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Inline Math with $ ... $

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which gives the roots of any quadratic equation $ax^2 + bx + c = 0$.

Einstein's famous equation $E = mc^2$ relates energy and mass. The speed of light $c \approx 3 \times 10^8$ m/s.

Display Math with $$ ... $$

The Pythagorean theorem:

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

Euler's identity:

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

LaTeX-style delimiters \( ... \) and \[ ... \]

Inline with backslash-paren: \(f(x) = x^2 + 2x + 1\) is a parabola.

Display with backslash-bracket:

\[\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}\]

Summations and Products

The sum of the first $n$ integers:

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

The factorial can be written as:

$$n! = \prod_{i=1}^n i$$

Matrices

A rotation matrix:

$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

Greek Letters and Symbols

In mathematics, we often use Greek letters like $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\theta$, $\lambda$, $\mu$, $\pi$, $\sigma$, $\omega$, and $\Omega$.

Common relations: $\le$, $\ge$, $\ne$, $\approx$, $\equiv$, $\in$, $\subset$, $\subseteq$.

Fractions and Roots

Nested fractions:

$$\frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}}$$

Square and nth roots: $\sqrt{2}$, $\sqrt[3]{8}$, $\sqrt[n]{x^n} = x$

Alignment Environments

System of equations:

$$\begin{align} x + y &= 5 \\ 2x - y &= 1 \end{align}$$

Multiple steps:

$$\begin{aligned} (a + b)^2 &= (a + b)(a + b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 \end{aligned}$$

Calculus

The fundamental theorem of calculus:

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

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

Partial derivatives:

$$\frac{\partial^2 f}{\partial x \partial y}$$

Limits

Definition of derivative:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Trigonometry

Basic identity:

$$\sin^2\theta + \cos^2\theta = 1$$

Tangent definition: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Text in Math Mode

$$\text{volume} = \frac{4}{3}\pi r^3$$

Complex Expressions

The Gaussian integral:

$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$

Cauchy's integral formula:

$$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a} dz$$

Aligned Expressions

\begin{aligned} \frac{\mathrm{d}\text{EP}}{\mathrm{d}t} &= I(t) − κ_d·\text{EP} - κ_{\text{EP}}·u_{\text{EP}}(t) & \text{inputs minus dissipation minus control} \\ \frac{\mathrm{d}f}{\mathrm{d}t} &= β_R·R(t) - β_d·D(t) + κ_f·u_f(t) & \text{repair signals minus damage plus training/therapy} \\ y(t) &= α·\text{GDF15}(t) + β·\text{EE}(t) + γ·(\text{VO}_2\text{max}(t))⁻¹ & \text{observable estimator } E[\text{eR}] \\ \end{aligned}