Difference between revisions of "User:Mingli/Wikitext"
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} | } | ||
</graphviz> | </graphviz> | ||
| + | |||
| + | ==Math example== | ||
| + | === The Lorenz Equations === | ||
| + | <source lang='html5'> | ||
| + | <math>\begin{align} | ||
| + | \dot{x} & = \sigma(y-x) \\ | ||
| + | \dot{y} & = \rho x - y - xz \\ | ||
| + | \dot{z} & = -\beta z + xy | ||
| + | \end{align}</math> | ||
| + | </source> | ||
| + | |||
| + | <math>\begin{align} | ||
| + | \dot{x} & = \sigma(y-x) \\ | ||
| + | \dot{y} & = \rho x - y - xz \\ | ||
| + | \dot{z} & = -\beta z + xy | ||
| + | \end{align}</math> | ||
| + | |||
| + | === The Cauchy-Schwarz Inequality === | ||
| + | <source lang='html5'> | ||
| + | <math>\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)</math> | ||
| + | </source> | ||
| + | <math>\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)</math> | ||
| + | |||
| + | === A Cross Product Formula === | ||
| + | <source lang='html5'> | ||
| + | <math>\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} | ||
| + | \mathbf{i} & \mathbf{j} & \mathbf{k} \\ | ||
| + | \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ | ||
| + | \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 | ||
| + | \end{vmatrix}</math> | ||
| + | </source> | ||
| + | |||
| + | <math>\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} | ||
| + | \mathbf{i} & \mathbf{j} & \mathbf{k} \\ | ||
| + | \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ | ||
| + | \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 | ||
| + | \end{vmatrix}</math> | ||
| + | |||
| + | === The probability of getting k heads when flipping n coins is === | ||
| + | <source lang='html5'> | ||
| + | <math>P(E) = {n \choose k} p^k (1-p)^{ n-k}</math> | ||
| + | </source> | ||
| + | |||
| + | <math>P(E) = {n \choose k} p^k (1-p)^{ n-k}</math> | ||
| + | |||
| + | === An Identity of Ramanujan === | ||
| + | <source lang='html5'> | ||
| + | <math>\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = | ||
| + | 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} | ||
| + | {1+\frac{e^{-8\pi}} {1+\ldots} } } }</math> | ||
| + | </source> | ||
| + | |||
| + | <math>\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = | ||
| + | 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} | ||
| + | {1+\frac{e^{-8\pi}} {1+\ldots} } } }</math> | ||
| + | |||
| + | === A Rogers-Ramanujan Identity === | ||
| + | <source lang='html5'> | ||
| + | <math>1 + \frac{q^2}{(1-q)} + \frac{q^6}{(1-q)(1-q^2)} + \cdots | ||
| + | = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, | ||
| + | \quad\quad for\,|q|<1.</math> | ||
| + | </source> | ||
| + | |||
| + | <math>1 + \frac{q^2}{(1-q)} + \frac{q^6}{(1-q)(1-q^2)} + \cdots | ||
| + | = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, | ||
| + | \quad\quad for\,|q|<1.</math> | ||
| + | |||
| + | === Maxwell’s Equations === | ||
| + | <source lang='html5'> | ||
| + | <math>\begin{align} | ||
| + | \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ | ||
| + | \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ | ||
| + | \nabla \cdot \vec{\mathbf{B}} & = 0 | ||
| + | \end{align}</math> | ||
| + | </source> | ||
| + | |||
| + | <math>\begin{align} | ||
| + | \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ | ||
| + | \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ | ||
| + | \nabla \cdot \vec{\mathbf{B}} & = 0 | ||
| + | \end{align}</math> | ||
Revision as of 03:53, 29 August 2018
Contents
[hide]Graphviz example
Graph image source changed. Reload page to display updated graph image.
Math example
The Lorenz Equations
<math>\begin{align}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{align}</math>
[math]\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}[/math]
The Cauchy-Schwarz Inequality
<math>\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)</math>
[math]\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)[/math]
A Cross Product Formula
<math>\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
\end{vmatrix}</math>
[math]\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}[/math]
The probability of getting k heads when flipping n coins is
<math>P(E) = {n \choose k} p^k (1-p)^{ n-k}</math>
[math]P(E) = {n \choose k} p^k (1-p)^{ n-k}[/math]
An Identity of Ramanujan
<math>\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } }</math>
[math]\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } }[/math]
A Rogers-Ramanujan Identity
<math>1 + \frac{q^2}{(1-q)} + \frac{q^6}{(1-q)(1-q^2)} + \cdots
= \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad for\,|q|<1.</math>
[math]1 + \frac{q^2}{(1-q)} + \frac{q^6}{(1-q)(1-q^2)} + \cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad for\,|q|\lt 1.[/math]
Maxwell’s Equations
<math>\begin{align}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0
\end{align}</math>
[math]\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}[/math]
