SearchOnMath

Euler's identity, \(e^{i\pi}+1=0\), relates the imaginary unit and two famous irrational numbers in a way that is so simple it has been called beautiful.

Maxwell's equations in a vacuum describe a wave propagating at speed \(c\) as the interaction of electric field \(E\) and magnetic field \(B\): \(\nabla \cdot E = 0\), \(\nabla \times E = -\frac{\partial B}{\partial t}\), \(\nabla \cdot B = 0\), \(\nabla \times B = \frac{1}{c^2} \frac{\partial E}{\partial t}\).

The wave equation, \(\frac{\partial^2u}{\partial t^2}= c^2\nabla^2 u\), relates how \(u\) varies with time and with the spatial coordinates as a wave propagates with speed \(c\).