#!/usr/bin/env python
# coding: utf-8

# The Markdown parser included in IPython is MathJax-aware.  This means that you can freely mix in mathematical expressions using the [MathJax subset of Tex and LaTeX](http://docs.mathjax.org/en/latest/tex.html#tex-support).  [Some examples from the MathJax site](http://www.mathjax.org/demos/tex-samples/) are reproduced below, as well as the Markdown+TeX source.

# # Motivating Examples
# 
# ---
# 
# ## The Lorenz Equations
# ### Source
# ```\begin{align}
# \dot{x} & = \sigma(y-x) \\
# \dot{y} & = \rho x - y - xz \\
# \dot{z} & = -\beta z + xy
# \end{align}
# ```
# ### Display
# \begin{align}
# \dot{x} & = \sigma(y-x) \\
# \dot{y} & = \rho x - y - xz \\
# \dot{z} & = -\beta z + xy
# \end{align}

# ## The Cauchy-Schwarz Inequality
# ### Source
# ```\begin{equation*}
# \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)
# \end{equation*}
# ```
# ### Display
# \begin{equation*}
# \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)
# \end{equation*}

# ## A Cross Product Formula
# ### Source
# ```\begin{equation*}
# \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}  
# \end{equation*}
# ```
# ### Display
# \begin{equation*}
# \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}  
# \end{equation*}

# ## The probability of getting \(k\) heads when flipping \(n\) coins is
# ### Source
# ```\begin{equation*}
# P(E)   = {n \choose k} p^k (1-p)^{ n-k} 
# \end{equation*}
# ```
# ### Display
# \begin{equation*}
# P(E)   = {n \choose k} p^k (1-p)^{ n-k} 
# \end{equation*}

# ## An Identity of Ramanujan
# ### Source
# ```\begin{equation*}
# \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} } } } 
# \end{equation*}
# ```
# ### Display
# \begin{equation*}
# \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} } } } 
# \end{equation*}

# ## A Rogers-Ramanujan Identity
# ### Source
# ```\begin{equation*}
# 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 $|q|<1$}. 
# \end{equation*}
# ```
# ### Display
# \begin{equation*}
# 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 $|q|<1$}. 
# \end{equation*}

# ## Maxwell's Equations
# ### Source
# ```\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}
# ```
# ### Display
# \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}

# # Equation Numbering and References
# 
# ---
# 
# Equation numbering and referencing will be available in a future version of IPython.

# # Inline Typesetting (Mixing Markdown and TeX)
# 
# ---
# 
# While display equations look good for a page of samples, the ability to mix math and *formatted* **text** in a paragraph is also important.
# 
# ## Source
# ``` This expression $\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a **[Markdown-formatted](http://daringfireball.net/projects/markdown/)** sentence.  
# ```
# ## Display
# This expression $\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a **[Markdown-formatted](http://daringfireball.net/projects/markdown/)** sentence.  

# # Other Syntax
# 
# ---
# 
# You will notice in other places on the web that `$$` are needed explicitly to begin and end MathJax typesetting.  This is **not** required if you will be using TeX environments, but the IPython notebook will accept this syntax on legacy notebooks.  
# 
# ### Source
# ```
# $$
# \begin{array}{c}
# y_1 \\\
# y_2 \mathtt{t}_i \\\
# z_{3,4}
# \end{array}
# $$
# ```
# 
# ```
# $$
# \begin{array}{c}
# y_1 \cr
# y_2 \mathtt{t}_i \cr
# y_{3}
# \end{array}
# $$
# ```
# 
# ```
# $$\begin{eqnarray} 
# x' &=& &x \sin\phi &+& z \cos\phi \\
# z' &=& - &x \cos\phi &+& z \sin\phi \\
# \end{eqnarray}$$
# ```
# 
# ```
# $$
# x=4
# $$
# ```
# 
# ### Display
# $$
# \begin{array}{c}
# y_1 \\\
# y_2 \mathtt{t}_i \\\
# z_{3,4}
# \end{array}
# $$
# 
# $$
# \begin{array}{c}
# y_1 \cr
# y_2 \mathtt{t}_i \cr
# y_{3}
# \end{array}
# $$
# 
# $$\begin{eqnarray} 
# x' &=& &x \sin\phi &+& z \cos\phi \\
# z' &=& - &x \cos\phi &+& z \sin\phi \\
# \end{eqnarray}$$
# 
# $$
# x=4
# $$