Align and Aligned
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\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}
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\begin{aligned}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{aligned}
Definitions by cases
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$$
f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}
$$
To get a larger vertical space between cases we can use \[2ex] instead of \. For example, you get this:
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f(n) =
\begin{cases}
\frac{n}{2}, & \text{if $n$ is even} \\[2ex]
3n+1, & \text{if $n$ is odd}
\end{cases}
iint
For double and triple integrals, don’t use \int\int or \int\int\int. Instead use the special forms \iint and \iiint:
Using \newcommand
It’s enough to insert something like
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$ \newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 } $
at the top of your post (remember the dollars!). Then you can just use your commands as you are used to do: in my example typing ` \SES{A}{B}{C} ` will produce the following:
$ \newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 } $
\[\SES{A}{B}{C}\]It’s also possible to use plain \def:
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\def\ses#1#2#3{0 \to #1 \to #2 \to #3 \to 0}
$\def\ses#1#2#3{0 \to #1 \to #2 \to #3 \to 0}$
and then \ses{A}{B}{C} will produce the same output.
Tags and References
For longer calculations (or referring to other post’s results) it is convenient to use the tagging/labelling/referencing system. To tag an equation use \tag{yourtag}, and if you want to refer to that tag later on, add \label{somelabel} right after the \tag. It is not necessary that yourtag and somelabel are the same, but it usually is more convenient to do so:
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$$ a := x^2-y^3 \tag{mmtag}\label{mmtag} $$
In order to refer to an equation, just use \eqref{somelabel}
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$$ a+y^3 \stackrel{\eqref{mmtag}}= x^2 $$
or \ref{somelabel}
$\ref{mmtag}$
or \eqref{somelabel}
$\eqref{mmtag}$
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Equations are usually referred to as $\eqref{*}$, but you can also use $\ref{*}$.
Multi-line equation
Multi-line equation is actually just one equation rather than several equations. So the correct environment is aligned instead of align.
\[\begin{equation}\begin{aligned} a &= b + c \\ &= d + e + f + g \\ &= h + i \end{aligned}\end{equation}\tag{2}\label{eq2}\]Equation $\eqref{eq2}$ is a multi-line equation. The code to produce equation $\eqref{eq2}$ is
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$$\begin{equation}\begin{aligned}
a &= b + c \\
&= d + e + f + g \\
&= h + i
\end{aligned}\end{equation}\tag{2}\label{eq2}$$
Multiple aligned equations
For multiple aligned equations, we use the align environment.
\[\begin{align} a &= b + c \tag{3}\label{eq3} \\ x &= yz \tag{4}\label{eq4}\\ l &= m - n \tag{5}\label{eq5} \end{align}\]Equation $\eqref{eq3}$, $\eqref{eq4}$ and $\eqref{eq5}$ are multiple equations aligned together. The code to produce these equations is,
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$$\begin{align}
a &= b + c \tag{3}\label{eq3} \\
x &= yz \tag{4}\label{eq4}\\
l &= m - n \tag{5}\label{eq5}
\end{align}$$
Reasoning
\implies (⟹) is a marginally preferable alternative to \Rightarrow (⇒) for implication.
There’s also \iff ⟺ and \impliedby ⟸.
\to (→) is preferable to \rightarrow or \longrightarrow for things like 𝑓:𝐴→𝐵. The reverse is \gets (←).
Linear programming
Formulation: A theoretical LPP can be typeset as
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\begin{array}{ll}
\text{maximize} & c^T x \\
\text{subject to}& d^T x = \alpha \\
&0 \le x \le 1.
\end{array}
To input a numerical LPP, use alignat instead of align to get better alignment between signs, variables and coefficients.
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\begin{alignat}{5}
\max \quad & z = & x_1 & + & 12 x_2 & & & && \\
\mbox{s.t.} \quad & & 13 x_1 & + & x_2 & + & 12x_3 & \geq 5 && \tag{constraint 1} \\
& & x_1 & & & + & x_3 & \leq 16 && \tag{constraint 2} \\
& & 15 x_1 & + & 201 x_2 & & & = 14 && \tag{constraint 3} \\
& & \rlap{x_i \ge 0, i = 1, 2, 3}
\end{alignat}
Latex Math Blocks
this is the lemma $\eqref{mylemma}$.
this is the theorem $\eqref{thm11}$.
The link to the lemma is the lemma
The link to the example is the example
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