{
“cells”: [
{

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“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](https://docs.mathjax.org/en/latest/input/tex/). [Some examples from the MathJax demos site](https://mathjax.github.io/MathJax-demos-web/) are reproduced below, as well as the Markdown+TeX source.”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“# Motivating Examplesn”, “n”, “## The Lorenz Equationsn”, “### Sourcen”, “\n", "\\begin{align}\n", "\\dot{x} & = \\sigma(y-x) \\\\\n", "\\dot{y} & = \\rho x - y - xz \\\\\n", "\\dot{z} & = -\\beta z + xy\n", "\\end{align}\n", "n”, “### Displayn”, “n”, “\begin{align}n”, “\dot{x} & = \sigma(y-x) \\n”, “\dot{y} & = \rho x - y - xz \\n”, “\dot{z} & = -\beta z + xyn”, “\end{align}”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## The Cauchy-Schwarz Inequalityn”, “### Sourcen”, “\n", "\\begin{equation*}\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)\n", "\\end{equation*}\n", "n”, “### Displayn”, “n”, “$\begin{equation*}n”, “\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)n”, “\end{equation*}$”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## A Cross Product Formulan”, “### Sourcen”, “\n", "\\begin{equation*}\n", "\\mathbf{V}_1 \\times \\mathbf{V}_2 =  \\begin{vmatrix}\n", "\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n", "\\frac{\\partial X}{\\partial u} &  \\frac{\\partial Y}{\\partial u} & 0 \\\\\n", "\\frac{\\partial X}{\\partial v} &  \\frac{\\partial Y}{\\partial v} & 0\n", "\\end{vmatrix}  \n", "\\end{equation*}\n", "n”, “### Displayn”, “n”, “$\begin{equation*}n”, “\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}n”, “\mathbf{i} & \mathbf{j} & \mathbf{k} \\n”, “\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\n”, “\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0n”, “\end{vmatrix} n”, “\end{equation*}$”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## The probability of getting $$k$$ heads when flipping $$n$$ coins isn”, “### Sourcen”, “\n", "\\begin{equation*}\n", "P(E)   = {n \\choose k} p^k (1-p)^{ n-k} \n", "\\end{equation*}\n", "n”, “### Displayn”, “n”, “$\begin{equation*}n”, “P(E) = {n \choose k} p^k (1-p)^{ n-k} n”, “\end{equation*}$”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## An Identity of Ramanujann”, “### Sourcen”, “\n", "\\begin{equation*}\n", "\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n", "1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n", "{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n", "\\end{equation*}\n", "n”, “### Displayn”, “$\begin{equation*}n”, “\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =n”, “1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}n”, “{1+\frac{e^{-8\pi}} {1+\ldots} } } } n”, “\end{equation*}$”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## A Rogers-Ramanujan Identityn”, “### Sourcen”, “\n", "\\begin{equation*}\n", "1 +  \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n", "\\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n", "\\quad\\quad \\text{for $|q|<1$}. \n", "\\end{equation*}\n", "n”, “### Displayn”, “n”, “$$\begin{equation*}n”, “1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =n”, “\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},n”, “\quad\quad \text{for |q|<1}. n”, “\end{equation*}$$”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## Maxwell’s Equationsn”, “### Sourcen”, “\n", "\\begin{align}\n", "\\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 \\\\\n", "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n", "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n", "\\end{align}\n", "n”, “### Displayn”, “n”, “\begin{align}n”, “\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 \\n”, “\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\n”, “\nabla \cdot \vec{\mathbf{B}} & = 0 n”, “\end{align}”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## Equation Numbering and Referencesn”, “n”, “Equation numbering and referencing will be available in a future version of the Jupyter notebook.”

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

“## Inline Typesetting (Mixing Markdown and TeX)n”, “n”, “While display equations look good for a page of samples, the ability to mix math and formatted text in a paragraph is also important.n”, “n”, “### Sourcen”, “\n", "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.  \n", "n”, “n”, “### Displayn”, “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. “

]

}, {

“cell_type”: “markdown”, “metadata”: {}, “source”: [

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

]

}

“kernelspec”: {

“display_name”: “Python 3”, “language”: “python”, “name”: “python3”

}, “language_info”: {

“codemirror_mode”: {

“name”: “ipython”, “version”: 3

}, “file_extension”: “.py”, “mimetype”: “text/x-python”, “name”: “python”, “nbconvert_exporter”: “python”, “pygments_lexer”: “ipython3”, “version”: “3.7.3”

}

}, “nbformat”: 4, “nbformat_minor”: 1

}