PoojaMoolya at 12:16, 20 February 2017

2017-02-20T12:16:42Z

Gaurav at 07:19, 10 July 2014

2014-07-10T07:19:53Z

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Sneha at 10:13, 11 March 2013

2013-03-11T10:13:40Z

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Minal: Created page with '{| border=1 !Timing !Narration |- | 0:00 | Welcome to the tutorial on 'Using Sage'. |- | 0:02 | At the end of this tutorial, you will be able to, # Learn the range of things fo…'

2013-01-02T12:00:47Z

Created page with '{| border=1 !Timing !Narration |- | 0:00 | Welcome to the tutorial on 'Using Sage'. |- | 0:02 | At the end of this tutorial, you will be able to, # Learn the range of things fo…'

New page

{| border=1
!Timing
!Narration
|-
| 0:00
| Welcome to the tutorial on 'Using Sage'.

|-
| 0:02
| At the end of this tutorial, you will be able to,

# Learn the range of things for which Sage can be used.
# Know the functions used for Calculus in Sage.
# Learn about graph theory and number theory using Sage.

|-
|0:16
| Before beginning this tutorial,we would suggest you to complete the tutorial on "Getting started with Sage".

|-
|0:22
|Let us begin with Calculus.

|-
|0:24
|We shall be looking at limits, differentiation, integration, and Taylor polynomial.

|-
| 0:30
| We have our Sage notebook running.

|-
|0:32
| In case, you don't have it running, start is using the command, <tt>sage --notebook</tt>.

|-
|0:39
|So type sage and specify notebook.

|-
| 0:45
|So to find the limit of the function x into sin(1/x), at x=0, we say by typing it lim(x*sin(1/x),x=0)

|-
|1:07
| We get the limit to be 0, as expected.

|-
|1:11
|It is also possible to limit a point from one direction. For example, let us find the limit of 1/x at x=0, when approaching from the positive side.

|-
|1:23
|lim within brackets (1/x, x=0, dir='right')

|-
| 1:32
| To find the limit from the negative side, we say,

|-
|1:36
|lim(1/x, x=0, dir='left')

|-
| 1:45
| Let us see how to perform differentiation, using Sage.

|-
|1:51
| We shall find the differential of the expression <tt>exp of (sin(x squared)) by x</tt> with respect to <tt>x</tt>.

|-
|2:11
| For this, we shall first define the expression, and then use the <tt>diff</tt> function to obtain the differential of the expression.

|-
|2:21
|So we can type var('x)
f=exp of (sin x squared)/x and then third line you can type
diff(f,x)

|-
| 2:44
| We can also obtain the partial differentiation of an expression w.r.t one of the variables.

|-
|2:51
| Let us differentiate the expression <tt>exp(sin (y - x squared))/x</tt> w.r.t x and y.

|-
|3:07
|that is with respect to x and y.

|-
|3:10
|so you can type var('x y')

|-
|3:15
|second line you can type f=exp(sin(y - x squared))by x

|-
|3:26
|then you can type diff(f,x) then next line you can type diff(f,y)

|-
| 3:43
| Thus we get our partial differential solution.

|-
|3:51
|Now, let us look at integration.

|-
|3:53
| We shall use the expression obtained from the differentiation that we did before, <tt>diff(f, y)</tt> which gave us the expression ---<tt>e^(sin(-x squared + y)) multiplied by cos(-x squared plus y) by x</tt>.

|-
|4:15
| The <tt>integrate</tt> command is used to obtain the integral of an expression or function.

|-
|4:21
|So you can type integrate(e^(sin(-x squared plus y))multiplied by cos(-x squared +y)by x,y)

|-
| 4:39
| As we can see,we get back the correct expression.

|-
|4:44
| The minus sign being inside or outside the <tt>sin</tt> function doesn't change much.

|-
|4:48
|Now, let us find the value of the integral between the limits 0 and pi/2.

|-
|4:55
|So for that you can type integral(e^(sin(-x squared plus y))multiplied by cos(-x squared plus y) by x,y,0,pi/2)

|-
| 5:11
| Hence we get our solution for definite integration.

|-
|5:15
|Now, let us see how to obtain the Taylor expansion of an expression using sage.

|-
|5:20
|Let us obtain the Taylor expansion of <tt>(x + 1) raised to n</tt> up to degree 4 about 0.

