Mathematical Equations

Euler's Identity

$$e^{i\pi} + 1 = 0$$

Connects five fundamental mathematical constants

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

Relationship in a right triangle

Basel Problem

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

Sum of reciprocals of squares

Maxwell's Equations

$$\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$

Fundamental equations of electromagnetism

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Solves quadratic equations

Fourier Transform

$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx$$

Decomposes functions into their frequency components

Navier-Stokes Equations

$$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}$$

Describes fluid motion

Fermat's Last Theorem

$$a^n + b^n = c^n$$

No solutions for n > 2

Ramsey's Theorem

$$R(s,t) \leq R(s-1,t) + R(s,t-1)$$

Graph coloring