# Test for Math Formulas v02

## Introduction

This test file is made to test math formulas with the MarkIt markup language.

## Greek letters

All greek letters are available, majuscules and minuscules are available:

### Minuscules

`\$%alpha\$` :
$\alpha$
`\$%beta\$` :
$\beta$
`\$%gamma\$` :
$\gamma$
`\$%delta\$` :
$\delta$
`\$%epsilon\$` :
$\epsilon$
`\$%zeta\$` :
$\zeta$
`\$%eta\$` :
$\eta$
`\$%theta\$` :
$\theta$
`\$%iota\$` :
$\iota$
`\$%kappa\$` :
$\kappa$
`\$%lambda\$` :
$\lambda$
`\$%mu\$` :
$\mu$
`\$%nu\$` :
$\nu$
`\$%xi\$` :
$\xi$
`\$%omicron\$` :
$ο$
`\$%pi\$` :
$\pi$
`\$%rho\$` :
$\rho$
`\$%sigma\$` :
$\sigma$
`\$%tau\$` :
$\tau$
`\$%upsilon\$` :
$\upsilon$
`\$%phi\$` :
$\phi$
`\$%khi\$` :
$\chi$
`\$%psi\$` :
$\psi$
`\$%omega\$` :
$\omega$

### Majuscules

`\$%ALPHA\$` :
$Α$
`\$%BETA\$` :
$Β$
`\$%GAMMA\$` :
$\Gamma$
`\$%DELTA\$` :
$\Delta$
`\$%EPSILON\$` :
$Ε$
`\$%ZETA\$` :
$Ζ$
`\$%ETA\$` :
$Η$
`\$%THETA\$` :
$\Theta$
`\$%IOTA\$` :
$Ι$
`\$%KAPPA\$` :
$Κ$
`\$%LAMBDA\$` :
$\Lambda$
`\$%MU\$` :
$Μ$
`\$%NU\$` :
$Ν$
`\$%XI\$` :
$\Xi$
`\$%OMICRON\$` :
$Ο$
`\$%PI\$` :
$\Pi$
`\$%RHO\$` :
$Ρ$
`\$%SIGMA\$` :
$\Sigma$
`\$%TAU\$` :
$Τ$
`\$%UPSILON\$` :
$Υ$
`\$%PHI\$` :
$\Phi$
`\$%KHI\$` :
$Χ$
`\$%PSI\$` :
$\Psi$
`\$%OMEGA\$` :
$\Omega$

## Basic operations

For additions simply use the common sign :

`\$a + b\$` :
$a+b$

#### Blocks

Several additions in block mode with spaces in their writing

$a+b$ $1+c=2+\alpha$ $\beta +\Gamma +\lambda$

The same additions in block mode without spaces in their writing

$a+b$ $1+c=2+\alpha$ $\beta +\Gamma +\lambda$

#### Inlines

The same operations in Inline mode

The first $a+b$, the second,$1+c=2+\alpha$, the third $\beta +\Gamma +\lambda$.

### Substraction

For substraction simply use the common sign :

`\$a - b\$` :
$a-b$

#### Blocks

Several substractions in block mode with spaces in their writing

$a-b$ $1-c=2-\alpha$ $\beta -\Gamma -\lambda$

The same substractions in block mode without spaces in their writing

$a-b$ $1-c=2-\alpha$ $\beta -\Gamma -\lambda$

#### Inlines

The same operations in Inline mode

The first $a-b$, the second,$1-c=2-\alpha$, the third $\beta -\Gamma -\lambda$.

### Multiplication

For multiplication you can use several signs :

`\$a * b\$` :
$a\phantom{\rule{0.1em}{0ex}}b$
`\$a times b\$` :
$a×b$
`\$a cdot b\$` :
$a\cdot b$

Every multiplication must be described and no spaces are allowed. If you wish to replace a multiplication sign by a space, use a `*` and use the appropriate document style option.

#### Blocks

Several multiplications in block mode with spaces in their writing

$a\phantom{\rule{0.1em}{0ex}}b$ $a\cdot b$ $a×b$ $1\phantom{\rule{0.1em}{0ex}}c=2\phantom{\rule{0.1em}{0ex}}\alpha$ $1\cdot c=2\cdot \alpha$ $1×c=2×\alpha$ $\beta \phantom{\rule{0.1em}{0ex}}\Gamma \phantom{\rule{0.1em}{0ex}}\lambda$ $\beta \cdot \Gamma \cdot \lambda$ $\beta ×\Gamma ×\lambda$

The same multiplications in block mode without spaces in their writing

$a\phantom{\rule{0.1em}{0ex}}b$ $1\phantom{\rule{0.1em}{0ex}}c=2\phantom{\rule{0.1em}{0ex}}\alpha$ $\beta \phantom{\rule{0.1em}{0ex}}\Gamma \phantom{\rule{0.1em}{0ex}}\lambda$