|-
|5:27
|So for that you can type var of ('x n') then type taylor within brackets((x+1) raised to n,x,0,4)

|-
| 5:42
| We easily got the Taylor expansion,using the taylor function <tt>taylor() function</tt>.

|-
| 5:49
| So this brings us to the end of the features of Sage for Calculus, that we will be looking at.

|-
|5:56
| For more, look at the Calculus quick-ref from the Sage Wiki.

|-
| 6:03
| Next let us move on to Matrix Algebra.

|-
|6:07
| Let us begin with solving the equation <tt>Ax = v</tt>, where A is the matrix <tt><nowiki>matrix ([[1,2], [3,4]])</nowiki></tt> and v is the vector <tt><nowiki>vector ([1,2])</nowiki></tt>.

|-
| 6:19
| So, to solve the equation, <tt>Ax = v</tt> we simply say

|-
|6:23
|A=matrix ([1,2] comma [3,4]) then v is equal to vector([1,2])

|-
|6:35
|then x=A dot solve underscore right(v)

|-
|6:50
|then you have to type

|-
|7:01
|then type x

|-
| 7:07
| To solve an equation, <tt>xA = v</tt> we simply say

|-
|7:14
|x=A dot solve underscore left(v)

|-
|7:25
|then type x

|-
| 7:32
| The left and right here, denote the position of <tt>A</tt>, relative to x.

|-
|7:36
|Now, let us look at Graph Theory in Sage.

|-
|7:39
|We shall look at some ways to create graphs and some of the graph families available in Sage.

|-
|7:45
|The simplest way to define an arbitrary graph is to use a dictionary of lists.

|-
|7:49
| We create a simple graph by using the <tt>Graph()</tt> function.

|-
|7:53
|So G=Graph({0:[1,2,3], 2:[4]}) and hit shift enter

|-
| 8:13
| To view the visualization of the graph, we say

|-
|8:17
|G.show()

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| 8:24
| Similarly, we can obtain a directed graph using the <tt>DiGraph</tt> function.

|-
| 8:31
| So ,you have to type G=DiGraph that is D and G are capital ({0 colon [1,2,3],2 colon[4]}) and hit shift enter.

|-
| 8:59
| Sage also provides a lot of graph families which can be viewed by typing <tt><nowiki>graph.<tab></nowiki></tt>.

|-
| 9:04
| Let us obtain a complete graph with 5 vertices and then show the graph.

|-
| 9:09
| So you can type there G=graphs dot CompleteGraph(5) then type G dot show()

|-
| 9:28
| Sage provides other functions for Number theory and Combinatorics.

|-
|9:35
|Let's have a glimpse of a few of them.

|-
|9:42
| So <tt>prime_range</tt> gives primes in the range 100 to 200.

|-
|9:46
|So you can type there prime_range within brackets 100,200.

|-
| 9:58
| <tt>is_prime</tt> checks if 1999 is a prime number or not.

|-
| 10:05
| So for that you can type if_prime of (1999) and hit shift enter.

|-
|10:13
|So you will get the answer.

|-
| 10:15
| <tt>factor(2001)</tt> gives the factorized form of 2001.

|-
|10:20
|So to see that you can type factor(2001) and hit shift enter.

|-
|10:33
|So you can see the value in the output.

|-
| 10:36
| So the <tt>Permutations()</tt> gives the permutations of <tt><nowiki>[1, 2, 3, 4]</nowiki></tt>

|-
|10:43
|So for that you can type C=Permutations([1,2,3,4]) and next you can type C.list()

|-
| 10:57
| And the <tt>Combinations()</tt> gives all the combinations of <tt><nowiki>[1, 2, 3, 4]</nowiki></tt>

|-
|11:02
|For that you can type C= Combinations([1,2,3,4]) and type C dot list()

|-
|11:17
|So now you can see the solution displayed

|-
| 11:26
| This brings us to the end of the tutorial.

|-
|11:29
| In this tutorial, we have learnt to,

|-
|11:32
|1. Use functions for calculus like -- - lim()-- to find out the limit of a function - diff()-- to find out the differentiation of an expression - integrate()-- to integrate over an expression - integral()-- to find out the definite integral of an expression by specifying the limits<br/>

|-
|11:52
|solve()-- to solve a function, relative to it's position.

|-
|11:56
|then create both a simple graph and a directed graph, using the functions <tt>graph</tt> and <tt>digraph</tt> respectively.

