1. Inline math and basic symbols
You can place math naturally inside text, like $e^{i\pi}+1=0$, $\alpha,\beta,\gamma$,
$x_1, x_2, \dots, x_n$, or set-builder notation $A = \set{x \in \R : x^2 < 2}$.
$f(x)=x^3-2x+1$, $\lim_{x\to 0}\frac{\sin x}{x}=1$, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
2. Display equations
Displayed math is ideal for larger formulas and derivations.
\[
\int_0^{\infty} e^{-x^2}\,\dd x = \frac{\sqrt{\pi}}{2}
\]
\[
\nabla \cdot \vect{E} = \frac{\rho}{\varepsilon_0},
\qquad
\nabla \times \vect{E} = -\frac{\partial \vect{B}}{\partial t}
\]
3. Alignment and multi-line derivations
The AMS alignment environments are one of the nicest features to preserve from LaTeX.
\[
\begin{aligned}
(x+y)^3
&= (x+y)(x+y)^2 \\
&= (x+y)(x^2+2xy+y^2) \\
&= x^3 + 3x^2y + 3xy^2 + y^3
\end{aligned}
\]
4. Cases and piecewise functions
Great for algorithms, semantics, and discrete math definitions.
\[
f(n)=
\begin{cases}
1, & \text{if } n=0,\\
n\,f(n-1), & \text{if } n>0.
\end{cases}
\]
\[
\abs{x}=
\begin{cases}
x, & x\ge 0,\\
-x, & x<0.
\end{cases}
\]
5. Matrices and determinants
Useful in linear algebra, computer graphics, and quantum computing.
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
0 & 0 & 1
\end{bmatrix},
\qquad
\det(A)=1
\]
\[
U = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\]
6. Number sets and custom macros
This page defines macros like $\R$, $\N$, $\Z$, $\Q$, $\C$,
$\vect{v}$, $\abs{x}$, and $\norm{\vect{x}}$.
\[
\Z \subset \Q \subset \R \subset \C,
\qquad
\vect{u}\cdot\vect{v} = \norm{\vect{u}}\,\norm{\vect{v}}\cos\theta
\]
7. Summation, products, and limits
MathJax handles large operators well in both inline and display forms.
\[
\sum_{k=0}^{n} ar^k = a\frac{1-r^{n+1}}{1-r},
\qquad
\prod_{p\text{ prime}} \frac{1}{1-p^{-s}} = \zeta(s)
\]
\[
\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e
\]
8. Logic and theoretical computer science
Nicely suited for formal reasoning and semantics.
\[
(P \to Q) \iff (\neg P \lor Q),
\qquad
\models \forall x\, (P(x) \to Q(x))
\]
\[
\lambda x.\,x : A \to A,
\qquad
\Gamma \vdash e : \tau
\]
9. Calculus and differential equations
Large fractions, partial derivatives, and vector notation render cleanly.
\[
\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)
\]
\[
y'' + 3y' + 2y = 0
\quad\Longrightarrow\quad
y(t)=c_1 e^{-t}+c_2 e^{-2t}
\]
10. Probability and statistics
Good support for expectation, variance, and distributions.
\[
\mathbb{E}[X] = \sum_x x\,P(X=x),
\qquad
\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X]^2
\]
\[
X \sim \mathcal{N}(\mu,\sigma^2)
\]
11. Commutative diagrams, arrows, and category flavor
Even without TikZ, many arrow-heavy expressions still look excellent.
\[
A \xrightarrow{f} B \xrightarrow{g} C,
\qquad
g \circ f : A \to C
\]
\[
F : \mathcal{C} \to \mathcal{D},
\qquad
\eta : \mathrm{Id}_{\mathcal{C}} \Rightarrow GF
\]
12. Decorations, cancellation, and color
These are often handy for teaching, derivations, and slide explanations.
\[
\cancel{x^2} + 2x + \cancel{1} - \cancel{x^2} = 2x + \cancel{1}
\]
\[
{\color{#38bdf8}a^2} + {\color{#a78bfa}b^2} = {\color{#22c55e}c^2}
\]
14. A compact “big features” demo
\[
\begin{aligned}
\oint_{\partial \Omega} \vect{F}\cdot \dd\vect{r}
&= \iint_{\Omega} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\dd A \\
\Pr(X \ge k)
&= \sum_{j=k}^{n} \binom{n}{j} p^j(1-p)^{n-j} \\
A^{-1}
&= \frac{1}{\det A}\operatorname{adj}(A), \qquad \det A \ne 0 \\
\forall f \in C([a,b])
&\;\exists c \in [a,b] \text{ such that } \int_a^b f(x)\,\dd x = f(c)(b-a)
\end{aligned}
\]
This section mixes contour integrals, probability, linear algebra, and analysis in one aligned block.