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# # Beautiful Mathematics Typesetting
# [Tex](https://en.wikipedia.org/wiki/TeX)
#
# [LaTex](https://www.latex-project.org/)
#
# [Motivating Examples](http://jupyter-notebook.readthedocs.io/en/latest/examples/Notebook/Typesetting%20Equations.html)

# ### The Lorenz Equations

# + active=""
# \begin{align}
# \dot{x} & = \sigma(y-x) \\
# \dot{y} & = \rho x - y - xz \\
# \dot{z} & = -\beta z + xy
# \end{align}
# -

# \begin{align}
# \dot{x} & = \sigma(y-x) \\
# \dot{y} & = \rho x - y - xz \\
# \dot{z} & = -\beta z + xy
# \end{align}

# ### The Cauchy-Schwarz Inequality

# + active=""
# \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*}
# -

# \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*}

# ### Cross Product Formula

# + active=""
# \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*}
# -

# \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*}

# ### Probability of getting (k) heads when flipping (n) coins

# + active=""
# \begin{equation*}
# P(E)   = {n \choose k} p^k (1-p)^{ n-k}
# \end{equation*}
# -

# \begin{equation*}
# P(E)   = {n \choose k} p^k (1-p)^{ n-k}
# \end{equation*}

# ### Identity of Ramanujan
# [Srinivasa Ramanujan](https://en.wikipedia.org/wiki/Srinivasa_Ramanujan)
#
# Self-taught, no formal training in mathematics, made contributions to:
# - mathematical analysis
# - number theory
# - infinite series
# - continued fractions

# + active=""
# \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*}
# -

# \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*}

# ### Maxwell’s Equations

# + active=""
# \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}
# -

# \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}



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