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1 | 1 | { |
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2 | 2 | "metadata": { |
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3 | 3 | "name": "" |
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4 | 4 | }, |
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5 | 5 | "nbformat": 3, |
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6 | 6 | "nbformat_minor": 0, |
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7 | 7 | "worksheets": [ |
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8 | 8 | { |
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9 | 9 | "cells": [ |
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10 | 10 | { |
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11 | 11 | "cell_type": "markdown", |
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12 | 12 | "metadata": {}, |
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13 | 13 | "source": [ |
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14 | 14 | "The Markdown parser included in IPython 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](http://www.mathjax.org/demos/tex-samples/) are reproduced below, as well as the Markdown+TeX source." |
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15 | 15 | ] |
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16 | 16 | }, |
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17 | 17 | { |
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18 | 18 | "cell_type": "markdown", |
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19 | 19 | "metadata": {}, |
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20 | 20 | "source": [ |
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21 | 21 | "# Motivating Examples\n", |
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22 | 22 | "\n", |
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23 | 23 | "---\n", |
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24 | 24 | "\n", |
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25 | 25 | "## The Lorenz Equations\n", |
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26 | 26 | "### Source\n", |
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27 | 27 | "```\\begin{align}\n", |
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28 | 28 | "\\dot{x} & = \\sigma(y-x) \\\\\n", |
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29 | 29 | "\\dot{y} & = \\rho x - y - xz \\\\\n", |
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30 | 30 | "\\dot{z} & = -\\beta z + xy\n", |
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31 | 31 | "\\end{align}\n", |
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32 | 32 | "```\n", |
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33 | 33 | "### Display\n", |
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34 | 34 | "\\begin{align}\n", |
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35 | 35 | "\\dot{x} & = \\sigma(y-x) \\\\\n", |
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36 | 36 | "\\dot{y} & = \\rho x - y - xz \\\\\n", |
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37 | 37 | "\\dot{z} & = -\\beta z + xy\n", |
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38 | 38 | "\\end{align}" |
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39 | 39 | ] |
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40 | 40 | }, |
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41 | 41 | { |
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42 | 42 | "cell_type": "markdown", |
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43 | 43 | "metadata": {}, |
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44 | 44 | "source": [ |
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45 | 45 | "## The Cauchy-Schwarz Inequality\n", |
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46 | 46 | "### Source\n", |
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47 | 47 | "```\\begin{equation*}\n", |
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48 | 48 | "\\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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49 | 49 | "\\end{equation*}\n", |
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50 | 50 | "```\n", |
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51 | 51 | "### Display\n", |
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52 | 52 | "\\begin{equation*}\n", |
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53 | 53 | "\\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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54 | 54 | "\\end{equation*}" |
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55 | 55 | ] |
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56 | 56 | }, |
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57 | 57 | { |
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58 | 58 | "cell_type": "markdown", |
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59 | 59 | "metadata": {}, |
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60 | 60 | "source": [ |
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61 | 61 | "## A Cross Product Formula\n", |
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62 | 62 | "### Source\n", |
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63 | 63 | "```\\begin{equation*}\n", |
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64 | 64 | "\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n", |
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65 | 65 | "\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n", |
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66 | 66 | "\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n", |
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67 | 67 | "\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n", |
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68 | 68 | "\\end{vmatrix} \n", |
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69 | 69 | "\\end{equation*}\n", |
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70 | 70 | "```\n", |
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71 | 71 | "### Display\n", |
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72 | 72 | "\\begin{equation*}\n", |
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73 | 73 | "\\mathbf{V}_1 \\times \\mathbf{V}_2 = \\begin{vmatrix}\n", |
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74 | 74 | "\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\n", |
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75 | 75 | "\\frac{\\partial X}{\\partial u} & \\frac{\\partial Y}{\\partial u} & 0 \\\\\n", |
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76 | 76 | "\\frac{\\partial X}{\\partial v} & \\frac{\\partial Y}{\\partial v} & 0\n", |
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77 | 77 | "\\end{vmatrix} \n", |
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78 | 78 | "\\end{equation*}" |
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79 | 79 | ] |
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80 | 80 | }, |
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81 | 81 | { |
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82 | 82 | "cell_type": "markdown", |
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83 | 83 | "metadata": {}, |
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84 | 84 | "source": [ |
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85 | 85 | "## The probability of getting \\(k\\) heads when flipping \\(n\\) coins is\n", |
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86 | 86 | "### Source\n", |
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87 | 87 | "```\\begin{equation*}\n", |
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88 | 88 | "P(E) = {n \\choose k} p^k (1-p)^{ n-k} \n", |
