.. 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
-------------------
.. L1
{{{ Show the first slide containing title, name of the production
team along with the logo of MHRD }}}
.. R1
Hello friends and welcome to the tutorial on "Symbolics with Sage".
.. L2
{{{ Show objectives slide }}}
.. R2
At the end of this tutorial, you will be able to,
1. Define symbolic expressions in sage.
#. Use built-in constants and functions.
#. Perform Integration, differentiation using sage.
#. Define matrices.
#. Define Symbolic functions.
#. Simplify0and solve symbolic expressions and functions.
.. L3
{{{ Switch to the pre-requisite slide }}}
.. R3
Before beginning this tutorial,we would suggest you to complete the
tutorial on "Getting started with sage notebook".
In addtion to a lot of other things, Sage can do Symbolic Math and
we shall start with defining symbolic expressions in Sage.
.. L4
{{{ Open the sage notebook }}}
.. R4
Have your Sage notebook opened. If not, pause the video and
start you Sage notebook right now.
.. R5
On the sage notebook type
.. L5
::
sin(y)
.. R6
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.
.. L6
::
var('y')
.. R7
Now if you type sin(y),Sage simply returns the expression.
.. L7
::
sin(y)
.. R8
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.
.. L8
::
var('x,alpha,y,beta')
x^2/alpha^2+y^2/beta^2
.. R9
We have defined 4 variables, ``x``, ``y``, ``alpha`` and ``beta`` and
have defined a symbolic expression using them.
Here is an expression in ``theta``
.. L9
::
var('theta')
sin(theta)*sin(theta)+cos(theta)*cos(theta)
.. R10
Now that you know how to define symbolic expressions in Sage, here is
an exercise.
Pause the video here, try out the following exercise and resume the video.
.. L10
.. L11
{{{ show slide showing exercise 1 }}}
.. R11
Define following expressions as symbolic expressions in Sage.
1. x^2+y^2
#. y^2-4ax
.. L12
{{{continue from paused state}}}
{{{ show slide showing solution 1 }}}
.. R12
The solution is on your screen.
<pause for sometime,then continue>
.. R13
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.
.. L13
::
n(pi)
n(e)
n(oo)
.. R14
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.
.. L14
::
n<Tab>
.. R15
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.
.. L15
::
n(pi, digits = 10)
.. R16
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.
.. L16
::
sin(pi/2)
arctan(oo)
log(e,e)
.. R17
Pause the video here, try out the following exercise and resume the video.
.. L17
.. L18
{{{ show slide showing exercise 2 }}}
.. R18
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)
.. L19
{{{continue from paused state}}}
{{{ show slide showing solution 2 }}}
.. R19
The solutions are on your screen.
<pause for sometime,then continue>
.. R20
Given that we have defined variables like x, y etc., we can define an
arbitrary function with desired name in the following way.
.. L20
::
var('x')
function('f',x)
.. R21
Here f is the name of the function and x is the independent variable .
Now we can define f(x)
.. L21
::
f(x) = x/2 + sin(x)
.. R22
Evaluating this function f for the value x=pi returns pi/2.
.. L22
::
f(pi)
.. R23
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
.. L23
::
var('x')
h(x)=x^2
g(x)=1
f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x)
f
.. R24
We can also define functions convergent series and other series.
We first define a function f(n) in the way discussed before.
.. L24
::
var('n')
function('f', n)
.. R25
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
.. L25
::
var('n')
function('f', n)
f(n) = 1/n^2
sum(f(n), n, 1, oo)
.. R26
Let us now try another series
.. L26
::
f(n) = (-1)^(n-1)*1/(2*n - 1)
sum(f(n), n, 1, oo)
.. R27
This series converges to pi/4.
Pause the video here, try out the following exercise and resume the video.
.. L27
.. L28
{{{ show slide showing exercise 3 }}}
.. R28
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.
