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}

The Tension-gradient of a Scenario:

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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GRADIENTS

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

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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