--- a/symbolics/script.rst Wed Oct 20 16:19:55 2010 +0530
+++ b/symbolics/script.rst Thu Oct 21 00:22:42 2010 +0530
@@ -3,43 +3,23 @@
Hello friends and welcome to the tutorial on symbolics with sage.
-
-.. #[Madhu: Sounds more or less like an ad!]
-
-{{{ Part of Notebook with title }}}
+{{{ Show welcome slide }}}
-.. #[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.
+{{{ Show outline slide }}}
-.. #[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 ..."]
+* 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.
-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]
+We can use Sage for symbolic maths.
On the sage notebook type::
@@ -48,7 +28,7 @@
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')
@@ -56,66 +36,56 @@
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
- learn. If you don't learn from your mistakes, I don't know what
- to say]
+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..
-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
+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)
-.. #[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
+
+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
- 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::
+{{{ Type n(pi)
+ n(e)
+ n(oo)
+ On the sage notebook }}}
- n(e)
-
- 2.71828182845905
+
-gives numerical value of e.
-
-If you look into the documentation of n by doing
+If you look into the documentation of function "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
+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.
-.. #[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
@@ -125,10 +95,10 @@
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,cos,log,factorial,gamma,exp,arcsin etc ...
+lets try some of them out on the sage notebook.
-.. #[Madhu: Here "a lot" makes sense]
+
::
sin(pi/2)
@@ -141,12 +111,9 @@
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)
+ var('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 ::
@@ -158,186 +125,153 @@
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
+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
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 }}}
+ var('x')
+ 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]
-
-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 can also define functions which are 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)
-
-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]
+ sum(f(n), n, 1, oo)
-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)
+This series converges to pi/4.
-
-We can perform simple calculus operation using sage
+Moving on let us see how to perform simple calculus operations 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 .
-
-.. #[Madhu: Full stop, Full stop, Full stop]
+independent variable.
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
-
-.. #[Madhu: Two separate lines.]
+ x = var('x')
+ s = integral(1/(1 + (tan(x))**2),x)
+ 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(y)
+ 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 ::
+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)
-
-.. #[Madhu: Separate lines?]
+ 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)
-
-.. #[Madhu: Separate lines?]
+ 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 .
+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 ::
- var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A
+Lets us now try some matrix algebra symbolically ::
+
+
-.. #[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.inverse()
-
-.. #[Madhu: Why don't you break the lines?]
-You can do ::
-
- A.<Tab>
+::
+ A.det()
+ A.inverse()
-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 .
+* 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 .
-.. #[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]