.. L29
{{{continue from paused state}}}
{{{ show slide showing solution 3 }}}
.. R29
The solution is on your screen
<pause for sometime,then continue>
.. R30
Moving on let us see how to perform simple calculus operations using Sage
For example lets try an expression first
.. L30
::
diff(x**2+sin(x),x)
.. R31
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
.. L31
::
f=exp(x^2)+arcsin(x)
diff(f(x),x)
.. R32
To get a higher order differential we need to add an extra third argument
for order
.. L32
::
diff(f(x),x,3)
.. R33
in this case it is 3.
Just like differentiation of expression you can also integrate them
.. L33
::
x = var('x')
s = integral(1/(1 + (tan(x))**2),x)
s
.. R34
Many a times we need to find factors of an expression, we can use the
"factor" function
.. L34
::
y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2)
f = factor(y)
.. R35
One can simplify complicated expression by using the function ``simplify``.
.. L35
::
f.simplify_full()
.. R36
This simplifies the expression fully. We can also do simplification of
just the algebraic part and the trigonometric part
.. L36
::
f.simplify_exp()
f.simplify_trig()
.. R37
One can also find roots of an equation by using ``find_root`` function
.. L37
::
phi = var('phi')
find_root(cos(phi)==sin(phi),0,pi/2)
.. R38
Let's substitute this solution into the equation and see we were
correct
.. L38
::
var('phi')
f(phi)=cos(phi)-sin(phi)
root=find_root(f(phi)==0,0,pi/2)
f.substitute(phi=root)
.. R39
as we can see when we substitute the value the answer is almost = 0 showing
the solution we got was correct.
Pause the video here, try out the following exercise and resume the video.
.. L39
.. L40
{{{ show slide showing exercise 4 }}}
.. R40
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.
.. L41
{{{continue from paused state}}}
{{{ show slide showing solution 4 }}}
.. R41
The solution is on your screen
<pause for sometime,then continue>
.. R42
Lets us now try some matrix algebra symbolically
.. L42
::
var('a,b,c,d')
A=matrix([[a,1,0],[0,b,0],[0,c,d]])
A
.. R43
Now lets do some of the matrix operations on this matrix
.. L43
::
A.det()
A.inverse()
.. R44
As we can see, we got the determinant and the inverse of the matrix
respectively.
Pause the video here, try out the following exercise and resume the video.
.. L44
.. L45
{{{ show slide showing exercise 5 }}}
.. R45
Find the determinant and inverse of
A=[[x,0,1][y,1,0][z,0,y]]
.. L46
{{{continue from paused state}}}
{{{ show slide showing solution 4 }}}
.. R47
The solution is on your screen
<pause for sometime,then continue>
.. L48
{{{ Show the summary slide }}}
.. R48
This brings us to the end of this tutorial. In this tutorial,
we have learnt to,
1. Define symbolic expression and functions using the method ``var``.
#. Use built-in constants like pi,e,oo and functions like
sum,sin,cos,log,exp and many more.
#. Use <Tab> to see the documentation of a function.
#. Do simple calculus using functions
- diff()--to find a differential of a function
- integral()--to integrate an expression
- simplify--to simplify complicated expression.
#. Substitute values in expressions using ``substitute`` function.
#. Create symbolic matrices and perform operations on them like--
- det()--to find out the determinant of a matrix
- inverse()--to find out the inverse of a matrix.
.. L49
{{{Show self assessment questions slide}}}
.. R49
Here are some self assessment questions for you to solve
1. How do you define a name 'y' as a symbol?
2. Get the value of pi upto precision 5 digits using sage?
3. Find third order differential function of
f(x)=sin(x^2)+exp(x^3)
.. L50
{{{solution of self assessment questions on slide}}}
.. R50
And the answers,
1. We define a symbol using the function ``var``.In this case it will be
::
var('y')
2. The value of pi upto precision 5 digits is given as,
::
n(pi,5)
3. The third order differential function can be found out by adding the
third argument which states the order.The syntax will be,
::
diff(f(x),x,3)
.. L51
{{{Show thank you slide}}}
.. R51
Hope you have enjoyed this tutorial and found it useful.
Thank You!