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+Symbolics with Sage
+-------------------
+
+This tutorial on using Sage for symbolic calculation is brought to you
+by Fossee group.
+
+{{{ Part of Notebook with title }}}
+
+We would be using simple mathematical functions on the sage notebook
+for this tutorial .
+
+During the course of the tutorial we will learn
+
+
+{{{ Part of Notebook with outline }}}
+
+To define symbolic expressions in sage .  Use built-in costants and
+function.  Integration , differentiation using sage .  Defining
+matrices.  Defining Symbolic functions .  Simplifying and solving
+symbolic expressions and functions
+    
+
+
+Using sage we can perform mathematical operations on symbols .
+
+On the sage notebook type::
+   
+    sin(y)
+
+It raises a name error saying that y is not defined . But in sage we
+can declare y as a symbol using var function. ::
+   
+    var('y')
+   
+Now if you type::
+
+    sin(y)
+
+     sage simply returns the expression .
+
+thus now sage treats sin(y) as a symbolic expression . You can use
+this to do a lot of symbolic maths using sage's built-in constants and
+expressions .
+
+Try out ::
+   
+   var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2
+ 
+Similarly , we can define many algebraic and trigonometric expressions
+using sage .
+
+
+
+Sage also provides a few built-in constants which are commonly used in
+mathematics .
+
+example : pi,e,oo , Function n gives the numerical values of all these
+    constants.
+
+For instance::
+
+   n(e)
+   
+   2.71828182845905
+
+gives numerical value of e.
+
+If you look into the documentation of n by doing ::
+
+   n(<Tab>
+
+You will see what all arguments it can take etc .. It will be very
+helpful if you look at the documentation of all functions introduced
+      
+
+Also we can define the no of digits we wish to use in the numerical
+value . For this we have to pass an argument digits.  Type::
+  
+   n(pi, digits = 10)
+
+Apart from the constants sage also has a lot of builtin functions like
+sin,cos,sinh,cosh,log,factorial,gamma,exp,arcsin,arccos,arctan etc ...
+lets try some out on the sage notebook. ::
+     
+   sin(pi/2)
+   
+   arctan(oo)
+     
+   log(e,e)
+
+
+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(<tab> {{{ Just to show the documentation
+       extend this line }}} 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 function that are not continuous but defined
+piecewise.  We will be using a function which is a parabola between 0
+to 1 and a constant from 1 to 2 .  type the following as given on the
+screen::
+      
+
+      var('x') h(x)=x^2 g(x)=1 f=Piecewise(<Tab> {{{ Just to show the
+      documentation extend this line }}}
+      f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) f
+
+Checking f at 0.4, 1.4 and 3 :: f(0.4) f(1.4) f(3)
+
+for f(3) it raises a value not defined in domain error .
+
+
+Apart from operations on expressions and functions one can also use
+them for 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)
+
+For the famous Madhava series :: var('n') function('f', n)
+
+    f(n) = (-1)^(n-1)*1/(2*n - 1)
+
+This series converges to pi/4. It was used by ancient Indians to
+interpret pi.
+
+For a divergent series, sum would raise a an error 'Sum is
+divergent' :: 
+	
+	var('n') 
+	function('f', n) 
+	f(n) = 1/n sum(f(n), n,1, oo)
+
+
+
+
+We can perform simple calculus operation using sage
+
+For example lets try an expression first ::
+
+    diff(x**2+sin(x),x) 2x+cos(x)
+
+The diff function differentiates an expression or a function . Its
+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 differentiation we need to add an extra argument
+for order ::
+ 
+   diff(<tab> 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
+
+
+
+To find factors of an expression use the function factor
+
+    factor(<tab> y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) f =
+    factor(y)
+
+One can also simplify complicated expression using sage ::
+    f.simplify_full()
+
+This simplifies the expression fully . You 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)
+
+Lets 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 the solution is almost equal to zero .
+
+
+We can also define symbolic matrices ::
+
+
+
+   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()
+
+You can do ::
+    
+    A.<Tab>
+
+To see what all operations are available
+
+
+{{{ Part of the notebook with summary }}}
+
+So in this tutorial we learnt how to
+
+
+We learnt about defining symbolic expression and functions .  
+And some built-in constants and functions .  
+Getting value of built-in constants using n function.  
+Using Tab to see the documentation.  
+Also we learnt how to sum a series using sum function.  
+diff() and integrate() for calculus operations .  
+Finding roots , factors and simplifying expression using find_root(), 
+factor() , simplify_full, simplify_exp , simplify_trig .
+Substituting values in expression using substitute function.
+And finally creating symbolic matrices and performing operation on them .