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

$%alpha$ : 

$$\alpha$$

$%beta$ : 

$$\beta$$

$%gamma$ : 

$$\gamma$$

$%delta$ : 

$$\delta$$

$%epsilon$ : 

$$\epsilon$$

$%zeta$ : 

$$\zeta$$

$%eta$ : 

$$\eta$$

$%theta$ : 

$$\theta$$

$%iota$ : 

$$\iota$$

$%kappa$ : 

$$\kappa$$

$%lambda$ : 

$$\lambda$$

$%mu$ : 

$$\mu$$

$%nu$ : 

$$\nu$$

$%xi$ : 

$$\xi$$

$%omicron$ : 

$$o$$

$%pi$ : 

$$\pi$$

$%rho$ : 

$$\rho$$

$%sigma$ : 

$$\sigma$$

$%tau$ : 

$$\tau$$

$%upsilon$ : 

$$\upsilon$$

$%phi$ : 

$$\phi$$

$%khi$ : 

$$\chi$$

$%psi$ : 

$$\psi$$

$%omega$ : 

$$\omega$$

Majuscules

$%ALPHA$ : 

$${\rm A}$$

$%BETA$ : 

$${\rm B}$$

$%GAMMA$ : 

$$\Gamma$$

$%DELTA$ : 

$$\Delta$$

$%EPSILON$ : 

$${\rm E}$$

$%ZETA$ : 

$${\rm Z}$$

$%ETA$ : 

$${\rm H}$$

$%THETA$ : 

$$\Theta$$

$%IOTA$ : 

$${\rm I}$$

$%KAPPA$ : 

$${\rm K}$$

$%LAMBDA$ : 

$$\Lambda$$

$%MU$ : 

$${\rm M}$$

$%NU$ : 

$${\rm N}$$

$%XI$ : 

$$\Xi$$

$%OMICRON$ : 

$${\rm O}$$

$%PI$ : 

$$\Pi$$

$%RHO$ : 

$${\rm P}$$

$%SIGMA$ : 

$$\Sigma$$

$%TAU$ : 

$${\rm T}$$

$%UPSILON$ : 

$${\rm Y}$$

$%PHI$ : 

$$\Phi$$

$%KHI$ : 

$${\rm X}$$

$%PSI$ : 

$$\Psi$$

$%OMEGA$ : 

$$\Omega$$

Basic operations

Addition

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 :

$a * b$ : 

$$ab$$

$a times b$ : 

$$a\times b$$

$a cdot b$ : 

$$a\cdot b$$

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

$$ab$$ $$a\cdot b$$ $$a\times b$$ $$1c=2\alpha$$ $$1\cdot c=2\cdot \alpha$$ $$1\times c=2\times \alpha$$ $$\beta \Gamma \lambda$$ $$\beta \cdot \Gamma \cdot \lambda$$ $$\beta \times \Gamma \times \lambda$$

The same multiplications in block mode without spaces in their writing

$$ab$$ $$1c=2\alpha$$ $$\beta \Gamma \lambda$$

Inlines

The same operations in Inline mode

$ab$ and $a\cdot b$ and $a\times b$ and $1c=2\alpha$ and $1\cdot c=2\cdot \alpha$ and $1\times c=2\times \alpha$ and $\beta \Gamma \lambda$ and $\beta \cdot \Gamma \cdot \lambda$ and $\beta \times \Gamma \times \lambda$

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

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

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 $\beta /\Gamma /\lambda /{\rm A}=1$ and $\frac{\frac{\frac{\beta }{\Gamma }}{\lambda }}{{\rm A}}=1$.

