st-scripts: comparison #symbolics.rst#

equal deleted inserted replaced

-:fc71d5c27ce6
+:8a42b4203f6d
+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]