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281 lines
8.2 KiB
281 lines
8.2 KiB
2 years ago
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"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."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Motivating Examples\n",
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"\n",
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"## The Lorenz Equations\n",
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"### Source\n",
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"```\n",
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"\\begin{align}\n",
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"\\dot{x} & = \\sigma(y-x) \\\\\n",
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"\\dot{y} & = \\rho x - y - xz \\\\\n",
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"\\dot{z} & = -\\beta z + xy\n",
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"\\end{align}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$\\begin{align}\n",
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"\\dot{x} & = \\sigma(y-x) \\\\\n",
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"\\dot{y} & = \\rho x - y - xz \\\\\n",
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"\\dot{z} & = -\\beta z + xy\n",
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"\\end{align}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## The Cauchy-Schwarz Inequality\n",
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"### Source\n",
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"```\n",
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"\\begin{equation*}\n",
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"\\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",
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"\\end{equation*}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$\\begin{equation*}\n",
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"\\left( \\sum_{k=1}^n a_k b_k \\right)^2 \\leq \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n",
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"\\end{equation*}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## A Cross Product Formula\n",
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"### Source\n",
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"```\n",
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"\\begin{equation*}\n",
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"\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n",
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"\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n",
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"\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n",
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"\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n",
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"\\end{vmatrix} \n",
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"\\end{equation*}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$\\begin{equation*}\n",
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"\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n",
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"\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n",
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"\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n",
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"\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n",
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"\\end{vmatrix} \n",
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"\\end{equation*}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## The probability of getting \\(k\\) heads when flipping \\(n\\) coins is\n",
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"### Source\n",
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"```\n",
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"\\begin{equation*}\n",
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"P(E) = {n \\choose k} p^k (1-p)^{ n-k} \n",
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"\\end{equation*}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$\\begin{equation*}\n",
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"P(E) = {n \\choose k} p^k (1-p)^{ n-k} \n",
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"\\end{equation*}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## An Identity of Ramanujan\n",
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"### Source\n",
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"```\n",
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"\\begin{equation*}\n",
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"\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n",
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"1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n",
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"{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n",
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"\\end{equation*}\n",
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"```\n",
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"### Display\n",
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"$\\begin{equation*}\n",
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"\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n",
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"1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n",
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"{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n",
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"\\end{equation*}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## A Rogers-Ramanujan Identity\n",
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"### Source\n",
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"```\n",
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"\\begin{equation*}\n",
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"1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n",
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"\\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n",
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"\\quad\\quad \\text{for $|q|<1$}. \n",
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"\\end{equation*}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$$\\begin{equation*}\n",
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"1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n",
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"\\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n",
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"\\quad\\quad \\text{for $|q|<1$}. \n",
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"\\end{equation*}$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Maxwell's Equations\n",
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"### Source\n",
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"```\n",
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"\\begin{align}\n",
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"\\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",
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"\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n",
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"\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n",
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"\\end{align}\n",
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"```\n",
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"### Display\n",
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"\n",
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"$\\begin{align}\n",
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"\\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\ \\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n",
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"\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n",
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"\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n",
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"\\end{align}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Equation Numbering and References\n",
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"\n",
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"Equation numbering and referencing will be available in a future version of the Jupyter notebook."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Inline Typesetting (Mixing Markdown and TeX)\n",
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"\n",
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"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",
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"\n",
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"### Source\n",
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"```\n",
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"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",
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"```\n",
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"\n",
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"### Display\n",
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"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. "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Other Syntax\n",
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"\n",
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"You will notice in other places on the web that `$$` are needed explicitly to begin and end MathJax typesetting. This is **not** required if you will be using TeX environments, but the Jupyter notebook will accept this syntax on legacy notebooks. \n",
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"\n",
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"## Source\n",
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"\n",
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"```\n",
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"$$\n",
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"\\begin{array}{c}\n",
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"y_1 \\\\\\\n",
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"y_2 \\mathtt{t}_i \\\\\\\n",
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"z_{3,4}\n",
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"\\end{array}\n",
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"$$\n",
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"```\n",
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"\n",
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"```\n",
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"$$\n",
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"\\begin{array}{c}\n",
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"y_1 \\cr\n",
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"y_2 \\mathtt{t}_i \\cr\n",
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"y_{3}\n",
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"\\end{array}\n",
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"$$\n",
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"```\n",
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"\n",
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"```\n",
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"$$\\begin{eqnarray} \n",
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"x' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\n",
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"z' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n",
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"\\end{eqnarray}$$\n",
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"```\n",
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"\n",
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"```\n",
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"$$\n",
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"x=4\n",
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"$$\n",
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"```\n",
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"\n",
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"## Display\n",
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"\n",
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"$$\n",
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"\\begin{array}{c}\n",
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"y_1 \\\\\\\n",
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"y_2 \\mathtt{t}_i \\\\\\\n",
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"z_{3,4}\n",
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"\\end{array}\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\begin{array}{c}\n",
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"y_1 \\cr\n",
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"y_2 \\mathtt{t}_i \\cr\n",
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"y_{3}\n",
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"\\end{array}\n",
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"$$\n",
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"\n",
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"$$\\begin{eqnarray} \n",
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"x' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\n",
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"z' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n",
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"\\end{eqnarray}$$\n",
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"\n",
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"$$\n",
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"x=4\n",
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"$$"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 1
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}
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