How to make special symbols $\alpha$, $\beta$, $\gamma$, $\delta$,$\pi$, $\sin x$, $\hbar$, $\lambda$, $\ldots$ We also can make subscripts $A_{x}$, $A_{xy}$ and superscripts, $e^x$, $e^{x^2}$, and $e^{a^b}$.

How to write equations. \begin{equation} \Delta =\sum_{i=1}^N w_i (x_i - \bar{x})^2 . \end{equation} It is a good idea to number equations, but we can have an equation without a number by writing \begin{equation} P(x) = \frac{x - a}{b - a} , \nonumber \end{equation} and \begin{equation} g = \frac{1}{2} \sqrt{2\pi} . \nonumber \end{equation}

Examples of more complicated equations: \begin{equation} I = \! \int_{-\infty}^\infty f(x)\,dx \label{eq:fine}. \end{equation}

How to align several equations: \begin{align} a & = b \\ c &= d , \end{align}

We can also have different cases: \begin{equation} \label{eq:mdiv} m(T) = \begin{cases} 0 & \text{$T > T_c$} \\ \bigl(1 - [\sinh 2 \beta J]^{-4} \bigr)^{\! 1/8} & \text{$T < T_c$} \end{cases} \end{equation}

Common Greek letters: $\alpha, \beta, \gamma, \Gamma, \delta,\Delta, \epsilon, \zeta, \eta, \theta, \Theta, \kappa, \lambda, \Lambda, \mu, \nu, \xi, \Xi, \pi, \Pi, \rho, \sigma, \tau, \phi, \Phi, \chi, \psi, \Psi, \omega, \Omega$

Special Symbols: The derivative is defined as \begin{equation} \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y} {\Delta x} \end{equation} \begin{equation} f(x) \to y \quad \mbox{as} \quad x \to x_{0} \end{equation} \begin{equation} f(x) \mathop {\longrightarrow} \limits_{x \to x_0} y \end{equation}

Order of magnitude: \begin{equation} \log_{10}f \simeq n \end{equation} \begin{equation} f(x)\sim 10^{n} \end{equation} Approximate equality: \begin{equation} f(x)\simeq g(x) \end{equation} Remember to keep everything in proportion: \begin{equation} f(x) \propto x^3 . \end{equation}

Chemical Formulas

\begin{equation} \textrm{NO}_{2}^{-} \rightarrow \textrm{N}_{2}\textrm{O} \end{equation}

Or you can have inline formulas if you want to talk about $\textrm{NH}_{4}^{+}$ being converted into $\textrm{N}_{2}\textrm{O}$, or whatever.