example5.html

<!DOCTYPE html><html><head><meta charset="utf-8"/><title>A second demo of MathJax with StrapDown.js</title></head><body><xmp>
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# A third example with [MathJax](http://www.mathjax.org/) activacted
> This *StrapDown*-flavored HTML page comes with MathJax enabled, simply because **we asked the script to do so**.

The last line of this page source (you can read it with "Ctrl+u" in Firefox) imports the script ``strapdown.min.js`` with this URL : ``"strapdown.min.js?src=example5&mathjax=y&theme=united&beacon=y"``.

This way, some options are avaible:

1. ``mathjax=y`` if you want to load MathJax, with ``$`` being the default separator for inline maths and ``$$`` for display,
2. ``theme=united`` or ``theme=united`` if you want to load a specific theme (without having to specify it as the theme attribute of the main ``xmp``),
3. ``beacon=y`` if you want to include an invisible [GA Beacon](http://besson.qc.to/beacon.en.html) image in your page.

***

> These examples are the same as [the previous example](example4.html) directly imported from [the samples from the mathjax.org website](http://www.mathjax.org/demos/tex-samples/).

> The following equations are included in the HTML source code as pure LaTeX code.

***

### The [Lorenz Equations](https://en.wikipedia.org/wiki/Lorenz_system#Overview)
### $$\left\\\{\begin{aligned}  \dot{x} & = \sigma(y-x) \\\\  \dot{y} & = \rho x - y - xz \\\\  \dot{z} & = -\beta z + xy \end{aligned}\right. $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page (*ie.* after the opening ``xmp`` tag and before its closing):
```latex
$$\left\\\{\begin{aligned}
\dot{x} & = \sigma(y-x) \\\\
\dot{y} & = \rho x - y - xz \\\\
\dot{z} & = -\beta z + xy
\end{aligned}\right. $$
```

### The [Cauchy-Schwarz Inequality](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Rn) (in $\mathbb{R}^n$)
### $$ \left( \sum\_{k=1}^n a\_k b\_k \right)^2 \leq \left( \sum\_{k=1}^n a\_k^2 \right) \left( \sum\_{k=1}^n b\_k^2 \right) $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$ \left( \sum\_{k=1}^n a\_k b\_k \right)^2 \leq \left( \sum\_{k=1}^n a\_k^2 \right) \left( \sum\_{k=1}^n b\_k^2 \right) $$
```

### A [Cross Product](https://en.wikipedia.org/wiki/Cross_product#Coordinate_notation) Formula
### $$\mathbf{V}\_1 \times \mathbf{V}\_2 =  \begin{vmatrix}  \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\  \frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\\\  \frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v} & 0  \end{vmatrix}  $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$\mathbf{V}\_1 \times \mathbf{V}\_2 =  \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\\\
\frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\\\
\frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v} & 0
\end{vmatrix}  $$
```

### The probability of [getting $k$ heads when flipping $n$ coins](https://en.wikipedia.org/wiki/Bernoulli_process#Binomial_distribution) is
### $$P(E) = {n \choose k} p^k (1-p)^{n-k} $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$P(E) = {n \choose k} p^k (1-p)^{n-k} $$
```

### An Identity of Ramanujan (obviously)
### $$ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =  1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}  {1+\frac{e^{-8\pi}} {1+\ldots} } } } $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } } $$
```

### A [Rogers-Ramanujan Identity](https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities)
### $$  1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =  \prod\_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},  \quad\quad \text{for $|q|&lt;1$}. $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$  1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod\_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \text{for $|q|&lt;1$}. $$
```

### [Maxwell's Equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations)
### $$  \left\\\{\begin{aligned}    \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\\\    \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\\\   \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\\\    \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}\right.  $$

> The previous equation corresponds to the following code, inserted *verbatim* in the Markdown part of this page:
```latex
$$  \left\\\{\begin{aligned}
    \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\\\
    \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\\\
    \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\\\
    \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}\right.
$$
```

As you can see, math environment (like ``aligned``) are supported by MathJax, even with the default configuration and no external plugins.

****

### Inline equations are also supported.
Finally, while **display equations** look good for a page of samples, the ability to mix math and text in a paragraph is also important.
This expression $\sqrt{3x-1}+(1+x)^2$ is an example of an **inline equation** (inserted with the code ``$\sqrt{3x-1}+(1+x)^2$``).
As you see, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.

### End of the examples
That's all for today!

---

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