--- a/getting-started-with-symbolics/script.rst Wed Nov 10 19:00:23 2010 +0530
+++ b/getting-started-with-symbolics/script.rst Thu Nov 11 02:04:14 2010 +0530
@@ -25,66 +25,65 @@
Symbolics with Sage
-------------------
-Hello friends and welcome to the tutorial on symbolics with sage.
+Hello friends and welcome to the tutorial on Symbolics with Sage.
{{{ Show welcome slide }}}
-
-.. #[Madhu: What is this line doing here. I don't see much use of it]
-
During the course of the tutorial we will learn
{{{ Show outline slide }}}
-* Defining symbolic expressions in sage.
+* Defining symbolic expressions in Sage.
* Using built-in constants and functions.
-* Performing Integration, differentiation using sage.
+* Performing Integration, differentiation using Sage.
* Defining matrices.
-* Defining Symbolic functions.
+* Defining symbolic functions.
* Simplifying and solving symbolic expressions and functions.
-We can use Sage for symbolic maths.
+Amongst a lot of other things, Sage can do Symbolic Math and we shall
+start with defining symbolic expressions in Sage.
+
+Hope you have your Sage notebook open. If not, pause the video and
+start you Sage notebook.
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.
+It raises a name error saying that ``y`` is not defined. We need to
+declare ``y`` as a symbol. We do it using the ``var`` function.
+::
-
-::
var('y')
Now if you type::
sin(y)
-sage simply returns the expression.
-
+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..
+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 ::
+Let us try out a few examples. ::
var('x,alpha,y,beta')
x^2/alpha^2+y^2/beta^2
+
+We have defined 4 variables, ``x``, ``y``, ``alpha`` and ``beta`` and
+have defined a symbolic expression using them.
-taking another example ::
+Here is an expression in ``theta`` ::
var('theta')
sin(theta)*sin(theta)+cos(theta)*cos(theta)
-Similarly, we can define many algebraic and trigonometric expressions using sage .
-
+Now that you know how to define symbolic expressions in Sage, here is
+an exercise.
-Following is an exercise that you must do.
+{{ show slide showing question 1 }}
-%% %% Define following expressions as symbolic expressions
-in sage?
+%% %% Define following expressions as symbolic expressions in Sage.
1. x^2+y^2
#. y^2-4ax
@@ -93,42 +92,37 @@
The solution is on your screen.
-
-Sage also provides a few built-in constants which are commonly used in mathematics .
-
-example : pi,e,infinity , Function n gives the numerical values of all these constants.
+{{ show slide showing solution 1 }}
-{{{ Type n(pi) n(e) n(oo) On the sage notebook }}}
-
-
-
-If you look into the documentation of function "n" by doing
-
-.. #[Madhu: "documentation of the function "n"?]
+Sage also provides built-in constants which are commonly used in
+mathematics, for instance pi, e, infinity. The function ``n`` gives
+the numerical values of all these constants.
+::
+ n(pi)
+ n(e)
+ n(oo)
+
+If you look into the documentation of function ``n`` by doing
::
n(<Tab>
-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.
-
-
+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 in the course of this 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
+Also we can define the number of digits we wish to have in the
+constants. 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,log,factorial,gamma,exp,arcsin etc ...
-lets try some of them out on the sage notebook.
+Apart from the constants Sage also has a lot of built-in functions
+like ``sin``, ``cos``, ``log``, ``factorial``, ``gamma``, ``exp``,
+``arcsin`` etc ...
-
+Lets try some of them out on the Sage notebook.
::
sin(pi/2)
@@ -137,9 +131,12 @@
log(e,e)
-Following is are exercises that you must do.
+Following are exercises that you must do.
-%% %% Find the values of the following constants upto 6 digits precision
+{{ show slide showing question 2 }}
+
+%% %% Find the values of the following constants upto 6 digits
+ precision
1. pi^2
#. euler_gamma^2
@@ -150,19 +147,18 @@
1. sin(pi/4)
#. ln(23)
-Please, pause the video here. Do the exercises and then continue.
+Please, pause the video here. Do the exercises and then continue.
-The solutions are on your screen.
+The solutions are on your screen
-
+{{ show slide showing solution 2 }}
-Given that we have defined variables like x,y etc .. , We can define
-an arbitrary function with desired name in the following way.::
+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('f',x)
-
Here f is the name of the function and x is the independent variable .
Now we can define f(x) to be ::
@@ -174,29 +170,18 @@
We can also define functions that are not continuous but defined
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
-
+to 1 and a constant from 1 to 2 . Type the following
::
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
-
-
-
-
-We can also define functions which are series
-
+We can also define functions convergent series and other series.
We first define a function f(n) in the way discussed above.::
@@ -221,11 +206,11 @@
f(n) = (-1)^(n-1)*1/(2*n - 1)
sum(f(n), n, 1, oo)
-
This series converges to pi/4.
+Following are exercises that you must do.
-Following are exercises that you must do.
+{{ show slide showing question 3 }}
%% %% Define the piecewise function.
f(x)=3x+2
@@ -237,14 +222,15 @@
Please, pause the video here. Do the exercise(s) and then continue.
+{{ show slide showing solution 3 }}
+
Moving on let us see how to perform simple calculus operations using Sage
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
+The diff function differentiates an expression or a function. It's
first argument is expression or function and second argument is the
independent variable.
@@ -256,44 +242,40 @@
To get a higher order differential we need to add an extra third argument
for order ::
- diff(<tab> diff(f(x),x,3)
+ diff(f(x),x,3)
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
-
-
-Many a times we need to find factors of an expression ,we can use the "factor" function
+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 ::
+One can simplify complicated expression ::
f.simplify_full()
-This simplifies the expression fully . We can also do simplification
-of just the algebraic part and the trigonometric part ::
+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()
-
-
-One can also find roots of an equation by using find_root function::
+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)
-Lets substitute this solution into the equation and see we were
+Let's substitute this solution into the equation and see we were
correct ::
var('phi')
@@ -322,18 +304,13 @@
Please, pause the video here. Do the exercises and then continue.
-
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
Now lets do some of the matrix operations on this matrix
-
-
::
A.det()
A.inverse()
@@ -348,17 +325,15 @@
Please, pause the video here. Do the exercise(s) and then continue.
-
-
{{{ Show the summary slide }}}
-So in this tutorial we learnt how to
-
+That brings us to the end of this tutorial. In this tutorial we learnt
+how to
-* 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 .
+* define symbolic expression and functions
+* use built-in constants and functions
+* use <Tab> to see the documentation of a function
+* do simple calculus
+* substitute values in expressions using ``substitute`` function
+* create symbolic matrices and perform operations on them