol{v} \cdot \nabla f + {\boldsymbol{p} \over m} \cdot \nabla f = ({\partial f \over \partial t})_{coll}$$
Principle of Least Action (or Principle of stationary action) $$\delta \int_{t_1}^{t_2} L(q,\dot{q},t) dt=0$$
De Broglie's equation - showing that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy.
Fourier Transformation - an integral transform that re-expresses a function in terms of sinusoidal basis functions.
Einstein's field equations for General Relativity. $$R{\mu\nu}-{1 \over 2}g{\mu\nu}R+g{\mu\nu}\Lambda={8\pi G \over c^4} T{\mu\nu}$$
也不知是不是原创者故意将几个不知名的公式替换了.