#### Inlines

The same operations in Inline mode

$a\phantom{\rule{0.1em}{0ex}}b$ and $a\cdot b$ and $a×b$ and $1\phantom{\rule{0.1em}{0ex}}c=2\phantom{\rule{0.1em}{0ex}}\alpha$ and $1\cdot c=2\cdot \alpha$ and $1×c=2×\alpha$ and $\beta \phantom{\rule{0.1em}{0ex}}\Gamma \phantom{\rule{0.1em}{0ex}}\lambda$ and $\beta \cdot \Gamma \cdot \lambda$ and $\beta ×\Gamma ×\lambda$

### Division and fraction

For division you can use several signs :

`\$a / b\$` :
$a/b$
`\$a over b\$` :
$\frac{a}{b}$

If you wish to replace a multiplication sign by a vertical division, use the appropriate document style option.

#### Block mode

$a/b$ $\frac{a}{b}$ $1/c=2/\alpha$ $\frac{1}{c}=\frac{2}{\alpha }$ $\beta /\Gamma /\lambda$ $\frac{\frac{\beta }{\Gamma }}{\lambda }$ $\beta /\Gamma /\lambda /Α=1$ $\frac{\frac{\frac{\beta }{\Gamma }}{\lambda }}{Α}=1$

#### Inline mode

The same operations in Inline mode

$a/b$ and $\frac{a}{b}$ and $1/c=2/\alpha$ and $\frac{1}{c}=\frac{2}{\alpha }$ and $\beta /\Gamma /\lambda$ and $\frac{\frac{\beta }{\Gamma }}{\lambda }$ and $\beta /\Gamma /\lambda /Α=1$ and $\frac{\frac{\frac{\beta }{\Gamma }}{\lambda }}{Α}=1$.

### Exponents

For exponent you can use the `^` sign :

`\$a ^ b\$` :
${a}^{b}$

#### Block mode

${a}^{n}×{a}^{m}={a}^{n+m}$ ${\left({a}^{n}\right)}^{m}={a}^{n\phantom{\rule{0.1em}{0ex}}m}$ ${a}^{-n}={\left(\frac{1}{a}\right)}^{n}$

#### Inline mode

${a}^{n}×{a}^{m}={a}^{n+m}$ ${\left({a}^{n}\right)}^{m}={a}^{n\phantom{\rule{0.1em}{0ex}}m}$ ${a}^{-n}={\left(\frac{1}{a}\right)}^{n}$

### Parenthesis and braces

It is possible to set priority to some parts by enclosing them with braces `{ … }`. These braces won't be shown in the document.

You can also set prioity by enclosing these parts with parenthesis `( ... )`. These parenthesis will be shown in the document.

This behaviour is the same with all functions and arrangements.

### Functions

All standards functions are supported. You can enclose the content either with braces `{ … }` (not shown) either with parenthesis (adjusted to the height of the content)

`sin{x} sin(x)` :

$sinx$ $sin\left(x\right)$

`cos{x} cos(x)` :

$cosx$ $cos\left(x\right)$

`tan{x} tan(x)` :

$tanx$ $tan\left(x\right)$

`asin{x} asin(x)` :

$asinx$ $asin\left(x\right)$

`acos{x} acos(x)` :

$acosx$ $acos\left(x\right)$

`atan{x} atan(x)` :

$atanx$ $atan\left(x\right)$

### Integrals

#### Basis

You can show integrals by using the main command `int`. Tou can complete the description of the integral by using the `from`, `to` and `d` commands.

`from a` :

Set the lower bound of the integral as a.

`to b` :

Set the upper bound of the integral as b.

`dx` :

Set the derivation variable as x (no space is required).

Bellow several kind of integral desciptions:

`int x` :
$\int x$ $\int x$
`int dx x` :
$\int x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$ $\int x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$
`int from 1 d{x} x` :
${\int }_{1}x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$ ${\int }_{1}x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$
`int from u to v dx x` :
${\int }_{u}^{v}x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$ ${\int }_{u}^{v}x\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$
`int from a to {b+c} d{x} {f(x)+g(x)}` :
${\int }_{a}^{b+c}f\left(x\right)+g\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$ ${\int }_{a}^{b+c}f\left(x\right)+g\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$

#### Integration brackets

It is possible to use integration brackets with a lower and upper bound to give more details to the integrations calculation.