|-
|12:02
|then use functions for number theory.

|-
|12:04
|So for eg: - primes_range()-- function to find out the prime numbers within the specified range.

|-
|12:11
|then factor()-- function to find out the factorized form of the specified number.

|-
|12:15
| Permutations(), Combinations()-- to obtain the required permutation and combinations for the given set of values.

|-
| 12:22
| So here are some self assessment questions for you to solve

|-
|12:25
|1. How do you find the limit of the function <tt>x/sin(x)</tt> as <tt>x</tt> tends to <tt>0</tt> from the negative side.

|-
|12:32
|2. List all the primes between 2009 and 2900

|-
|12:37
|3. Solve the system of linear equations<br/> x-2y+3z = 7 2x+3y-z = 5 x+2y+4z = 9

|-
| 12:57
| So now we can look at the answers,

|-
| 13:02
| 1. To find out the limit of an expression from the negative side,we add an argument dir="left" as

|-
|13:09
|lim of(x/sin(x), x=0, dir="left")

|-
|13:19
|2. The prime numbers from 2009 and 2900 can be obtained as,

prime_range(2009, 2901)

|-
|13:32
|3. We shall first write the equations in matrix form and then use the solve() function

|-
| 13:39
| So you can type <nowiki>A = Matrix of within brackets([[1, -2, 3] comma</nowiki>
<nowiki>[2, 3, -1] comma</nowiki>
<nowiki>[1, 2, 4]])</nowiki>

|-
|13:48
|<nowiki>b = vector within brackets([7, 5, 9])</nowiki>

|-
|13:52
|then x = A dot solve_right(b)

|-
|13:58
|Then type x so that you can view the output of x.

|-
| 14:03
| So we hope that you have enjoyed this tutorial and found it useful.

|-
|14:06
| Thank you!

|}

@@ Line 10: / Line 10: @@
 | 00:02
 | At the end of this tutorial, you will be able to,
 Learn the range of things for which Sage can be used.
-# Learn the range of things for which Sage can be used.
+Know the functions used for Calculus in Sage.
-# Know the functions used for Calculus in Sage.
+Learn about graph theory and number theory using Sage.
-# Learn about graph theory and number theory using Sage.
 |-
@@ Line 54: / Line 53: @@
 |01:23
 |lim within brackets (1/x, x=0, dir='right')
 |-
@@ Line 78: / Line 76: @@
 |-
 |02:21
-|So we can type var('x)
+|So we can type var('x) f=exp of (sin x squared)/x and then third line you can type diff(f,x)
- f=exp of (sin x squared)/x and then third line you can type
- diff(f,x)
 |-
@@ Line 364: / Line 360: @@
 |-
 |12:25
-|1. How do you find the limit of the function x/sin(x) as x tends to 0 from the negative side.
+|How do you find the limit of the function x/sin(x) as x tends to 0 from the negative side.
 |-
 |12:32
-|2. List all the primes between 2009 and 2900
+|List all the primes between 2009 and 2900
 |-
 |12:37
-|3. Solve the system of linear equations x-2y+3z = 7 2x+3y-z = 5 x+2y+4z = 9
+| Solve the system of linear equations x-2y+3z = 7 2x+3y-z = 5 x+2y+4z = 9
 |-
@@ Line 380: / Line 376: @@
 |-
 | 13:02
-| 1. To find out the limit of an expression from the negative side,we add an argument dir="left" as
+|To find out the limit of an expression from the negative side,we add an argument dir="left" as
 |-
@@ Line 388: / Line 384: @@
 |-
 |13:19
-|2. The prime numbers from 2009 and 2900 can be obtained as,
+|The prime numbers from 2009 and 2900 can be obtained as, prime_range(2009, 2901)
-−
- prime_range(2009, 2901)
 |-
 |13:32
-|3. We shall first write the equations in matrix form and then use the solve() function
+|We shall first write the equations in matrix form and then use the solve() function
 |-

Python/C2/Using-Sage/English-timed - Revision history

PoojaMoolya at 12:16, 20 February 2017

Gaurav at 07:19, 10 July 2014

Sneha at 10:13, 11 March 2013

Minal: Created page with '{| border=1 !Timing !Narration |- | 0:00 | Welcome to the tutorial on 'Using Sage'. |- | 0:02 | At the end of this tutorial, you will be able to, # Learn the range of things fo…'