# Math Rendering Demo

## Inline math

Einstein's famous equation $E = mc^2$ changed physics forever.

The quadratic formula gives $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for any quadratic $ax^2 + bx + c = 0$.

A sum: $\sum_{i=1}^{n} x_i$ and an integral: $\int_0^\infty e^{-x} dx = 1$.

## Display math

The Gaussian distribution:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Euler's identity:

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

Matrix multiplication:

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$$

Maxwell's equations in differential form:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$

$$\nabla \cdot \mathbf{B} = 0$$

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

## Mixed content

The Taylor series of $e^x$ is given by:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

This converges for all $x \in \mathbb{R}$.
