# Formula Repository

This page is a sub-page of the page on our Learning Object Repository

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WordPress Latex support

AMS TeX Collections: Distribution

Latex documentation

Latex math symbols

Examples:

$e^{\i \pi} + 1 = 0$
$sin{x} + 1 = 0$
$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$
$A \textbf{bold \textit{Hello \LaTeX}} to start!$

This is an in-text $z=x+y$ math equation

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Generic function notation:

${f : \mathcal X \, \rightarrow \, \mathcal Y}$,

${\mathcal X \ni x \, \mapsto \, f(x )\in \mathcal Y}$,

$\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y$,

${x \, \longmapsto \, f(x)}$,

${{\mathcal X \, \stackrel {f} {\longrightarrow} \, \mathcal Y \:}\atop {\: x \, \longmapsto \, f(x) } } {\,}$.

Inverse fourier transform:

$f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x} \,d\xi$,

Binary composition:

${\mathcal S \times \mathcal S \ni (x, y) \, \mapsto \, x \ast y \in \mathcal S}$,

$z = x^{y^2}$,

${\oplus \atop {x \in \mathcal{P} } }$,

${\bigoplus \atop {x \in \mathcal{P} } } \mathbf {R}$,

$\mathrm{supp} f = \{ x \in X \, : \, f(x) \neq 0 \}$,

$R \subseteq \mathcal X \times \mathcal X$

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$x \pmod a$
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$\varphi ( x \ast y) = \varphi (x) {\ast}' \varphi (y), \, \forall x, y \in \mathcal G$

$\varphi ( x \ast y) = \varphi (x) {\ast}' \varphi (y), \, \forall x, y \in \mathcal G$

Community: $C = (A, P)$

Activities: $A = \{ A_1, A_2, \ldots, A_{n} \}$

Participators: $P = \{ P_1, P_2, \ldots, P_{m} \}$

Possible participator grouping (of an activity $A_{i}$):

$\mathrm{P_{GI}}(A_{i}) = \sum_{k=1}^m (1-P_{k})(P_{k} \in A_{i}) A_{i}$

Actual participator grouping (of an activity $A_{i}$):

$\mathrm{G_{I}}(A_{i}) = \sum_{j=1}^m \sum_{s \in {\prod_{}^j}m} P_{s}(P_{s} \in A_{i}) A_{i}$

Possible activity grouping (of a participator $P_{k}$):

$\mathrm{P_{GI}}(P_{k}) = \sum_{i=1}^n (1-A_{i})(A_{i} \in P_{k}) P_{k}$

Actual activity grouping (of a participator $P_{k}$):

$\mathrm{G_{I}}(P_{k}) = \sum_{i=1}^n \sum_{s \in {\prod_{}^i}n} A_{s}(A_{s} \in P_{k}) P_{k}$

Cardinality formulas:

Total number of participators:

$|P| = \sum_{k=1}^n (-1)^{k-1} \sum_{s \in {\prod_{}^k}n}|\cap P_{A_{s}}| = m$

where

$\cap P_{A_{(1,2)}} = P_{A_1} \cap P_{A_2}$

Total number of activities:

$|A| = \sum_{k=1}^m (-1)^{k-1} \sum_{s \in {\prod_{}^k}m}|\cap A_{P_{s}}| = n$

where

$\cap A_{P_{(1,2)}} = A_{P_1} \cap A_{P_2}$

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checking some expressions:

$\mathbf N \times \mathbf N \, \stackrel {+} {\longrightarrow} \, \mathbf N$

${\mathbf N \times \mathbf N \ni (x, y) \, \mapsto \, x + y \in \mathbf N}$

$dS_{cenario} = \sum_{T_{ensions}}{\frac{\partial{S_{cenario}}}{\partial{T_{ension}}}} dT_{ension}$

$\ldots$

$dS_{cenario} = \bigoplus_{T_{ensions}}{\frac{\partial{S_{cenario}}}{\partial{T_{ension}}}} dT_{ension}$

$dI_{ssue} = \sum_{T_{ensions}}{\frac{\partial{I_{ssue}}}{\partial{T_{ension}}}} dT_{ension}$

$dI_{ssue} = \bigoplus_{T_{ensions}}{\frac{\partial{I_{ssue}}}{\partial{T_{ension}}}} dT_{ension}$

$\int\limits_{a}^{b} f(x) \, dx$

$\int_a^b f(x) \, dx$

$f = {\sum\limits_{m \in M}^{ \text {} }}{f_m} m \, , \text{ and } \, g = {\sum\limits_{m \in m}^{ \text {} }}{g_m} m \, ,$

$R\{X\} \times R\{X\} \ni (f, g) \, \mapsto \, f \ast g \in {R\{X\}\, , \text{where} \, (f \ast g)(m) \stackrel {\mathrm{def}}{=}{\sum\limits_{m'm'' = m}^{ \text {} }f(m')g(m'')}}$

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$\mathcal F \, \stackrel {\chi_{apple}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{pear}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{banana}} {\longrightarrow} \, \mathbf N$

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A function $f$ from $\mathbf{R}^2$ to $\mathbf{R}$ can be described by:

${{\mathbf{R}^2 \, \stackrel {f} {\longrightarrow} \, \mathbf{R} \:}\atop {\: (x,y) \, \longmapsto \, f(x,y) } } {\,}.$

The differential $df$ of the function $f$ at the point $(a,b) \in \mathbf{R}^2$ is given by:

$df_{(a,b)} = \frac{\partial f}{\partial x}_{(a,b)} dx + \frac{\partial f}{\partial y}_{(a,b)} dy.$

The equation of the level curve ($\, f = const \,$) of the function $f$ at the point $(a,b)$ is given by:

$f(x,y)=f(a,b).$

The equation of the tangent to the level curve of the function $f$ at the point $(a,b)$ is given by:

$\frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b) = 0.$

The gradient of the function $f$ is the function ${\triangledown f}$ defined by:

${{\mathbf{R}^2 \, \stackrel {\triangledown f} {\longrightarrow} \, \mathbf{R}^2 \:}\atop {\: (x,y) \, \longmapsto \, (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}) } } {\,}.$

Hence the value of ${\triangledown f}$ at the point $(x,y)$ is:

${\triangledown f}_{(x,y)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}).$

The value of ${\triangledown f}_{(x,y)}$ at the point $x=a,y=b$ is obtained by evaluating the function ${\triangledown f}_{(x,y)}$ at the point $(a,b):$

${\triangledown f}_{(a,b)} = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y})_{(a,b)}.$

Hence we see that the gradient ${\triangledown f}_{(a,b)}$ of the function $f$ at the point $(a,b)$ is perpendicular to the tangent of the level curve of the function $f$ at the point $(a,b).$

Flying carpets and level surfaces

The “flying carpet equation” of the function $f$ is given by:

$z = f(x, y).$

The level surface equation of the function $f$ is given by:

$g(x,y,z) \stackrel {\mathrm{def}}{=} f(x,y) - z = 0.$

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$\mathrm{supp} f \, \stackrel {\mathrm{def}}{=} \{ x \in X \, : \, f(x) \neq 0 \} \, .$
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In the animation below, the “input” function $f$ is given by:

$f(x, y) = \frac{1}{3} (x^2 + 2 y^2) + \frac {3}{4}$

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$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

$\begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{x(p)}$

$df(x(u))_{f(x(p))} = f'(x(u))_{x(p)} x'(u)_p du = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}_{x(p)} \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \end{pmatrix}_p \begin{pmatrix} du_1 \\ du_2 \end{pmatrix}$

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