st-scripts: comparison getting_started_with

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+.. Objectives
+.. ----------
+.. By the end of this tutorial, you will be able to
+.. 1. Defining symbolic expressions in sage.
+.. # Using built-in constants and functions.
+.. # Performing Integration, differentiation using sage.
+.. # Defining matrices.
+.. # Defining Symbolic functions.
+.. # Simplifying and solving symbolic expressions and functions.
+.. Prerequisites
+.. -------------
+..   1. getting started with sage notebook
+.. Author              : Amit
+Internal Reviewer   :
+External Reviewer   :
+Language Reviewer   : Bhanukiran
+Checklist OK?       : <, if OK> [2010-10-05]
+Symbolics with Sage
+-------------------
+Hello friends and welcome to the tutorial on Symbolics with Sage.
+{{{ Show welcome slide }}}
+During the course of the tutorial we will learn
+{{{ Show outline slide  }}}
+* Defining symbolic expressions in Sage.
+* Using built-in constants and functions.
+* Performing Integration, differentiation using Sage.
+* Defining matrices.
+* Defining symbolic functions.
+* Simplifying and solving symbolic expressions and functions.
+In addtion to a lot of other things, Sage can do Symbolic Math and we shall
+start with defining symbolic expressions in Sage.
+Have your Sage notebook opened. If not, pause the video and
+start you Sage notebook right now.
+On the sage notebook type::
+sin(y)
+It raises a name error saying that ``y`` is not defined. We need to
+declare ``y`` as a symbol. We do it using the ``var`` function.
+::
+var('y')
+Now if you type::
+sin(y)
+Sage simply returns the expression.
+Sage treats ``sin(y)`` as a symbolic expression. We can use this to do
+symbolic math using Sage's built-in constants and expressions.
+Let us try out a few examples. ::
+var('x,alpha,y,beta')
+x^2/alpha^2+y^2/beta^2
+We have defined 4 variables, ``x``, ``y``, ``alpha`` and ``beta`` and
+have defined a symbolic expression using them.
+Here is an expression in ``theta``  ::
+var('theta')
+sin(theta)*sin(theta)+cos(theta)*cos(theta)
+Now that you know how to define symbolic expressions in Sage, here is
+an exercise.
+{{ show slide showing question 1 }}
+%% %% Define following expressions as symbolic expressions in Sage.
+1. x^2+y^2
+#. y^2-4ax
+Please, pause the video here. Do the exercise and then continue.
+The solution is on your screen.
+{{ show slide showing solution 1 }}
+Sage also provides built-in constants which are commonly used in
+mathematics, for instance pi, e, infinity. The function ``n`` gives
+the numerical values of all these constants.
+::
+n(pi)
+n(e)
+n(oo)
+If you look into the documentation of function ``n`` by doing
+::
+n(<Tab>
+You will see what all arguments it takes and what it returns. It will
+be very helpful if you look at the documentation of all functions
+introduced in the course of this script.
+Also we can define the number of digits we wish to have in the
+constants. For this we have to pass an argument -- digits.  Type
+::
+n(pi, digits = 10)
+Apart from the constants Sage also has a lot of built-in functions
+like ``sin``, ``cos``, ``log``, ``factorial``, ``gamma``, ``exp``,
+``arcsin`` etc ...
+Lets try some of them out on the Sage notebook.
+::
+sin(pi/2)
+arctan(oo)
+log(e,e)
+Following are exercises that you must do.
+{{ show slide showing question 2 }}
+%% %% Find the values of the following constants upto 6 digits
+precision
+1. pi^2
+#. euler_gamma^2
+%% %% Find the value of the following.
+1. sin(pi/4)
+#. ln(23)
+Please, pause the video here. Do the exercises and then continue.
+The solutions are on your screen
+{{ show slide showing solution 2 }}
+Given that we have defined variables like x, y etc., we can define an
+arbitrary function with desired name in the following way.::
+var('x')
+function('f',x)
+Here f is the name of the function and x is the independent variable .
