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| LaTeX Note 007                                  jodys@helluin.org |
| Equations                                                3/2/2014 |
|                                                             v0.05 |
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Let's just dive in and pick it up as we go. The inline notation to 
start math mode is 

$ ... $

anything in between that will get typeset as an equation.

$ a + b = c $

To make a fraction

\frac{a}{b}

If you need a percent sign, you need to escape it; LaTeX uses % as a
comment delimiter.

\%

To add subscripts

$ a_1 + b_2 = c_3 $

Superscripts

$ a^2 + b^2 = c^2 $

If you need more than one expression, or more than one digit for that
matter, in your subscripts or superscripts surround them with braces

$ a^{n-1} + b^27 = n $

If you would like to be able to typeset the notation for the statement 
"x is an element of the set of real numbers", you can use

$x \in \mathbb{R}$

The \mathbb (blackboard) uses the amsfont package which is called in 
the amsmath package

Square roots are easy!
$\sqrt{x}$

So are cube roots (and n!)
$\sqrt[n]{x}$

Topology Stuff

 Closure
	In your preamble define
	\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}

	Then $\overbar{\mathbb{R}}$ is the closure of the reals.
	
 Interior
	$\mathring{A}$

 Boundary
	$\partial{S}$
Analysis Stuff

Integral Spacing

Be sure to do integrals with the proper spacing using a \, to add just
a bit of space after the dt.

$\in_0^1 f(t) \, dt$

Function names when using function names in equations be sure to ask
LaTeX to typset them correctly by prefacing the function name with \

$\sin(x)$ looks substantially better than $sin(x)$ 


Misc. Symbols

ellipsis		\cdot 
right arrow  		\rightarrow 
implication arrow	\Rightarrow

Page Elements

Beyond the basics of setting the mathematical type, which is mostly just
a matter of getting to know TeX, there is also the issue of how your 
mathematics will fit into the rest of the page. Equations can be 
inline, by using $$ as shown above, but if you need to write more than
one line, then you will want to use different markup.

Aligned equations

Without numbering
\begin{align*}
	x &= y^2\\
	\sqrt{x} &= y
\end{align*}

With numbering (a (1) after the first line, (2) after the second)
\begin{align}
	x &= y^2\\
	\sqrt{x} &= y
\end{align}

With numbering on one line only (\notag after end of line you 
 don't want numbered)
\begin{align}
	x &= y^2 \notag\\
	\sqrt{x} &= y
\end{align}

Piecewise Function 

The key to this is the combination of the \left\{ (remember the 
backslash before the {) and the array enviroment.

\begin{align*} 
	f(x, y)= \left\{ 
	\begin{array}{rr}
		\frac{4 x^2 y}{x^4 + y^2}&: x^2+y^2 \neq 0\\
		0&: x^2+y^2=0
	\end{array}
	\right.
\end{align*}