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 Symbolics with Sage
 -------------------
 Hello friends and welcome to the tutorial on symbolics with sage.
+{{{ Show welcome slide }}}
-.. #[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 }}}
+{{{ Show outline slide  }}}
-To define symbolic expressions in sage.  Use built-in costants and
+* Defining symbolic expressions in sage.
-function. Integration, differentiation using sage. Defining
+* Using built-in costants and functions.
-matrices. Defining Symbolic functions. Simplifying and solving
+* Performing Integration, differentiation using sage.
-symbolic expressions and functions.
+* Defining matrices.
+* Defining Symbolic functions.
-.. #[Nishanth]: The formatting is all messed up
+* Simplifying and solving symbolic expressions and functions.
-First fix the formatting and compile the rst
-The I shall review
+We can use Sage for symbolic maths.
-.. #[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 .
+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
+Thus sage treats sin(y) as a symbolic expression . We can use
-learn. If you don't learn from your mistakes, I don't know what
+this to do  symbolic maths using sage's built-in constants and
-to say]
+expressions..
-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
+So let us try ::
-expressions .
+var('x,alpha,y,beta')
-.. #[Madhu: "Thus now"? It sounds like Dus and Nou, i.e 10 and 9 in
+x^2/alpha^2+y^2/beta^2
-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
+taking another example
+var('theta')
+sin^2(theta)+cos^2(theta)
+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
+example : pi,e,infinity , Function n gives the numerical values of all these
 constants.
-.. #[Madhu: This doesn't sound like scripts. How will I read this
+{{{ Type n(pi)
-while recording. Also if I were recording I would have read your
+	n(e)
-third constant as Oh-Oh i.e. double O. It took me at least 30
+	n(oo)
-seconds to figure out it is infinity]
+On the sage notebook }}}
-For instance::
-n(e)
+If you look into the documentation of function "n" by doing
-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
+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
+helpful if you look at the documentation of all functions introduced through
+this script.
-.. #[Madhu: What does etc .. mean in a script?]
-Also we can define the no of digits we wish to use in the numerical
+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 ...
+sin,cos,log,factorial,gamma,exp,arcsin etc ...
-lets try some out on the sage notebook.
+lets try some of them out on the sage notebook.
-.. #[Madhu: Here "a lot" makes sense]
 ::
 sin(pi/2)
 arctan(oo)
 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
+var('x')
-extend this line }}} function('f',x)
+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
+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
+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
+var('x')
-documentation extend this line }}}
+h(x)=x^2 g(x)=1
+f=Piecewise(<Tab>
+{{{ Show the documentation of Piecewise }}}
+::
 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]
+We can also define functions which are series
-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)
+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)
+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)
+Lets us now try another series ::
-.. #[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)
+sum(f(n), n, 1, oo)
-This series converges to pi/4. It was used by ancient Indians to
-interpret pi.
+This series converges to pi/4.
-.. #[Madhu: I am losing the context. Please add something to bring
-this thing to the context]
+Moving on let us see how to perform simple calculus operations using Sage
-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)
+diff(x**2+sin(x),x)
+2x+cos(x)
-The diff function differentiates an expression or a function . Its
+The diff function differentiates an expression or a function. Its
 first argument is expression or function and second argument is the
-independent variable .
+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)
+f=exp(x^2)+arcsin(x)
+diff(f(x),x)
-To get a higher order differentiation we need to add an extra argument
+To get a higher order differential we need to add an extra third 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
+x = var('x')
+s = integral(1/(1 + (tan(x))**2),x)
-.. #[Madhu: Two separate lines.]
+s
-To find the factors of an expression use the "factor" function
-.. #[Madhu: See the diff]
+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(<tab>
-factor(y)
+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 ::
+One can  simplify complicated expression ::
 f.simplify_full()
-This simplifies the expression fully . You can also do simplification
+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()
+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)
+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)
+var('phi')
-root=find_root(f(phi)==0,0,pi/2) f.substitute(phi=root)
+f(phi)=cos(phi)-sin(phi)
+root=find_root(f(phi)==0,0,pi/2)
-.. #[Madhu: Separate lines?]
+f.substitute(phi=root)
-as we can see the solution is almost equal to zero .
+as we can see when we substitute the value the answer is almost = 0 showing
+the solution we got was correct.
-.. #[Madhu: So what?]
-We can also define symbolic matrices ::
+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
-.. #[Madhu: Why don't you break the lines?]
+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
-.. #[Madhu: Why don't you break the lines? Also how do you connect
-this up? Use some transformation keywords in English]
+::
-::
+A.det()
-A.det() A.inverse()
+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 .
+* We learnt about defining symbolic expression and functions.
-And some built-in constants and functions .
+* Using built-in constants and functions.
-Getting value of built-in constants using n function.
+* Using <Tab>  to see the documentation of a function.
-Using Tab to see the documentation.
+* Simple calculus operations .
-Also we learnt how to sum a series using sum function.
+* Substituting values in expression using substitute function.
-diff() and integrate() for calculus operations .
+* Creating symbolic matrices and performing operation on them .
-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]

changeset 342	588b681e70c6
parent 321	2e49b1b72996