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* Defining symbolic expressions in sage.  

* Using built-in costants and functions. 

* Performing Integration, differentiation using sage. 

* Defining matrices. 

* Defining Symbolic functions.  

* Simplifying and solving symbolic expressions and functions.

We can use Sage for symbolic maths. 

sage simply returns the expression.

Thus sage treats sin(y) as a symbolic expression . We can use

this to do  symbolic maths using sage's built-in constants and

expressions..

So let us try ::

   var('x,alpha,y,beta') 

   x^2/alpha^2+y^2/beta^2

taking another example

   var('theta')

   sin^2(theta)+cos^2(theta)

Similarly, we can define many algebraic and trigonometric expressions

example : pi,e,infinity , Function n gives the numerical values of all these

{{{ Type n(pi)

   	n(e)

	n(oo) 

    On the sage notebook }}}  

If you look into the documentation of function "n" by doing

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 through

this script.

Also we can define the no. of digits we wish to use in the numerical

sin,cos,log,factorial,gamma,exp,arcsin etc ...

lets try some of them out on the sage notebook.

       var('x') 

       function('f',x)

piecewise.  Let us define a function which is a parabola between 0

to 1 and a constant from 1 to 2 .  Type the following as given on the

      var('x') 

      h(x)=x^2 g(x)=1 

      f=Piecewise(<Tab>

{{{ Show the documentation of Piecewise }}} 

::

We can also define functions which are series 

   var('n') 

   function('f', n)

   var('n') 

   function('f', n)

Lets us now try another series ::

    sum(f(n), n, 1, oo)

This series converges to pi/4. 

Moving on let us see how to perform simple calculus operations using Sage

    diff(x**2+sin(x),x) 

    2x+cos(x)

The diff function differentiates an expression or a function. Its

independent variable.

   f=exp(x^2)+arcsin(x) 

   diff(f(x),x)

To get a higher order differential we need to add an extra third argument

     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

    factor(<tab> 

    y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) 

    f = factor(y)

One can  simplify complicated expression ::

This simplifies the expression fully . We can also do simplification

    f.simplify_exp() 

    f.simplify_trig()

    phi = var('phi') 

    find_root(cos(phi)==sin(phi),0,pi/2)

     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.

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

::

    A.det() 

    A.inverse()

* We learnt about defining symbolic expression and functions.  

* Using built-in constants and functions.  

* Using <Tab>  to see the documentation of a function.  

* Simple calculus operations .  

* Substituting values in expression using substitute function.

* Creating symbolic matrices and performing operation on them .