{ "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", "" ] } ], "metadata": { "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 }