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