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89 | 89 | "\\end{equation*}\n", |
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90 | 90 | "```\n", |
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91 | 91 | "### Display\n", |
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92 | 92 | "\\begin{equation*}\n", |
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93 | 93 | "P(E) = {n \\choose k} p^k (1-p)^{ n-k} \n", |
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94 | 94 | "\\end{equation*}" |
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95 | 95 | ] |
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96 | 96 | }, |
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97 | 97 | { |
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98 | 98 | "cell_type": "markdown", |
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99 | 99 | "metadata": {}, |
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100 | 100 | "source": [ |
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101 | 101 | "## An Identity of Ramanujan\n", |
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102 | 102 | "### Source\n", |
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103 | 103 | "```\\begin{equation*}\n", |
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104 | 104 | "\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n", |
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105 | 105 | "1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n", |
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106 | 106 | "{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n", |
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107 | 107 | "\\end{equation*}\n", |
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108 | 108 | "```\n", |
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109 | 109 | "### Display\n", |
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110 | 110 | "\\begin{equation*}\n", |
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111 | 111 | "\\frac{1}{\\Bigl(\\sqrt{\\phi \\sqrt{5}}-\\phi\\Bigr) e^{\\frac25 \\pi}} =\n", |
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112 | 112 | "1+\\frac{e^{-2\\pi}} {1+\\frac{e^{-4\\pi}} {1+\\frac{e^{-6\\pi}}\n", |
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113 | 113 | "{1+\\frac{e^{-8\\pi}} {1+\\ldots} } } } \n", |
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114 | 114 | "\\end{equation*}" |
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115 | 115 | ] |
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116 | 116 | }, |
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117 | 117 | { |
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118 | 118 | "cell_type": "markdown", |
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119 | 119 | "metadata": {}, |
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120 | 120 | "source": [ |
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121 | 121 | "## A Rogers-Ramanujan Identity\n", |
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122 | 122 | "### Source\n", |
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123 | 123 | "```\\begin{equation*}\n", |
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124 | 124 | "1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n", |
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125 | 125 | "\\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n", |
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126 | 126 | "\\quad\\quad \\text{for $|q|<1$}. \n", |
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127 | 127 | "\\end{equation*}\n", |
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128 | 128 | "```\n", |
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129 | 129 | "### Display\n", |
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130 | 130 | "\\begin{equation*}\n", |
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131 | 131 | "1 + \\frac{q^2}{(1-q)}+\\frac{q^6}{(1-q)(1-q^2)}+\\cdots =\n", |
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132 | 132 | "\\prod_{j=0}^{\\infty}\\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\n", |
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133 | 133 | "\\quad\\quad \\text{for $|q|<1$}. \n", |
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134 | 134 | "\\end{equation*}" |
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135 | 135 | ] |
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136 | 136 | }, |
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137 | 137 | { |
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138 | 138 | "cell_type": "markdown", |
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139 | 139 | "metadata": {}, |
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140 | 140 | "source": [ |
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141 | 141 | "## Maxwell's Equations\n", |
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142 | 142 | "### Source\n", |
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143 | 143 | "```\\begin{align}\n", |
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144 | 144 | "\\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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145 | 145 | "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n", |
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146 | 146 | "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n", |
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147 | 147 | "\\end{align}\n", |
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148 | 148 | "```\n", |
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149 | 149 | "### Display\n", |
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150 | 150 | "\\begin{align}\n", |
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151 | 151 | "\\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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152 | 152 | "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n", |
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153 | 153 | "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n", |
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154 | 154 | "\\end{align}" |
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155 | 155 | ] |
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156 | 156 | }, |
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157 | 157 | { |
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158 | 158 | "cell_type": "markdown", |
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159 | 159 | "metadata": {}, |
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160 | 160 | "source": [ |
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161 | 161 | "# Equation Numbering and References\n", |
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162 | 162 | "\n", |
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163 | 163 | "---\n", |
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164 | 164 | "\n", |
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165 | 165 | "Equation numbering and referencing will be available in a future version of IPython." |
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166 | 166 | ] |
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167 | 167 | }, |
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168 | 168 | { |
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169 | 169 | "cell_type": "markdown", |
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170 | 170 | "metadata": {}, |
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171 | 171 | "source": [ |
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172 | 172 | "# Inline Typesetting (Mixing Markdown and TeX)\n", |
