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{| style="border-spacing:0;"
| style="border-top:0.05pt double #808080;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| <center>'''Visual Cue'''</center>
| style="border:0.05pt double #808080;padding:0.049cm;"| <center>'''Narration'''</center>
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 1
Title Slide
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Hello Friends and Welcome to the tutorial on 'Using Sage'.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 2
Objectives
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| # At the end of this tutorial, you will be able to,
#
# 1. Learn the range of things for which Sage can be used.
# 2. Perform integrations & other Calculus in Sage.
# 3. Perform matrix algebra in sage.
Let us begin with Calculus. We shall be looking at limits, differentiation, integration, and Taylor polynomial.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Open sage notebook
lim(x*sin(1/x), x=0)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| We have our '''Sage''' '''notebook''' running. In case, you don't have it running,
start is using the command, '''sage''' ''space hyphen hyphen'' '''notebook.
'''
To begin with, let us find the limit of the function '''x*sin(1/x)''', at '''x=0'''.
To do this we can use the '''lim''' '''funtcion''' as, '''lim''' ''within brackets'' '''x''' ''star '''''sin''' ''within brackets'' '''one''' ''divided by'' '''x '''''coma '''''x '''''is equal to '''''zero'''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| lim(1/x, x=0, dir='right')
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| We get the limit to be 0, as expected.
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. So we say '''lim '''''within brackets '''''one by x, x=0, dir '''''is equal to in single quotes '''''right.'''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| lim(1/x, x=0, dir='left')
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| The same way we can even find the limit from the negative side, we say, '''lim '''''within brackets '''''one by x, x=0, dir '''''is equal to in single quotes '''''left.'''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 3
Differential Expression
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Let us now see how to perform '''differentiation''', using '''Sage'''. We shall find the '''differential''' of the expression '''sin''' '''square''' '''by x''' with reference to '''x '''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| var('x')
f = exp(sin(x^2))/x
diff(f, x)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| So switch to the sage notebook.
We shall first define the expression, and then use the '''diff '''function to obtain the differential of the expression.
So, type '''var '''''within round brackets in single quotes '''''x. '''Now, '''f '''''is equal to '''''exp '''''within brackets '''''sin '''''within brackets '''''x '''''to the power '''''two by x.'''
We have the expression now and will obtain the differential using the '''diff function.'''
Type '''diff '''''within brackets '''''f '''''coma '''''x.'''
We get the differential.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show slide 4
Partial Differential Expression
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| We can also obtain the partial differentiation of an expression with one of the vriables.
Let us '''differentiate''' the '''expression
'''shown on the slide with
reference to '''x''' and '''y'''. Switch to sage notebook
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| var('x y')
f = exp(sin(y - x^2))/x
diff(f, x)
diff(f, y)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| We first define the expression.
So type, '''var '''''within round brackets in single quotes '''''x y'''
Then, '''f '''''is equal to '''''exp '''''in brackets '''''sin '''''in brackets '''''y '''''minus '''''x '''''to the power''''' two by x.'''
So the expression is ready now to get the partial differential of the expression we say '''diff '''''in brackets '''''f, x.'''
Similarly for '''y '''we say '''diff '''''in brackets '''''f, y''.'''''
Thus we get our partial differential solution.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 5
Integration
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Now, let us look at integration. We shall use the expression obtained from the differentiation that we calculated before, diff(f, y) which gave us the expression shown on the slide.
The integrate command is used to obtain the integral of an expression or function.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| integrate(cos(-x^2 + y)*e^(sin(-x^2 + y))/x, y)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| So, switch to sage notebook and type, '''integrate '''and the expression we got from the previous calculation. As we can see, we get back the correct expression.
The minus sign being
inside or outside the '''sin function''' doesn't change much.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| As we can see,we get back the correct expression. The minus sign being inside or outside the sin function doesn't change much.
Now, let us find the value of the integral between the limits 0 and pi/2.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| var('x n')
taylor((x+1)^n, x, 0, 4)
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Hence we get our solution for the definite integration. Let us now see how to obtain the Taylor expansion of an expression using sage using '''taylor function'''.