${\int }_{0}^{2}\left(4-{x}^{2}\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x=4\phantom{\rule{0.1em}{0ex}}{\int }_{0}^{2}1\mathrm{d}\phantom{\rule{0.0em}{0ex}}x-{\int }_{0}^{2}{x}^{2}\mathrm{d}\phantom{\rule{0.0em}{0ex}}x=4\phantom{\rule{0.1em}{0ex}}{\left[x\right]}_{0}^{2}-{\left[\frac{1}{3}\phantom{\rule{0.1em}{0ex}}{x}^{3}\right]}_{0}^{2}$

\$ lbrace 1 over b rbrace \$

$\left[\frac{1}{b}\right]$ $|\frac{1}{b}|$ $||\frac{1}{b}||$

\$ lparent 1 over b rparent \$

### Limits

You can show limits of functions by using the main command `lim`. Tou can complete the description of the limit by using the `as` and `approach` commands together.

`as a approach b` Set the variable and the limit it reach.

Bellow several kind of limits descriptions:

`lim x` :
$limx$ $limx$
`lim as x approach 0 sin(x)` :
$\underset{x\to 0}{lim}sin\left(x\right)$ $\underset{x\to 0}{lim}sin\left(x\right)$
`lim as x approach 0 {1 over x}` :
$\underset{x\to 0}{lim}\frac{1}{x}$ $\underset{x\to 0}{lim}\frac{1}{x}$
`lim as x approach infinity {1 over x}` :
$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}$ $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}$

### Matrix and Array

#### Array

To make an array, you can use the `array` function an describe each array cell by separating them with `#`. To create another line, use `##` and restart describing the new line.

An array is a simple tabular representation without parenthesis or lines.

`\$array(a # b # c ## d # e # f ## g # h # i)\$` :
$\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}$ $\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}$

#### Matrix

Matrix can be created the same way than arrays by using the `matrix` function instead of `array`. A matrix is an array representation surrounded by parenthesis

`\$matrix(a # b # c ## d # e # f ## g # h # i)\$` :
$\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$ $\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$

#### Determinant

You can also describe a determinant by using the `det` function. A determinant is an array representation surrounded by vertical lines.

`\$det(a # b # c ## d # e # f ## g # h # i)\$` :
$|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|$ $|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|$
$|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}|=a\phantom{\rule{0.1em}{0ex}}|\begin{array}{cc}e& f\\ h& i\end{array}|-d\phantom{\rule{0.1em}{0ex}}|\begin{array}{cc}b& c\\ h& i\end{array}|+g\phantom{\rule{0.1em}{0ex}}|\begin{array}{cc}b& c\\ e& f\end{array}|$

## Exemples of formulas

### Trigonometry

#### Exact values

$sin\left(\frac{\pi }{6}\right)=\frac{1}{2}$ $cos\left(\frac{\pi }{6}\right)=\frac{\sqrt{\left(3\right)}}{2}$ $tan\left(\frac{\pi }{6}\right)=\frac{\sqrt{\left(3\right)}}{3}$ $sin\left(\frac{\pi }{4}\right)=\frac{\sqrt{\left(2\right)}}{2}$ $cos\left(\frac{\pi }{4}\right)=\frac{\sqrt{\left(2\right)}}{2}$ $tan\left(\frac{\pi }{4}\right)=1$ $sin\left(\frac{\pi }{3}\right)=\frac{\sqrt{\left(3\right)}}{2}$ $cos\left(\frac{\pi }{3}\right)=\frac{1}{2}$ $tan\left(\frac{\pi }{3}\right)=\sqrt{\left(3\right)}$

#### Sum formulas

$sin\left(a+b\right)=sin\left(a\right)\phantom{\rule{0.1em}{0ex}}cos\left(b\right)+sin\left(b\right)\phantom{\rule{0.1em}{0ex}}cos\left(a\right)$ $sin\left(a+b\right)=sina×cosb+sinb×cosa$ $cos\left(a+b\right)=cosa×cosb-sina×sinb$ $tan\left(a+b\right)=\frac{tana+tanb}{1-tana×tanb}$

### Circumference of a circle

#### Block mode

The formula for the circumference of a circle is :

$C=2\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}r$

#### Inline mode

The formula for the circumference of a circle is : $C=2\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}r$.

### Volume of a sphere

#### Block mode

The formula for the volume of a sphere is :

$V=\frac{4}{3}\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}{r}^{3}$

#### Inline mode

The formula for the volume of a sphere is : $V=\frac{4}{3}\phantom{\rule{0.1em}{0ex}}\pi \phantom{\rule{0.1em}{0ex}}{r}^{3}$.

### Slope of a line

#### Block mode

$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

#### Inline mode

The same formula in inline mode : $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$.

$y=a\phantom{\rule{0.1em}{0ex}}{x}^{2}+b\phantom{\rule{0.1em}{0ex}}x+c$

#### Block mode

$x=\frac{-b±\sqrt{{b}^{2}-4\phantom{\rule{0.1em}{0ex}}a\phantom{\rule{0.1em}{0ex}}c}}{2\phantom{\rule{0.1em}{0ex}}a}$

#### Inline mode

The same formula in inline mode : $x=\frac{-b±\sqrt{{b}^{2}-4\phantom{\rule{0.1em}{0ex}}a\phantom{\rule{0.1em}{0ex}}c}}{2\phantom{\rule{0.1em}{0ex}}a}$.

