1\documentclass[12pt]{article}
2\usepackage{amsmath, amssymb, amsthm}
3
4\newtheorem{theorem}{Theorem}[section]
5\newtheorem{lemma}[theorem]{Lemma}
6\newtheorem{corollary}[theorem]{Corollary}
7\newtheorem{definition}{Definition}[section]
8
9\title{Mathematics in LaTeX}
10\author{sublimated}
11\date{}
12
13\begin{document}
14\maketitle
15
16\section{Inline and Display Math}
17
18Inline math: the quadratic formula gives $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
19
20Display math (unnumbered):
21\[ e^{i\pi} + 1 = 0 \]
22
23Display math (numbered):
24\begin{equation}
25  \label{eq:euler}
26  e^{i\theta} = \cos\theta + i\sin\theta
27\end{equation}
28
29Reference: Equation~\eqref{eq:euler} is Euler's formula.
30
31\section{Fractions and Roots}
32
33Nested fractions: $\frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}$.
34
35Display fraction:
36\[ \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u \]
37
38Roots: $\sqrt{2}$, $\sqrt[3]{8} = 2$, $\sqrt[n]{x^n} = |x|$.
39
40\section{Sums, Products, Integrals}
41
42\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]
43
44\[ \prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdots x_n \]
45
46\[ \int_0^\infty e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2} \]
47
48\[ \oint_{\partial\Omega} \mathbf{F} \cdot d\mathbf{s} = \iint_\Omega (\nabla \times \mathbf{F}) \cdot d\mathbf{A} \]
49
50\section{Matrices}
51
52\[
53A = \begin{pmatrix}
54  a_{11} & a_{12} & a_{13} \\
55  a_{21} & a_{22} & a_{23} \\
56  a_{31} & a_{32} & a_{33}
57\end{pmatrix}
58\]
59
60Determinant notation:
61\[
62\det(A) = \begin{vmatrix}
63  1 & 2 \\
64  3 & 4
65\end{vmatrix} = -2
66\]
67
68\section{Aligned Equations}
69
70\begin{align}
71  \nabla \cdot \mathbf{E}  &= \frac{\rho}{\varepsilon_0} \label{eq:gauss} \\
72  \nabla \cdot \mathbf{B}  &= 0 \label{eq:nomonopole} \\
73  \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \label{eq:faraday} \\
74  \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \label{eq:ampere}
75\end{align}
76
77Maxwell's equations: \eqref{eq:gauss}--\eqref{eq:ampere}.
78
79\section{Cases}
80
81\[
82|x| = \begin{cases}
83  x  & \text{if } x \geq 0, \\
84  -x & \text{if } x < 0.
85\end{cases}
86\]
87
88\section{Theorems and Proofs}
89
90\begin{definition}
91  A function $f: \mathbb{R} \to \mathbb{R}$ is \emph{continuous} at $c$ if
92  $\lim_{x \to c} f(x) = f(c)$.
93\end{definition}
94
95\begin{theorem}[Intermediate Value Theorem]
96  \label{thm:ivt}
97  If $f$ is continuous on $[a, b]$ and $y$ is between $f(a)$ and $f(b)$,
98  then there exists $c \in (a, b)$ such that $f(c) = y$.
99\end{theorem}
100
101\begin{proof}
102  Let $S = \{x \in [a,b] : f(x) \leq y\}$. Since $S$ is bounded above by $b$
103  and non-empty ($a \in S$), let $c = \sup S$. By continuity of $f$ at $c$,
104  we have $f(c) = y$.
105\end{proof}
106
107\begin{lemma}
108  Every convergent sequence is bounded.
109\end{lemma}
110
111\begin{corollary}
112  If $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, then $f$ has a zero in $(a,b)$.
113\end{corollary}
114
115\section{Greek and Symbols}
116
117Lowercase: $\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \lambda, \mu, \pi, \sigma, \phi, \omega$.
118
119Uppercase: $\Gamma, \Delta, \Theta, \Lambda, \Pi, \Sigma, \Phi, \Omega$.
120
121Operators: $\times, \div, \pm, \mp, \cdot, \circ, \otimes, \oplus$.
122
123Relations: $\leq, \geq, \neq, \approx, \equiv, \sim, \propto, \in, \subset, \subseteq$.
124
125Arrows: $\to, \leftarrow, \Rightarrow, \Leftrightarrow, \mapsto, \hookrightarrow$.
126
127\end{document}