Large operators: inline vs display
Inline: $\sum_{k=1}^{n} k$ and $\int_0^1 x^2\,dx$ and $\prod_{k=1}^{n} k$
Display: $$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$ $$\int_0^\infty e^{-x}\,dx = 1 \qquad \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$ $$\prod_{p\;\text{prime}} \frac{1}{1-p^{-s}} = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
Algebra
Quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Pythagorean theorem: $a^2 + b^2 = c^2$
Binomial theorem: $$\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} = (x + y)^n$$
Calculus
Definition of derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Gaussian integral: $$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$
Product rule: $\dfrac{d}{dx}\bigl[f(x)\,g(x)\bigr] = f'(x)\,g(x) + f(x)\,g'(x)$
Greek letters
Euler's identity: $e^{i\pi} + 1 = 0$
$\alpha,\ \beta,\ \gamma,\ \delta,\ \varepsilon,\ \zeta,\ \eta,\ \theta,\ \iota,\ \kappa,\ \lambda,\ \mu,\ \nu,\ \xi,\ \pi,\ \rho,\ \sigma,\ \tau,\ \upsilon,\ \phi,\ \chi,\ \psi,\ \omega$
Fourier transform: $$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\,e^{-2\pi i x \xi}\,dx$$
Physics
Einstein: $E = mc^2$
Schrödinger equation: $$i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi$$
Maxwell in free space: $$\nabla \cdot \mathbf{E} = 0 \qquad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
Piecewise definitions (stretchy {)
Two rows (short brace): $$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$
Four rows (tall brace): $$f(x) = \begin{cases} \dfrac{\sin x}{x}, & \text{if } x > 0 \\[6pt] \displaystyle\sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k+1)!}, & \text{if } x = 0 \\[6pt] -\sqrt{|x|}, & \text{if } -1 \leq x < 0 \\[4pt] \infty, & \text{otherwise} \end{cases}$$
Stretchy delimiters
Two-high (\binom):
$$\binom{n}{k}$$
Around a fraction: $$\left( \dfrac{\frac{a}{b}}{\frac{c}{d}} \right)^{n} \qquad \left[ \dfrac{\frac{a}{b}}{\frac{c}{d}} \right]^{n}$$
3×3 matrix: $$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \qquad \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$
Stretchy arrows
Vector accents: $\vec{v},\ \vec{F},\ \vec{p},\ \vec{\mathbf{0}},\ \vec{\mathbf{B}},\ \vec{ABC},\ \overrightarrow{ABC},\ \overleftarrow{ABC}$
Stretchy operators with labels: $A \xrightarrow{f} B \xrightarrow{n+\mu-1} C \qquad X \xleftarrow{g} Y$
AMScd commutative diagram: $$\begin{CD} A @>f>> B \\ @VgVV @VVhV \\ C @>>k> D \end{CD}$$