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     2  Script

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     4 

     5 {{{ show the welcome slide }}}

     6 

     7 Welcome to this tutorial on using Sage.

     8 

     9 {{{ show the slide with outline }}} 

    10 

    11 In this tutorial we shall quickly look at a few examples of the areas

    12 (name the areas, here) in which Sage can be used and how it can be

    13 used.

    14 

    15 {{{ show the slide with Calculus outline }}} 

    16 

    17 Let us begin with Calculus. We shall be looking at limits,

    18 differentiation, integration, and Taylor polynomial.

    19 

    20 {{{ show sage notebook }}}

    21 

    22 We have our Sage notebook running. In case, you don't have it running,

    23 start is using the command, ``sage --notebook``.

    24 

    25 To find the limit of the function x*sin(1/x), at x=0, we say

    26 ::

    27 

    28    lim(x*sin(1/x), x=0)

    29 

    30 We get the limit to be 0, as expected. 

    31 

    32 It is also possible to the limit at a point from one direction. For

    33 example, let us find the limit of 1/x at x=0, when approaching from

    34 the positive side.

    35 ::

    36 

    37     lim(1/x, x=0, dir='above')

    38 

    39 To find the limit from the negative side, we say,

    40 ::

    41 

    42     lim(1/x, x=0, dir='above')   

    43 

    44 Let us now see how to differentiate, using Sage. We shall find the

    45 differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. We

    46 shall first define the expression, and then use the ``diff`` function

    47 to obtain the differential of the expression.

    48 ::

    49 

    50     var('x')

    51     f = exp(sin(x^2))/x

    52 

    53     diff(f, x)

    54 

    55 We can also obtain the partial differentiation of an expression w.r.t

    56 one of the variables. Let us differentiate the expression

    57 ``exp(sin(y - x^2))/x`` w.r.t x and y.

    58 ::

    59 

    60     var('x y')

    61     f = exp(sin(y - x^2))/x

    62 

    63     diff(f, x)

    64 

    65     diff(f, y)

    66 

    67 Now, let us look at integration. We shall use the expression obtained

    68 from the differentiation that we did before, ``diff(f, y)`` ---

    69 ``e^(sin(-x^2 + y))*cos(-x^2 + y)/x``. The ``integrate`` command is

    70 used to obtain the integral of an expression or function.

    71 ::

    72 

    73     integrate(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y)

    74 

    75 We get back the correct expression. The minus sign being inside or

    76 outside the ``sin`` function doesn't change much. 

    77 

    78 Now, let us find the value of the integral between the limits 0 and

    79 pi/2. 

    80 ::

    81 

    82     integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2)

    83 

    84 Let us now see how to obtain the Taylor expansion of an expression

    85 using sage. Let us obtain the Taylor expansion of ``(x + 1)^n`` up to

    86 degree 4 about 0.

    87 ::

    88 

    89     var('x n')

    90     taylor((x+1)^n, x, 0, 4)

    91 

    92 This brings us to the end of the features of Sage for Calculus, that

    93 we will be looking at. For more, look at the Calculus quick-ref from

    94 the Sage Wiki. 

    95 

    96 Next let us move on to Matrix Algebra. 

    97 

    98 {{{ show the equation on the slides }}}

    99 

   100 Let us begin with solving the equation ``Ax = v``, where A is the

   101 matrix ``matrix([[1,2],[3,4]])`` and v is the vector

   102 ``vector([1,2])``. 

   103 

   104 To solve the equation, ``Ax = v`` we simply say

   105 ::

   106 

   107     x = solve_right(A, v)

   108 

   109 To solve the equation, ``xA = v`` we simply say

   110 ::

   111 

   112     x = solve_left(A, v)

   113 

   114 The left and right here, denote the position of ``A``, relative to x. 

   115 

   116 #[Puneeth]: any suggestions on what more to add?

   117 

   118 Now, let us look at Graph Theory in Sage. 

   119 

   120 We shall look at some ways to create graphs and some of the graph

   121 families available in Sage. 

   122 

   123 The simplest way to define an arbitrary graph is to use a dictionary

   124 of lists. We create a simple graph by

   125 ::

   126 

   127   G = Graph({0:[1,2,3], 2:[4]})

   128 

   129 We say 

   130 ::

   131 

   132   G.show()

   133 

   134 to view the visualization of the graph. 

   135 

   136 Similarly, we can obtain a directed graph using the ``DiGraph``

   137 function. 

   138 ::

   139 

   140   G = DiGraph({0:[1,2,3], 2:[4]})

   141 

   142 

   143 Sage also provides a lot of graph families which can be viewed by

   144 typing ``graph.<tab>``. Let us obtain a complete graph with 5 vertices

   145 and then show the graph. 

   146 ::

   147 

   148   G = graphs.CompleteGraph(5)

   149 

   150   G.show()

   151 

   152 

   153 Sage provides other functions for Number theory and

   154 Combinatorics. Let's have a glimpse of a few of them.  

   155 

   156 

   157 ::

   158 

   159   prime_range(100, 200)

   160 

   161 gives primes in the range 100 to 200. 

   162 

   163 ::

   164 

   165   is_prime(1999) 

   166 

   167 checks if 1999 is a prime number or not. 

   168 

   169 ::

   170 

   171   factor(2001)

   172 

   173 gives the factorized form of 2001. 

   174 

   175 ::

   176 

   177   C = Permutations([1, 2, 3, 4])

   178   C.list()

   179 

   180 gives the permutations of ``[1, 2, 3, 4]``

   181 

   182 ::

   183 

   184   C = Combinations([1, 2, 3, 4])

   185   C.list()

   186 

   187 gives all the combinations of ``[1, 2, 3, 4]``

   188   

   189 That brings us to the end of this session showing various features

   190 available in Sage. 

   191 

   192 {{{ Show summary slide }}}

   193 

   194 We have looked at some of the functions available for Linear Algebra,

   195 Calculus, Graph Theory and Number theory.   

   196 

   197 Thank You!