Exponents

For exponent you can use the ^ sign :

$a ^ b$ : 

$$a^b$$

Block mode

$$a^n\times a^m=a^{n+m}$$ $${\left(a^n\right)}^m=a^{nm}$$ $$a^{-n}={\left(\frac{1}{a}\right)}^n$$

Inline mode

$a^n\times a^m=a^{n+m}$ ${\left(a^n\right)}^m=a^{nm}$ $a^{-n}={\left(\frac{1}{a}\right)}^n$

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 : 

$$\int x\mathrm{d}x$$ $$\int x\mathrm{d}x$$

int from 1 d{x} x : 

$${\int }_{1}x\mathrm{d}x$$ $${\int }_{1}x\mathrm{d}x$$

int from u to v dx x : 

$${\int }_u^{v}x\mathrm{d}x$$ $${\int }_u^{v}x\mathrm{d}x$$

int from a to {b+c} d{x} {f(x)+g(x)} : 

$${\int }_a^{b+c}f\left(x\right)+g\left(x\right)\mathrm{d}x$$ $${\int }_a^{b+c}f\left(x\right)+g\left(x\right)\mathrm{d}x$$

Integration brackets

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

$${\int }_0^2(4-x^2)\mathrm{d}x=4{\int }_0^{2}1\mathrm{d}x-{\int }_0^2{x}^2\mathrm{d}x=4{\left[x\right]}_0^2-{\left[\frac{1}{3}{x}^3\right]}_0^2$$

$ lbrace 1 over b rbrace $

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

$ 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.

$det(a # b # c ## d # e # f ## g # h # i)$ : 

$$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$$ $$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$$

$$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-d\left|\begin{array}{cc}b& c\\ h& i\end{array}\right|+g\left|\begin{array}{cc}b& c\\ e& f\end{array}\right|$$

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

$$sin(a+b)=sin\left(a\right)cos\left(b\right)+sin\left(b\right)cos\left(a\right)$$ $$sin(a+b)=sina\times cosb+sinb\times cosa$$ $$cos(a+b)=cosa\times cosb-sina\times sinb$$ $$tan(a+b)=\frac{tana+tanb}{1-tana\times tanb}$$

Circumference of a circle

Block mode

The formula for the circumference of a circle is :

$$C=2\pi r$$

Inline mode

The formula for the circumference of a circle is : $C=2\pi r$.

Volume of a sphere

Block mode

The formula for the volume of a sphere is :

$$V=\frac{4}{3}\pi r^3$$

Inline mode

The formula for the volume of a sphere is : $V=\frac{4}{3}\pi r^3$.

Slope of a line

Block mode

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Inline mode

The same formula in inline mode : $m=\frac{y_2-y_1}{x_2-x_1}$.

Quadratic equation

$$y=a{x}^2+bx+c$$

Quadratic formula

Block mode

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Inline mode

The same formula in inline mode : $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

Derivatives

Basic formula

$$f\left(x_0\right)\text{'}=\underset{x\to x_0}{lim}\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}$$ $$(u+v)\text{'}=u\text{'}+v\text{'}$$ $$(u\times v)\text{'}=u\text{'}\times v+u\times v\text{'}$$

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

$${\int }_a^b(f\left(x\right)+g\left(x\right))\mathrm{d}x={\int }_a^{b}f\left(x\right)\mathrm{d}x+{\int }_a^{b}g\left(x\right)\mathrm{d}x$$ $${\int }_a^{b}k\cdot f\left(x\right)\mathrm{d}x=k\cdot {\int }_a^{b}f\left(x\right)\mathrm{d}x$$

Inline mode

The first formula : ${\int }_a^b(f\left(x\right)+g\left(x\right))\mathrm{d}x={\int }_a^{b}f\left(x\right)\mathrm{d}x+{\int }_a^{b}g\left(x\right)\mathrm{d}x$, the second : ${\int }_a^{b}k\cdot f\left(x\right)\mathrm{d}x=k\cdot {\int }_a^{b}f\left(x\right)\mathrm{d}x$.

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

Additions

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

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

Gometric symbols

$$a\parallel b$$ $$a⟂b$$

Logic

$$a\cup b$$ $$a\cap b$$ $$a\in b$$ $$a\notin b$$