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1 ======== |
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2 Script |
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3 ======== |
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4 |
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5 {{{ show the welcome slide }}} |
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6 |
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7 Welcome to this tutorial on using Sage. |
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8 |
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9 {{{ show the slide with outline }}} |
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10 |
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11 In this tutorial we shall quickly look at a few examples of the areas |
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12 (name the areas, here) in which Sage can be used and how it can be |
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13 used. |
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14 |
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15 {{{ show the slide with Calculus outline }}} |
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16 |
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17 Let us begin with Calculus. We shall be looking at limits, |
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18 differentiation, integration, and Taylor polynomial. |
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19 |
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20 {{{ show sage notebook }}} |
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21 |
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22 We have our Sage notebook running. In case, you don't have it running, |
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23 start is using the command, ``sage --notebook``. |
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24 |
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25 To find the limit of the function x*sin(1/x), at x=0, we say:: |
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26 |
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27 lim(x*sin(1/x), x=0) |
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28 |
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29 We get the limit to be 0, as expected. |
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30 |
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31 It is also possible to the limit at a point from one direction. For |
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32 example, let us find the limit of 1/x at x=0, when approaching from |
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33 the positive side.:: |
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34 |
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35 lim(1/x, x=0, dir='above') |
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36 |
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37 To find the limit from the negative side, we say,:: |
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38 |
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39 lim(1/x, x=0, dir='above') |
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40 |
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41 Let us now see how to differentiate, using Sage. We shall find the |
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42 differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. We |
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43 shall first define the expression, and then use the ``diff`` function |
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44 to obtain the differential of the expression.:: |
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45 |
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46 var('x') |
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47 f = exp(sin(x^2))/x |
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48 |
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49 diff(f, x) |
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50 |
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51 We can also obtain the partial differentiation of an expression w.r.t |
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52 one of the variables. Let us differentiate the expression |
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53 ``exp(sin(y - x^2))/x`` w.r.t x and y.:: |
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54 |
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55 var('x y') |
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56 f = exp(sin(y - x^2))/x |
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57 |
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58 diff(f, x) |
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59 |
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60 diff(f, y) |
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61 |
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62 Now, let us look at integration. We shall use the expression obtained |
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63 from the differentiation that we did before, ``diff(f, y)`` --- |
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64 ``e^(sin(-x^2 + y))*cos(-x^2 + y)/x``. The ``integrate`` command is |
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65 used to obtain the integral of an expression or function.:: |
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66 |
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67 integrate(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y) |
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68 |
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69 We get back the correct expression. The minus sign being inside or |
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70 outside the ``sin`` function doesn't change much. |
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71 |
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72 Now, let us find the value of the integral between the limits 0 and |
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73 pi/2. :: |
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74 |
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75 integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2) |
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76 |
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77 Let us now see how to obtain the Taylor expansion of an expression |
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78 using sage. Let us obtain the Taylor expansion of ``(x + 1)^n`` up to |
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79 degree 4 about 0.:: |
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80 |
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81 var('x n') |
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82 taylor((x+1)^n, x, 0, 4) |
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83 |
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84 This brings us to the end of the features of Sage for Calculus, that |
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85 we will be looking at. For more, look at the Calculus quick-ref from |
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86 the Sage Wiki. |
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87 |
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88 Next let us move on to Matrix Algebra. |
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89 |
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90 {{{ show the equation on the slides }}} |
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91 |
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92 Let us begin with solving the equation ``Ax = v``, where A is the |
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93 matrix ``matrix([[1,2],[3,4]])`` and v is the vector |
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94 ``vector([1,2])``. |
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95 |
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96 To solve the equation, ``Ax = v`` we simply say:: |
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97 |
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98 x = solve_right(A, v) |
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99 |
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100 To solve the equation, ``xA = v`` we simply say:: |
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101 |
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102 x = solve_left(A, v) |
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103 |
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104 The left and right here, denote the position of ``A``, relative to x. |
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105 |
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106 |
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107 |
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108 Now, let us look at Graph Theory in Sage. |
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109 |
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110 Graph: G = Graph({0:[1,2,3], 2:[4]}) |
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111 Directed Graph: DiGraph(dictionary) |
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112 Graph families: graphs. tab |
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113 Invariants: G.chromatic polynomial(), G.is planar() |
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114 Paths: G.shortest path() |
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115 Visualize: G.plot(), G.plot3d() |
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116 Automorphisms: G.automorphism group(), G1.is isomorphic(G2), G1.is subgraph(G2) |
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117 |
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118 Now let us look at bits and pieces of Number theory, combinatorics, |
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119 |