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+.. Objectives
+.. ----------
+
+.. By the end of this tutorial you will --
+
+.. 1. Get an idea of the range of things for which Sage can be used.
+.. #. Know some of the functions for Calculus
+.. #. Get some insight into Graphs in Sage.
+
+
+.. Prerequisites
+.. -------------
+
+.. Getting Started -- Sage
+
+.. Author : Puneeth
+ Internal Reviewer : Anoop Jacob Thomas<anoop@fossee.in>
+ External Reviewer :
+ Language Reviewer : Bhanukiran
+ Checklist OK? : <06-11-2010, Anand, OK> [2010-10-05]
+
+Script
+------
+
+{{{ show the welcome slide }}}
+
+Hello Friends. Welcome to this tutorial on using Sage.
+
+{{{ show the slide with outline }}}
+
+In this tutorial we shall quickly look at a few examples of using Sage
+for Linear Algebra, Calculus, Graph Theory and Number theory.
+
+{{{ 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='below')
+
+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.
+
+.. #[[Anoop: I feel we should add more slides, a possibility is to add
+ the code which they are required to type in, I also feel we should
+ add some review problems for them to try out.]]
+
+{{{ Show summary slide }}}
+
+We have looked at some of the functions available for Linear Algebra,
+Calculus, Graph Theory and Number theory.
+
+This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India
+
+Hope you have enjoyed and found it useful.
+Thank you!
+