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-========
- Script
-========
-
-{{{ show the welcome slide }}}
-
-Welcome to this tutorial on using Sage.
-
-{{{ show the slide with outline }}}
-
-In this tutorial we shall quickly look at a few examples of the areas
-(name the areas, here) in which Sage can be used and how it can be
-used.
-
-{{{ show the slide with Calculus outline }}}
-
-Let us begin with Calculus. We shall be looking at limits,
-differentiation, integration, and Taylor polynomial.
-
-{{{ show sage notebook }}}
-
-We have our Sage notebook running. In case, you don't have it running,
-start is using the command, ``sage --notebook``.
-
-To find the limit of the function x*sin(1/x), at x=0, we say
-::
-
- lim(x*sin(1/x), x=0)
-
-We get the limit to be 0, as expected.
-
-It is also possible to the limit at a point from one direction. For
-example, let us find the limit of 1/x at x=0, when approaching from
-the positive side.
-::
-
- lim(1/x, x=0, dir='above')
-
-To find the limit from the negative side, we say,
-::
-
- lim(1/x, x=0, dir='above')
-
-Let us now see how to differentiate, using Sage. We shall find the
-differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. We
-shall first define the expression, and then use the ``diff`` function
-to obtain the differential of the expression.
-::
-
- var('x')
- f = exp(sin(x^2))/x
-
- diff(f, x)
-
-We can also obtain the partial differentiation of an expression w.r.t
-one of the variables. Let us differentiate the expression
-``exp(sin(y - x^2))/x`` w.r.t x and y.
-::
-
- var('x y')
- f = exp(sin(y - x^2))/x
-
- diff(f, x)
-
- diff(f, y)
-
-Now, let us look at integration. We shall use the expression obtained
-from the differentiation that we did before, ``diff(f, y)`` ---
-``e^(sin(-x^2 + y))*cos(-x^2 + y)/x``. The ``integrate`` command is
-used to obtain the integral of an expression or function.
-::
-
- integrate(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y)
-
-We get back the correct expression. The minus sign being inside or
-outside the ``sin`` function doesn't change much.
-
-Now, let us find the value of the integral between the limits 0 and
-pi/2.
-::
-
- integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2)
-
-Let us now see how to obtain the Taylor expansion of an expression
-using sage. Let us obtain the Taylor expansion of ``(x + 1)^n`` up to
-degree 4 about 0.
-::
-
- var('x n')
- taylor((x+1)^n, x, 0, 4)
-
-This brings us to the end of the features of Sage for Calculus, that
-we will be looking at. For more, look at the Calculus quick-ref from
-the Sage Wiki.
-
-Next let us move on to Matrix Algebra.
-
-{{{ show the equation on the slides }}}
-
-Let us begin with solving the equation ``Ax = v``, where A is the
-matrix ``matrix([[1,2],[3,4]])`` and v is the vector
-``vector([1,2])``.
-
-To solve the equation, ``Ax = v`` we simply say
-::
-
- x = solve_right(A, v)
-
-To solve the equation, ``xA = v`` we simply say
-::
-
- x = solve_left(A, v)
-
-The left and right here, denote the position of ``A``, relative to x.
-
-#[Puneeth]: any suggestions on what more to add?
-
-Now, let us look at Graph Theory in Sage.
-
-We shall look at some ways to create graphs and some of the graph
-families available in Sage.
-
-The simplest way to define an arbitrary graph is to use a dictionary
-of lists. We create a simple graph by
-::
-
- G = Graph({0:[1,2,3], 2:[4]})
-
-We say
-::
-
- G.show()
-
-to view the visualization of the graph.
-
-Similarly, we can obtain a directed graph using the ``DiGraph``
-function.
-::
-
- G = DiGraph({0:[1,2,3], 2:[4]})
-
-
-Sage also provides a lot of graph families which can be viewed by
-typing ``graph.<tab>``. Let us obtain a complete graph with 5 vertices
-and then show the graph.
-::
-
- G = graphs.CompleteGraph(5)
-
- G.show()
-
-
-Sage provides other functions for Number theory and
-Combinatorics. Let's have a glimpse of a few of them.
-
-
-::
-
- prime_range(100, 200)
-
-gives primes in the range 100 to 200.
-
-::
-
- is_prime(1999)
-
-checks if 1999 is a prime number or not.
-
-::
-
- factor(2001)
-
-gives the factorized form of 2001.
-
-::
-
- C = Permutations([1, 2, 3, 4])
- C.list()
-
-gives the permutations of ``[1, 2, 3, 4]``
-
-::
-
- C = Combinations([1, 2, 3, 4])
- C.list()
-
-gives all the combinations of ``[1, 2, 3, 4]``
-
-That brings us to the end of this session showing various features
-available in Sage.
-
-{{{ Show summary slide }}}
-
-We have looked at some of the functions available for Linear Algebra,
-Calculus, Graph Theory and Number theory.
-
-Thank You!