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-Symbolics with Sage
--------------------
-
-Hello friends and welcome to this tutorial on symbolics with sage.
-
-
-.. #[Madhu: Sounds more or less like an ad!]
-
-{{{ Part of Notebook with title }}}
-
-.. #[Madhu: Please make your instructions, instructional. While
-     recording if I have to read this, think what you are actually
-     meaning it will take a lot of time]
-
-We would be using simple mathematical functions on the sage notebook
-for this tutorial.
-
-.. #[Madhu: What is this line doing here. I don't see much use of it]
-
-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.
-
-.. #[Nishanth]: The formatting is all messed up
-                First fix the formatting and compile the rst
-                The I shall review
-.. #[Madhu: Please make the above items full english sentences, not
-     the slides like points. The person recording should be able to
-     read your script as is. It can read something like "we will learn
-     how to define symbolic expressions in Sage, using built-in ..."]
-
-Using sage we can perform mathematical operations on symbols.
-
-.. #[Madhu: Same mistake with period symbols! Please get the
-     punctuation right. Also you may have to rephrase the above
-     sentence as "We can use Sage to perform sybmolic mathematical
-     operations" or such]
-
-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.
-
-.. #[Madhu: But is not required]
-::
-    var('y')
-   
-Now if you type::
-
-    sin(y)
-
-    sage simply returns the expression .
-
-.. #[Madhu: Why is this line indented? Also full stop. When will you
-     learn? Yes we can correct you. But corrections are for you to
-     learn. If you don't learn from your mistakes, I don't know what
-     to say]
-
-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 .
-
-.. #[Madhu: "Thus now"? It sounds like Dus and Nou, i.e 10 and 9 in
-     Hindi! Full stop again. "a lot" doesn't mean anything until you
-     quantify it or give examples.]
-
-Try out
-
-.. #[Madhu: "So let us try" sounds better]
- ::
-   
-   var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2
- 
-Similarly , we can define many algebraic and trigonometric expressions
-using sage .
-
-.. #[Madhu: comma again. Show some more examples?]
-
-
-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.
-
-.. #[Madhu: This doesn't sound like scripts. How will I read this
-     while recording. Also if I were recording I would have read your
-     third constant as Oh-Oh i.e. double O. It took me at least 30
-     seconds to figure out it is infinity]
-
-For instance::
-
-   n(e)
-   
-   2.71828182845905
-
-gives numerical value of e.
-
-If you look into the documentation of n by doing
-
-.. #[Madhu: "documentation of the function "n"?]
-
-::
-   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
-
-.. #[Madhu: What does etc .. mean in a script?]
-
-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
-
-.. #[Madhu: "no of digits"? Also "We wish to obtain" than "we wish to
-     use"?]
-::
-
-   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.
-
-.. #[Madhu: Here "a lot" makes sense]
-::
-     
-   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)
-
-.. #[Madhu: What will the person recording show in the documentation
-     without a script for it? Please don't assume recorder can cook up
-     things while recording. It is impractical]
-
-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.  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
-
-.. #[Madhu: Instead of "We will be using ..." how about "Let us define
-     a function ..."]
-::
-      
-
-      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)
-
-.. #[Madhu: Again this doesn't sound like a script]
-
-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 .
-
-.. #[Madhu: I am not able to understand this line. "Use them as
-.. series". Use what as series?]
-
-We first define a function f(n) in the way discussed above.::
-
-   var('n') function('f', n)
-
-.. #[Madhu: Shouldn't this be on 2 separate lines?]
-
-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)
-
-.. #[Madhu: What is this? your double colon says it must be code block
-     but where is the indentation and other things. How will the
-     recorder know about it?]
-
-    f(n) = (-1)^(n-1)*1/(2*n - 1)
-
-This series converges to pi/4. It was used by ancient Indians to
-interpret pi.
-
-.. #[Madhu: I am losing the context. Please add something to bring
-     this thing to the context]
-
-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
-
-.. #[Madhu: When you switch to irrelevant topics make sure you use
-    some connectors in English like "Moving on let us see how to
-    perform simple calculus operations using Sage" or something like
-    that]
-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 .
-
-.. #[Madhu: Full stop, Full stop, Full stop]
-
-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)
-
-.. #[Madhu: Please try to be more explicit saying third argument]
-
-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
-
-.. #[Madhu: Two separate lines.]
-
-To find the factors of an expression use the "factor" function
-
-.. #[Madhu: See the diff]
-
-::
-    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()
-    
-.. #[Madhu: Separate lines?]
-
-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)
-
-.. #[Madhu: Separate lines?]
-
-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)
-
-.. #[Madhu: Separate lines?]
-
-as we can see the solution is almost equal to zero .
-
-.. #[Madhu: So what?]
-
-We can also define symbolic matrices ::
-
-
-
-   var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A
-
-.. #[Madhu: Why don't you break the lines?]
-
-Now lets do some of the matrix operations on this matrix
-
-.. #[Madhu: Why don't you break the lines? Also how do you connect
-     this up? Use some transformation keywords in English]
-::
-    A.det() A.inverse()
-
-.. #[Madhu: Why don't you break the lines?]
-
-You can do ::
-    
-    A.<Tab>
-
-To see what all operations are available
-
-.. #[Madhu: Sounds very abrupt]
-
-{{{ 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 .
-
-.. #[Madhu: See what Nishanth is doing. He has written this as
-     points. So easy to read out while recording. You may want to
-     reorganize like that]
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