The Markdown parser included in the Jupyter Notebook 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](https://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 the Jupyter notebook.

## 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](https://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](https://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 Jupyter 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

$$
\begin{array}{c}
y_1 \\\
y_2 \mathtt{t}_i \\\
z_{3,4}
\end{array}
$$

$$
\begin{array}{c}
y_1 \cr
y_2 \mathtt{t}_i \cr
y_{3}
\end{array}
$$

$$\begin{eqnarray} 
x' &=& &x \sin\phi &+& z \cos\phi \\
z' &=& - &x \cos\phi &+& z \sin\phi \\
\end{eqnarray}$$

$$
x=4
$$

548a69f612667ecccf76e1eb70bb4d22bbb24b16
ipynb
Jupyter Notebook
jupyter-notebook-viewer/src/notebooks.ipynb
walkwel/problem-components
57b20300ce48e8d6c6f0da8034e7215c6d08b04e
MIT
jupyter-notebook-viewer/src/notebooks.ipynb
walkwel/problem-components
57b20300ce48e8d6c6f0da8034e7215c6d08b04e
MIT
jupyter-notebook-viewer/src/notebooks.ipynb
walkwel/problem-components
57b20300ce48e8d6c6f0da8034e7215c6d08b04e
MIT
2018-07-08T07:36:17.000Z
2018-07-08T07:36:17.000Z
Qwen/Qwen-72B
1. YES
2. YES