+Now we can define f(x) to be ::
+f(x) = x/2 + sin(x)
+Evaluating this function f for the value x=pi returns pi/2.::
+	   f(pi)
+We can also define functions that are not continuous but defined
+piecewise.  Let us define a function which is a parabola between 0
+to 1 and a constant from 1 to 2 .  Type the following
+::
+var('x')
+h(x)=x^2
+g(x)=1
+f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x)
+f
+We can also define functions convergent series and other series.
+We first define a function f(n) in the way discussed above.::
+var('n')
+function('f', n)
+To sum the function for a range of discrete values of n, we use the
+sage function sum.
+For a convergent series , f(n)=1/n^2 we can say ::
+var('n')
+function('f', n)
+f(n) = 1/n^2
+sum(f(n), n, 1, oo)
+Lets us now try another series ::
+f(n) = (-1)^(n-1)*1/(2*n - 1)
+sum(f(n), n, 1, oo)
+This series converges to pi/4.
+Following  are exercises that you must do.
+{{ show slide showing question 3 }}
+%% %% Define the piecewise function.
+f(x)=3x+2
+when x is in the closed interval 0 to 4.
+f(x)=4x^2
+between 4 to 6.
+%% %% Sum  of 1/(n^2-1) where n ranges from 1 to infinity.
+Please, pause the video here. Do the exercise(s) and then continue.
+{{ show slide showing solution 3 }}
+Moving on let us see how to perform simple calculus operations using Sage
+For example lets try an expression first ::
+diff(x**2+sin(x),x)
+The diff function differentiates an expression or a function. It's
+first argument is expression or function and second argument is the
+independent variable.
+We have already tried an expression now lets try a function ::
+f=exp(x^2)+arcsin(x)
+diff(f(x),x)
+To get a higher order differential we need to add an extra third argument
+for order ::
+diff(f(x),x,3)
+in this case it is 3.
+Just like differentiation of expression you can also integrate them ::
+x = var('x')
+s = integral(1/(1 + (tan(x))**2),x)
+s
+Many a times we need to find factors of an expression, we can use the
+"factor" function
+::
+y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2)
+f = factor(y)
+One can simplify complicated expression ::
+f.simplify_full()
+This simplifies the expression fully. We can also do simplification of
+just the algebraic part and the trigonometric part ::
+f.simplify_exp()
+f.simplify_trig()
+One can also find roots of an equation by using ``find_root`` function::
+phi = var('phi')
+find_root(cos(phi)==sin(phi),0,pi/2)
+Let's substitute this solution into the equation and see we were
+correct ::
+var('phi')
+f(phi)=cos(phi)-sin(phi)
+root=find_root(f(phi)==0,0,pi/2)
+f.substitute(phi=root)
+as we can see when we substitute the value the answer is almost = 0 showing
+the solution we got was correct.
+Following are a few exercises that you must do.
+%% %% Differentiate the following.
+1. sin(x^3)+log(3x)  , degree=2
+#. x^5*log(x^7)      , degree=4
+%% %% Integrate the given expression
+sin(x^2)+exp(x^3)
+%% %% Find x
+cos(x^2)-log(x)=0
+Does the equation have a root between 1,2.
+Please, pause the video here. Do the exercises and then continue.
+Lets us now try some matrix algebra symbolically ::
+var('a,b,c,d')
+A=matrix([[a,1,0],[0,b,0],[0,c,d]])
+A
+Now lets do some of the matrix operations on this matrix
+::
+A.det()
+A.inverse()
+Following is an (are) exercise(s) that you must do.
+%% %% Find the determinant and inverse of :
+A=[[x,0,1][y,1,0][z,0,y]]
+Please, pause the video here. Do the exercise(s) and then continue.
+{{{ Show the summary slide }}}
+That brings us to the end of this tutorial. In this tutorial we learnt
+how to
+* define symbolic expression and functions
+* use built-in constants and functions
+* use <Tab> to see the documentation of a function
+* do simple calculus
+* substitute values in expressions using ``substitute`` function
+* create symbolic matrices and perform operations on them