17 Equations That Changed the World – A Website for Physics Lovers

In Pursuit of the Unknown: 17 Equations That Changed the World is a 2012 nonfiction book by British mathematician Ian Stewart FRS CMath FIMA, published by Basic Books. In the book Stewart traced a history of the role of mathematics in human history, beginning with the Pythagorean theorem to the equation that transformed the twenty-first-century financial market, the Black–Scholes model.

Seventeen equations are described in the book as follows:

Pythagorean equation,
$a^{2}+b^{2}=c^{2}$
Logarithm product identity,
$\log {xy}=\log {x}+\log {y}$
Derivative,
${\frac {df}{dt}}=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}$
Newton’s law of universal gravitation,
$F=G{\frac {m_{1}m_{2}}{r^{2}}}$
Imaginary unit,
$i^{2}=-1$
Euler’s polyhedron formula,
$V-E+F=2$
Normal distribution,
$\mathbf {\Phi } (x)={\frac {1}{\sqrt {2\pi \rho }}}e^{\frac {(x-\mu )^{2}}{2\rho ^{2}}}$
Wave equation in one space dimension,
${\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}$
Fourier transform,
$f(\omega )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\omega }dx$
Navier–Stokes momentum equation,
$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\nabla \cdot \mathbf {T} +\mathbf {f}$
Maxwell’s equations,
$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$ , $\nabla \cdot {\mathbf {H}}=0$ , $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {H} }{\partial t}}$ , $\nabla \times \mathbf {H} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {H} }{\partial t}}\right)$
Entropy and the second law of thermodynamics,
$dS\geq 0$
Mass–energy equivalence,
$E=mc^{2}$
Time-dependent Schrödinger equation,
$i\hbar {\frac {\partial }{\partial t}}\mathbf {\Psi } =H\mathbf {\Psi }$
Entropy in information theory,
$H=-\sum p(x)\log p(x)$
Logistic map,
$x_{t+1}=kx_{t}\left(1-x_{t}\right)$
Black–Scholes equation,
${\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}+{\frac {\partial V}{\partial t}}-rV=0$