Math
Examples below are from https://www.overleaf.com/learn
Expressions
The well known Pythagorean theorem \(x^2 + y^2 = z^2\) was proved to be invalid for other exponents. Meaning the next equation has no integer solutions:
\[ x^n + y^n = z^n \]
\begin{equation} E=mc^2 \end{equation}
Greek letters \(\alpha \beta \gamma \rho \sigma \delta \epsilon\)
Binary operators \(\times \otimes \oplus \cup \cap \)
Relation operators \(< > \subset \supset \subseteq \supseteq\)
Others \( \int \oint \sum \prod \)
Subscripts and superscripts
\( a_1^2 + a_2^2 = a_3^2 \)
\( x^{2 \alpha} - 1 = y_{ij} + y_{ij} \)
\( \sum_{i=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} \)
\( \sum_{i=1}^{\infty} \)
\( \cup_{i=1}^n \)
\( \cap_{i=1}^n \)
Brackets and Parentheses
\((x+y) [x+y] \{ x+y \} \langle x + y \rangle |x+y| \|x+y\| \)
\( F = G \left( \frac{m_1 m_2}{r^2} \right) \)
\( \left[ \frac{ N } { \left( \frac{L}{p} \right) - (m+n) } \right] \)
\( \begin{align*} y = 1 + & \left( \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \ldots \right. \\ &\left. \quad + \frac{1}{x^{n-1}} + \frac{1}{x^n} \right) \end{align*} \)
Matrices
\( \begin{matrix} 1 & 2 & 3\\ a & b & c \end{matrix} \)
\( \begin{pmatrix} 1 & 2 & 3\\ a & b & c \end{pmatrix} \)
\( \begin{bmatrix} 1 & 2 & 3\\ a & b & c \end{bmatrix} \)
\( \begin{Bmatrix} 1 & 2 & 3\\ a & b & c \end{Bmatrix} \)
\( \begin{vmatrix} 1 & 2 & 3\\ a & b & c \end{vmatrix} \)
\( \begin{Vmatrix} 1 & 2 & 3\\ a & b & c \end{Vmatrix} \)
Fractions and Binomials
\( \frac{1}{2} \)
\( f(x)=\frac{P(x)}{Q(x)} \ \ \textrm{and} \ \ f(x)=\textstyle\frac{P(x)}{Q(x)} \)
\( \frac{1+\frac{a}{b}}{1+\frac{1}{1+\frac{1}{a}}} \)
\( a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} \)
\( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
Operators
\( \sin(a + b) = \sin a \cos b + \cos b \sin a \)
\( \lim_{h \to 0 } \frac{f(x+h)-f(x)}{h} \)
Integrals, sums and limits
\( \int_{a}^{b} x^2 ,dx \)
\( \iint_V \mu(u,v) ,du,dv \)
\( \sum_{n=1}^{\infty} 2^{-n} = 1 \)
\( \prod_{i=a}^{b} f(i) \)
\( \lim_{x\to\infty} f(x) \)
\(
\)
\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}
The Cauchy-Schwarz Inequality
\[ \left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
A Cross Product Formula
\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix} \]
The probability of getting (k) heads when flipping (n) coins is:
\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]
An Identity of Ramanujan
\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
A Rogers-Ramanujan Identity
\[ 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for $|q| < 1$}. \]
Maxwell’s Equations
\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}
In-line Mathematics
Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an inline equation. As you see, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.