## Derivatives

### Basic formula

$f\left({x}_{0}\right)\text{'}=\underset{x\to {x}_{0}}{lim}\frac{f\left(x\right)-f\left({x}_{0}\right)}{x-{x}_{0}}$ $\left(u+v\right)\text{'}=u\text{'}+v\text{'}$ $\left(u×v\right)\text{'}=u\text{'}×v+u×v\text{'}$

## Limits

### Block mode

$\underset{x\to +\mathrm{\infty }}{lim}\frac{sinx}{x}=0$ $\underset{x\to 0}{lim}\frac{sinx}{x}=1$

### Inline mode

The first : $\underset{x\to +\mathrm{\infty }}{lim}\frac{sinx}{x}=0$, the second : $\underset{x\to 0}{lim}\frac{sinx}{x}=1$.

## Integrals

### Basic rules

#### Block mode

${\int }_{a}^{b}\left(f\left(x\right)+g\left(x\right)\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x={\int }_{a}^{b}f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x+{\int }_{a}^{b}g\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$ ${\int }_{a}^{b}k\cdot f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x=k\cdot {\int }_{a}^{b}f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$

#### Inline mode

The first formula : ${\int }_{a}^{b}\left(f\left(x\right)+g\left(x\right)\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x={\int }_{a}^{b}f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x+{\int }_{a}^{b}g\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$, the second : ${\int }_{a}^{b}k\cdot f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x=k\cdot {\int }_{a}^{b}f\left(x\right)\mathrm{d}\phantom{\rule{0.0em}{0ex}}x$.

## Matrixes

### Bases

#### Block mode

$A=\left(\begin{array}{ccc}{a}_{\mathrm{11}}& {a}_{\mathrm{12}}& {a}_{\mathrm{13}}\\ {a}_{\mathrm{21}}& {a}_{\mathrm{22}}& {a}_{\mathrm{23}}\\ {a}_{\mathrm{31}}& {a}_{\mathrm{32}}& {a}_{\mathrm{33}}\end{array}\right)$

$B=\left(\begin{array}{ccc}{b}_{\mathrm{11}}& {b}_{\mathrm{12}}& {b}_{\mathrm{13}}\\ {b}_{\mathrm{21}}& {b}_{\mathrm{22}}& {b}_{\mathrm{23}}\\ {b}_{\mathrm{31}}& {b}_{\mathrm{32}}& {b}_{\mathrm{33}}\end{array}\right)$

#### Inline mode

The first : $A=\left(\begin{array}{ccc}{a}_{\mathrm{11}}& {a}_{\mathrm{12}}& {a}_{\mathrm{13}}\\ {a}_{\mathrm{21}}& {a}_{\mathrm{22}}& {a}_{\mathrm{23}}\\ {a}_{\mathrm{31}}& {a}_{\mathrm{32}}& {a}_{\mathrm{33}}\end{array}\right)$, the second : $B=\left(\begin{array}{ccc}{b}_{\mathrm{11}}& {b}_{\mathrm{12}}& {b}_{\mathrm{13}}\\ {b}_{\mathrm{21}}& {b}_{\mathrm{22}}& {b}_{\mathrm{23}}\\ {b}_{\mathrm{31}}& {b}_{\mathrm{32}}& {b}_{\mathrm{33}}\end{array}\right)$.

$A+B=\left(\begin{array}{ccc}{a}_{\mathrm{11}}+{b}_{\mathrm{11}}& {a}_{\mathrm{12}}+{b}_{\mathrm{12}}& {a}_{\mathrm{13}}+{b}_{\mathrm{13}}\\ {a}_{\mathrm{21}}+{b}_{\mathrm{21}}& {a}_{\mathrm{22}}+{b}_{\mathrm{22}}& {a}_{\mathrm{23}}+{b}_{\mathrm{23}}\\ {a}_{\mathrm{31}}+{b}_{\mathrm{31}}& {a}_{\mathrm{32}}+{b}_{\mathrm{32}}& {a}_{\mathrm{33}}+{b}_{\mathrm{33}}\end{array}\right)$
$A=|\begin{array}{ccc}{a}_{\mathrm{11}}& {a}_{\mathrm{12}}& {a}_{\mathrm{13}}\\ {a}_{\mathrm{21}}& {a}_{\mathrm{22}}& {a}_{\mathrm{23}}\\ {a}_{\mathrm{31}}& {a}_{\mathrm{32}}& {a}_{\mathrm{33}}\end{array}|$
$a\parallel b$ $a⟂b$
$a\cup b$ $a\cap b$ $a\in b$ $a\notin b$