Mathematical Equations Test

This page demonstrates various mathematical notations using MathJax.

Inline Math

$$\begin{equation*} |x| = \begin{cases} -x & \text{if } x < 0,\\ 0 & \text{if } x = 0,\\ x & \text{if } x > 0\\ \end{cases} \end{equation*}$$

Here’s an inline equation: $E = mc^2$ represents Einstein’s famous equation.

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Display Math

The normal distribution probability density function:

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

Matrix Example

A simple matrix multiplication:

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

Calculus

The fundamental theorem of calculus:

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Where $F'(x) = f(x)$.

Summation and Products

Sum notation:

$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$

Product notation:

$$\prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdot \ldots \cdot x_n$$

Complex Equations

The Schrödinger equation:

$$i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$$

Set Theory

$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$ $$A \cap B = \{x : x \in A \text{ and } x \in B\}$$

Greek Letters

• Alpha: $\alpha$, Beta: $\beta$, Gamma: $\gamma$
• Delta: $\Delta$, Theta: $\theta$, Lambda: $\lambda$
• Sigma: $\Sigma$, Pi: $\Pi$, Omega: $\Omega$

Limits

$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$

Vectors

Vector notation: $\vec{v} = \langle v_1, v_2, v_3 \rangle$

Dot product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$