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    # A third example with [MathJax](http://www.mathjax.org/) activated
    > 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 available:

    1. ``mathjax=y`` if you want to load MathJax, with ``$ ... $`` being the default syntax for inline maths and ``$$ ... $$`` for display,
    2. ``theme=united`` or ``theme=cyborg`` if you want to load a specific theme (without having to specify it as the ``theme`` attribute of the main ``xmp`` tag containing your Markdown code),
    3. ``beacon=y`` if you want to include an invisible [GA Beacon](http://perso.crans.org/besson/beacon.en.html) image in your page,
    4. ``pedantic=y`` if you want to force the use of the ["pedantic" option](https://github.com/chjj/marked#pedantic) for [marked](https://github.com/chjj/marked), the Markdown parser used by StrapDown.js (note: you probably don't want that, [pedantic=false is usually better](https://github.com/chuckhoupt/strapdown/commit/1a090729fb717059be0689aa65025de6de67b3cd)). Default is pedantic=false.
    5. ``nonavbarfixed=y`` if you prefer the [top navigation bar](https://bootswatch.com/united/#navbar) to not be fixed, but scroll as the rest of the page (*new!*). See [example #8](example8.html),
    6. ``keepicon=y`` if you prefer StrapDown.js to keep the favicon included in your HTML document (*new!*). See [example #9](example9.html).
    7. ``nonavbar=y`` if you prefer to not have any [top navigation bar](https://bootswatch.com/united/#navbar) (*new!*). See [example #10](example10.html),

    ***

    > 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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