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173 | 173 | "\n", |
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174 | 174 | "---\n", |
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175 | 175 | "\n", |
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176 | 176 | "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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177 | 177 | "\n", |
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178 | 178 | "## Source\n", |
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179 | 179 | "``` This expression $\\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a **[Markdown-formatted](http://daringfireball.net/projects/markdown/)** sentence. \n", |
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180 | 180 | "```\n", |
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181 | 181 | "## Display\n", |
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182 | 182 | "This expression $\\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a **[Markdown-formatted](http://daringfireball.net/projects/markdown/)** sentence. " |
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183 | 183 | ] |
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184 | 184 | }, |
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185 | 185 | { |
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186 | 186 | "cell_type": "markdown", |
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187 | 187 | "metadata": {}, |
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188 | 188 | "source": [ |
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189 | 189 | "# Other Syntax\n", |
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190 | 190 | "\n", |
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191 | 191 | "---\n", |
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192 | 192 | "\n", |
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193 | 193 | "You will notice in other places on the web that `$$` are needed explicitly to begin and end MathJax typesetting. This is **not** required if you will be using TeX environments, but the IPython notebook will accept this syntax on legacy notebooks. \n", |
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194 | 194 | "\n", |
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195 | 195 | "### Source\n", |
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196 |
"``` |
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196 | "```\n", | |
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197 | "$$\n", | |
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197 | 198 | "\\begin{array}{c}\n", |
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198 | 199 | "y_1 \\\\\\\n", |
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199 | 200 | "y_2 \\mathtt{t}_i \\\\\\\n", |
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200 | 201 | "z_{3,4}\n", |
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201 | 202 | "\\end{array}\n", |
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202 | 203 | "$$\n", |
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203 | 204 | "```\n", |
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204 | 205 | "\n", |
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205 | 206 | "```\n", |
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206 | 207 | "$$\n", |
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207 | 208 | "\\begin{array}{c}\n", |
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208 | 209 | "y_1 \\cr\n", |
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209 | 210 | "y_2 \\mathtt{t}_i \\cr\n", |
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210 | 211 | "y_{3}\n", |
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211 | 212 | "\\end{array}\n", |
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212 | 213 | "$$\n", |
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213 | 214 | "```\n", |
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214 | 215 | "\n", |
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215 | 216 | "```\n", |
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216 | 217 | "$$\\begin{eqnarray} \n", |
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217 | 218 | "x' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\n", |
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218 | 219 | "z' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n", |
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219 | 220 | "\\end{eqnarray}$$\n", |
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220 | 221 | "```\n", |
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221 | 222 | "\n", |
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222 | 223 | "```\n", |
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223 | 224 | "$$\n", |
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224 | 225 | "x=4\n", |
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225 | 226 | "$$\n", |
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226 | 227 | "```\n", |
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227 | 228 | "\n", |
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228 | 229 | "### Display\n", |
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229 | 230 | "$$\n", |
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230 | 231 | "\\begin{array}{c}\n", |
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231 | 232 | "y_1 \\\\\\\n", |
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232 | 233 | "y_2 \\mathtt{t}_i \\\\\\\n", |
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233 | 234 | "z_{3,4}\n", |
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234 | 235 | "\\end{array}\n", |
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235 | 236 | "$$\n", |
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236 | 237 | "\n", |
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237 | 238 | "$$\n", |
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238 | 239 | "\\begin{array}{c}\n", |
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239 | 240 | "y_1 \\cr\n", |
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240 | 241 | "y_2 \\mathtt{t}_i \\cr\n", |
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241 | 242 | "y_{3}\n", |
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242 | 243 | "\\end{array}\n", |
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243 | 244 | "$$\n", |
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244 | 245 | "\n", |
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245 | 246 | "$$\\begin{eqnarray} \n", |
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246 | 247 | "x' &=& &x \\sin\\phi &+& z \\cos\\phi \\\\\n", |
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247 | 248 | "z' &=& - &x \\cos\\phi &+& z \\sin\\phi \\\\\n", |
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248 | 249 | "\\end{eqnarray}$$\n", |
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249 | 250 | "\n", |
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250 | 251 | "$$\n", |
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251 | 252 | "x=4\n", |
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252 | 253 | "$$" |
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253 | 254 | ] |
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254 | 255 | } |
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255 | 256 | ], |
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256 | 257 | "metadata": {} |
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257 | 258 | } |
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258 | 259 | ] |
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259 | 260 | } No newline at end of file |
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