st-scripts: comparison getting-started-with-symbolics/script.rst

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-:2ce824b5adf4
+:9a1c5d134feb
 Checklist OK?       : <put date stamp here, if OK> [2010-10-05]
 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
+It raises a name error saying that ``y`` is not defined. We need to
-can declare y as a symbol using var function.
+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.
+Sage treats ``sin(y)`` as a symbolic expression. We can use this to do
-Thus sage treats sin(y) as a symbolic expression . We can use
+symbolic maths using Sage's built-in constants and expressions.
-this to do  symbolic maths using sage's built-in constants and
-expressions..
+Let us try out a few examples. ::
-So let us try ::
 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
+%% %% Define following expressions as symbolic expressions in Sage.
-in sage?
 1. x^2+y^2
 #. y^2-4ax
 Please, pause the video here. Do the exercise and then continue.
 The solution is on your screen.
+{{ show slide showing solution 1 }}
-Sage also provides a few built-in constants which are commonly used in mathematics .
+Sage also provides built-in constants which are commonly used in
-example : pi,e,infinity , Function n gives the numerical values of all these constants.
+mathematics, for instance pi, e, infinity. The function ``n`` gives
+the numerical values of all these constants.
-{{{ Type n(pi) n(e) n(oo) On the sage notebook }}}
+::
+n(pi)
+n(e)
+n(oo)
-If you look into the documentation of function "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 takes and what it returns. It will be very
+You will see what all arguments it takes and what it returns. It will
-helpful if you look at the documentation of all functions introduced through
+be very helpful if you look at the documentation of all functions
-this script.
+introduced in the course of this script.
+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
-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
+Apart from the constants Sage also has a lot of built-in functions
-sin,cos,log,factorial,gamma,exp,arcsin etc ...
+like ``sin``, ``cos``, ``log``, ``factorial``, ``gamma``, ``exp``,
-lets try some of them out on the sage notebook.
+``arcsin`` etc ...
+Lets try some of them out on the Sage notebook.
 ::
 sin(pi/2)
 arctan(oo)
 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
 %% %% Find the value of the following.
 1. sin(pi/4)
 #. ln(23)
 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
+Given that we have defined variables like x, y etc., we can define an
-an arbitrary function with desired name in the following way.::
+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 ::
 f(x) = x/2 + sin(x)
 	   f(pi)
 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
+to 1 and a constant from 1 to 2 .  Type the following
-screen
 ::
 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 convergent series and other series.
-We can also define functions which are series
 We first define a function f(n) in the way discussed above.::
 var('n')
 function('f', n)
 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.
+{{ show slide showing question 3 }}
 %% %% Define the piecewise function.
 f(x)=3x+2
 when x is in the closed interval 0 to 4.
 f(x)=4x^2
 %% %% Sum  of 1/(n^2-1) where n ranges from 1 to infinity.
 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. It's
-The diff function differentiates an expression or a function. Its
 first argument is expression or function and second argument is the
 independent variable.
 We have already tried an expression now lets try a function ::
 diff(f(x),x)
 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
+This simplifies the expression fully. We can also do simplification of
-of just the algebraic part and the trigonometric part ::
+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')
 f(phi)=cos(phi)-sin(phi)
 root=find_root(f(phi)==0,0,pi/2)
 Does the equation have a root between 1,2.
 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()
 A=[[x,0,1][y,1,0][z,0,y]]
 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.
+* define symbolic expression and functions
-* Using built-in constants and functions.
+* use built-in constants and functions
-* Using <Tab>  to see the documentation of a function.
+* use <Tab> to see the documentation of a function
-* Simple calculus operations .
+* do simple calculus
-* Substituting values in expression using substitute function.
+* substitute values in expressions using ``substitute`` function
-* Creating symbolic matrices and performing operation on them .
+* create symbolic matrices and perform operations on them

changeset 458	9a1c5d134feb
parent 442	a9b71932cbfa
child 491	ebfe3a675882