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:

$$x^n+y^n=z^n$$

$$E=m{c}^2$$

Greek letters $\alpha \beta \gamma \rho \sigma \delta ϵ$

Binary operators $\times \otimes \oplus \cup \cap$

Relation operators $<>\subset \supset \subseteq \supseteq$

Others $\int \oint \sum \prod$

Subscripts and superscripts

$a_1^2+a_2^2=a_3^2$

$x^{2\alpha }-1=y_{ij}+y_{ij}$

$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{n^s}=\prod _p\frac{1}{1-p^{-s}}$

$\sum _{i=1}^{\mathrm{\infty }}$

${\cup }_{i=1}^n$

${\cap }_{i=1}^n$

Brackets and Parentheses

$(x+y)[x+y]\{x+y\}⟨x+y⟩|x+y|\parallel x+y\parallel$

$F=G\left(\frac{m_1{m}_2}{r^2}\right)$

$\left[\frac{N}{\left(\frac{L}{p}\right)-(m+n)}\right]$

$\begin{array}{rl}y=1+& (\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\dots \\ & +\frac{1}{x^{n-1}}+\frac{1}{x^n})\end{array}$

Matrices

$\begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}$

$\left(\begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}\right)$

$\left[\begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}\right]$

$\left\{\begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}\right\}$

$\left|\begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}\right|$

$\Vert \begin{array}{ccc}1& 2& 3\\ a& b& c\end{array}\Vert$

Fractions and Binomials

$\frac{1}{2}$

$f(x)=\frac{P(x)}{Q(x)}\text{and}f(x)=\frac{P(x)}{Q(x)}$

$\frac{1+\frac{a}{b}}{1+\frac{1}{1+\frac{1}{a}}}$

$a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\cdots }}}$

$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}$

Operators

$\sin (a+b)=\sin a\cos b+\cos b\sin a$

$\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

Integrals, sums and limits

${\int }_a^b{x}^2,dx$

${\iint }_V\mu (u,v),du,dv$

$\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}=1$

$\prod _{i=a}^{b}f(i)$

$\underset{x\to \mathrm{\infty }}{lim}f(x)$

\(

\)

$$\begin{array}{rl}\dot{x}& =\sigma (y-x)\\ \dot{y}& =\rho x-y-xz\\ \dot{z}& =-\beta z+xy\end{array}$$

The Cauchy-Schwarz Inequality

$${\left(\sum _{k=1}^n{a}_k{b}_k\right)}^2\le \left(\sum _{k=1}^n{a}_k^2\right)\left(\sum _{k=1}^n{b}_k^2\right)$$

A Cross Product Formula

$${\mathbf{V}}_1\times {\mathbf{V}}_2=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}\right|$$

The probability of getting (k) heads when flipping (n) coins is:

$$P(E)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$$

An Identity of Ramanujan

$$\frac{1}{(\sqrt{\varphi \sqrt{5}}-\varphi )e^{\frac{2}{5}\pi }}=1+\frac{e^{-2\pi }}{1+\frac{e^{-4\pi }}{1+\frac{e^{-6\pi }}{1+\frac{e^{-8\pi }}{1+\dots }}}}$$

A Rogers-Ramanujan Identity

$$1+\frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =\prod _{j=0}^{\mathrm{\infty }}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\text{for }|q|<1.$$

Maxwell’s Equations

$$\begin{array}{rl}\mathrm{\nabla }\times \overrightarrow{\mathbf{B}}-\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{E}}}{\partial t}& =\frac{4\pi }{c}\overrightarrow{\mathbf{j}}\\ \mathrm{\nabla }\cdot \overrightarrow{\mathbf{E}}& =4\pi \rho \\ \mathrm{\nabla }\times \overrightarrow{\mathbf{E}}+\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{B}}}{\partial t}& =\overrightarrow{\mathbf{0}}\\ \mathrm{\nabla }\cdot \overrightarrow{\mathbf{B}}& =0\end{array}$$

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.