{
 "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 Examples\n",
    "\n",
    "## The Lorenz Equations\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "\n",
    "$\\begin{align}\n",
    "\\dot{x} & = \\sigma(y-x) \\\\\n",
    "\\dot{y} & = \\rho x - y - xz \\\\\n",
    "\\dot{z} & = -\\beta z + xy\n",
    "\\end{align}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Cauchy-Schwarz Inequality\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "\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 Formula\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "\n",
    "$\\begin{equation*}\n",
    "\\mathbf{V}_1 \\times \\mathbf{V}_2 =  \\begin{vmatrix}\n",
    "\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n",
    "\\frac{\\partial X}{\\partial u} &  \\frac{\\partial Y}{\\partial u} & 0 \\\\\n",
    "\\frac{\\partial X}{\\partial v} &  \\frac{\\partial Y}{\\partial v} & 0\n",
    "\\end{vmatrix}  \n",
    "\\end{equation*}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The probability of getting \\(k\\) heads when flipping \\(n\\) coins is\n",
    "### Source\n",
    "```\n",
    "\\begin{equation*}\n",
    "P(E)   = {n \\choose k} p^k (1-p)^{ n-k} \n",
    "\\end{equation*}\n",
    "```\n",
    "### Display\n",
    "\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 Ramanujan\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "$\\begin{equation*}\n",
    "\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n",
    "1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n",
    "{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n",
    "\\end{equation*}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Rogers-Ramanujan Identity\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "\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 Equations\n",
    "### Source\n",
    "```\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",
    "### Display\n",
    "\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 References\n",
    "\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",
    "### Source\n",
    "```\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",
    "### Display\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.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Other Syntax\n",
    "\n",
    "You will notice in other places on the web that `$$` are needed explicitly to begin and end MathJax typesetting.  This is **not** required if you will be using TeX environments, but the Jupyter notebook will accept this syntax on legacy notebooks.  \n",
    "\n",
    "## Source\n",
    "\n",
    "```\n",
    "$$\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",
    "$$\n",
    "```\n",
    "\n",
    "## Display\n",
    "\n",
    "$$\n",
    "\\begin{array}{c}\n",
    "y_1 \\\\\\\n",
    "y_2 \\mathtt{t}_i \\\\\\\n",
    "z_{3,4}\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\begin{array}{c}\n",
    "y_1 \\cr\n",
    "y_2 \\mathtt{t}_i \\cr\n",
    "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=4\n",
    "$$"
   ]
  }
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