Let us obtain the Taylor expansion of(x+1)^nup to degree 4 about 0.
For this, type, '''var '''''in brackets '''''x n'''
Now, '''taylor '''''in brackets again in brackets '''''x + one '''''the whole to the power '''''n '''''coma '''''x, zero, four.'''
We easily got the Taylor expansion,using the function taylor().
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 6
More on Calculus
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| That we will be all about the features of '''Sage '''for calculus we will be looking at. For more, look at the Calculus quick-ref from the Sage Wiki.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 7
Equation
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Next let us move on to Matrix Algebra. Let us begin with solving the equation. Ax = v, where A is the matrix <nowiki>matrix([[1,2],[3,4]]) </nowiki>and v is the vector <nowiki>vector([1,2])</nowiki>.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Switch back to sage notebook page
<nowiki>A = matrix([[1,2],</nowiki>
<nowiki>[3,4]])</nowiki>
<nowiki>v = vector([1,2])</nowiki>
x = A.solve_right(v)
x
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| To solve the equation, Ax = v we simply say,
'''A '''''is equal to '''''matrix '''''within round brackets in square brackets '''''one, two, '''''again in square brackets '''''three, four'''
'''v '''''is equal to '''''vector '''''within round brackets in square brackets '''''one, two'''
'''x '''''is equal to '''''A '''''dot '''''solve_right '''''in brackets '''''v'''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| x = A.solve_left(v)
x
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| To solve the equation, xA = v we simply say,
'''x '''''is equal to '''''A '''''dot '''''solve_left '''''in brackets '''''v'''
'''The left and right here, denote the position of A, relative to x.'''
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 8
Summary slide
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| # This brings us to the end of this tutorial. In this tutorial we have learned to
#
# 1. Use functions like '''lim(), integrate(), integral(), solve()'''
# <nowiki>#. Use </nowiki>'''sage''' for performing '''matrix algebra, integrations''' & other calculus
operations using the above mentioned functions.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 9
Evaluation
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Here are some self assessment questions for you to solve
1. How do you find the limit of the function '''x/sin(x)''' as '''x''' tends to '''0''' from the
negative side.
2. Solve the system of linear equations
x-2y+3z = 7
2x+3y-z = 5
x+2y+4z = 9
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 10
Solutions
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| And the answers,
# 1. To find the limit of the function x/sin(x) as x tends to 0 from negative
side, use the lim function as:
#
lim(x/sin(x), x=0, dir="left")
# We shall first write the equations in '''matrix''' form and then use the '''solve()''' function
#
<nowiki>A = Matrix([[1, -2, 3],</nowiki>
<nowiki>[2, 3, -1],</nowiki>
<nowiki>[1, 2, 4]])</nowiki>
<nowiki>b = vector([7, 5, 9])</nowiki>
x = A.solve_right(b)
To view the output type x
x
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 11
FOSSEE
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| FOSSEE is Free and Open-source Software for Science and Engineering Education. The goal of this project is to enable all to use open source software tools. For more details, please visit the given link.
|}
{| style="border-spacing:0;"
| style="border-top:0.05pt double #808080;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 12
About the Spoken Tutorial Project
| style="border:0.05pt double #808080;padding:0.049cm;"| Watch the video available at the following link. It summarizes the Spoken Tutorial project. If you do not have good bandwidth, you can download and watch it.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 13
Spoken Tutorial Workshop
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| The Spoken Tutorial Project Team conducts SELF workshops using spoken tutorials, gives certificates to those who pass an online test. For more details, you may write to contact@spoken-tutorial.org
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 14
Acknowledgements
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Spoken Tutorial Project is a part of the "Talk to a Teacher" project. It is supported by the National Mission on Education through ICT, MHRD, Government of India. More information on this mission is available at the given link.
|-
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:none;padding:0.049cm;"| Show Slide 15
Thank You
| style="border-top:none;border-bottom:0.05pt double #808080;border-left:0.05pt double #808080;border-right:0.05pt double #808080;padding:0.049cm;"| Hope you have enjoyed this tutorial and found it useful. Thank you!
|}
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