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-.. Objectives
-.. ----------
-
-.. By the end of this tutorial, you will be able to
-
-.. 1. Defining symbolic expressions in sage.
-.. # Using built-in constants and functions.
-.. # Performing Integration, differentiation using sage.
-.. # Defining matrices.
-.. # Defining Symbolic functions.
-.. # Simplifying and solving symbolic expressions and functions.
-
-
-.. Prerequisites
-.. -------------
-
-.. 1. getting started with sage notebook
-
-
-.. Author : Amit
- Internal Reviewer :
- External Reviewer :
- Language Reviewer : Bhanukiran
- Checklist OK? : <, if OK> [2010-10-05]
-
-Symbolics with Sage
--------------------
-
-Hello friends and welcome to the tutorial on Symbolics with Sage.
-
-{{{ Show welcome slide }}}
-
-During the course of the tutorial we will learn
-
-{{{ Show outline slide }}}
-
-* Defining symbolic expressions in Sage.
-* Using built-in constants and functions.
-* Performing Integration, differentiation using Sage.
-* Defining matrices.
-* Defining symbolic functions.
-* Simplifying and solving symbolic expressions and functions.
-
-In addtion to a lot of other things, Sage can do Symbolic Math and we shall
-start with defining symbolic expressions in Sage.
-
-Have your Sage notebook opened. If not, pause the video and
-start you Sage notebook right now.
-
-On the sage notebook type::
-
- sin(y)
-
-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 treats ``sin(y)`` as a symbolic expression. We can use this to do
-symbolic math using Sage's built-in constants and expressions.
-
-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.
-
-Here is an expression in ``theta`` ::
-
- var('theta')
- sin(theta)*sin(theta)+cos(theta)*cos(theta)
-
-Now that you know how to define symbolic expressions in Sage, here is
-an exercise.
-
-{{ show slide showing question 1 }}
-
-%% %% Define following expressions as symbolic expressions 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 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 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
-
-::
-
- n(pi, digits = 10)
-
-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)
-
- arctan(oo)
-
- log(e,e)
-
-Following are exercises that you must do.
-
-{{ 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
-
-{{ 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.::
-
- 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)
-
-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. Let us define a function which is a parabola between 0
-to 1 and a constant from 1 to 2 . Type the following
-::
-
-
- var('x')
- h(x)=x^2
- g(x)=1
-
- f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x)
- f
-
-We can also define functions convergent series and other series.
-
-We first define a function f(n) in the way discussed above.::
-
- var('n')
- function('f', n)
-
-
-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)
-
-
-Lets us now try another series ::
-
-
- 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
- between 4 to 6.
-
-%% %% 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)
-
-The diff function differentiates an expression or a function. It's
-first argument is expression or function and second argument is the
-independent variable.
-
-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 differential we need to add an extra third argument
-for order ::
-
- 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
-
-::
-
- y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2)
- f = factor(y)
-
-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 ::
-
- f.simplify_exp()
- f.simplify_trig()
-
-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)
-
-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)
- f.substitute(phi=root)
-
-as we can see when we substitute the value the answer is almost = 0 showing
-the solution we got was correct.
-
-Following are a few exercises that you must do.
-
-%% %% Differentiate the following.
-
- 1. sin(x^3)+log(3x) , degree=2
- #. x^5*log(x^7) , degree=4
-
-%% %% Integrate the given expression
-
- sin(x^2)+exp(x^3)
-
-%% %% Find x
- cos(x^2)-log(x)=0
- 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()
-
-
-Following is an (are) exercise(s) that you must do.
-
-%% %% Find the determinant and inverse of :
-
- 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 }}}
-
-That brings us to the end of this tutorial. In this tutorial we learnt
-how to
-
-* 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
-