Python/C2/Getting-started-with-symbolics/English-timed - Revision history

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Pravin1389 at 06:41, 11 November 2013

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Minal: Created page with '{| border=1 !Timing !Narration |- | 0:02 | Hello friends and welcome to the tutorial on "Symbolics with Sage". |- | 0:07 | At the end of this tutorial, you will be able to, # D…'

2013-01-02T11:59:42Z

Created page with '{| border=1 !Timing !Narration |- | 0:02 | Hello friends and welcome to the tutorial on "Symbolics with Sage". |- | 0:07 | At the end of this tutorial, you will be able to, # D…'

New page

{| border=1
!Timing
!Narration
|-
| 0:02
| Hello friends and welcome to the tutorial on "Symbolics with Sage".

|-
| 0:07
| At the end of this tutorial, you will be able to,

# Define symbolic expressions in sage.
# Use built-in constants and functions.
# Perform Integration, differentiation using sage.
# Define matrices.
# Define Symbolic functions.
# Simplify and solve symbolic expressions and functions.

|-
| 0:24
| Before beginning this tutorial,we would suggest you to complete the tutorial on "Getting started with sage notebook".

|-
|0:31
|In addition to a lot of other things, Sage can do Symbolic Math and we shall start with defining symbolic expressions in Sage.

|-
| 0:42
| Have your Sage notebook opened.

|-
|0:44
|If not, pause the video and start you Sage notebook.

|-
|0:49
| On the sage notebook type sine y.

|-
|1:08
|Then click shift enter.

|-
|1:12
| It raises a name error saying that <tt>y</tt> is not defined.

|-
|1:14
| We need to declare <tt>y</tt> as a symbol.

|-
|1:17
| We do it using the <tt>var</tt> function.

|-
|1:19
|So type var within brackets and single quotes y.

|-
| 1:28
| Now if you type sin within brackets y,Sage simply returns the expression.

|-
|1:32
|So type sine y.

|-
| 1:37
| Now, sage treats <tt>sin of y</tt> as a symbolic expression.

|-
|1:42
|We can use this to do symbolic math using Sage's built-in constants and expressions.

|-
|1:47
|Let us try out a few examples.

|-
|1:50
|So let us type var within brackets and single quotes x comma alpha comma y comma beta

|-
|1:59
|Then next line you can type x charat 2 by alpha charat 2 plus y charat 2 by beta charat 2

|-
|2:10
|That is x squared by alpha squared plus y squared by beta squared.

|-
| 2:17
| We have defined 4 variables, <tt>x</tt>, <tt>y</tt>, <tt>alpha</tt> and <tt>beta</tt> and have defined a symbolic expression using them.

|-
|2:25
|Here is an expression in <tt>theta</tt>

|-
|2:29
|So you can type var within brackets and single quotes theta

|-
|2:38
|then sine within brackets theta multiplied by sine within brackets theta plus cos within brackets theta multiplied by cos within brackets theta

|-
| 2:55
| Now that you know how to define symbolic expressions in Sage, here is an exercise.

|-
|3:01
|Pause the video here, try out the following exercise and resume the video.

|-
|3:05
| Define following expressions as symbolic expressions in Sage.

|-
|3:11
|that is x squared plus y squared

|-
|3:13
|and next one is. y squared minus 4 ax

|-
| 3:18
| The solution is on your screen.

|-
|3:25
|that is var within brackets and single quotes x,y then x squared plus y squared that is x charat 2 plus y charat 2.

|-
|3:33
|then next is var within brackets and single quotes a,x,y then y charat 2 minus 4 into a into x

|-
| 3:49
| Sage also provides built-in constants which are commonly used in mathematics, for instance pi, e, infinity.

|-
|3:56
|The function <tt>n</tt> gives the numerical values of all these constants.

|-
|4:00
|So you can type n within brackets pi then n within brackets e then n within brackets zero zero that is oo.

|-
| 4:18
| If you look into the documentation of function <tt>n</tt><nowiki> by doing n<tab>, You will see what all arguments it takes and what it returns.</nowiki>

|-
| 4:26
| So you can type n and hit tab.

|-
| 4:30
| It will be very helpful if you look at the documentation of all functions introduced in the course of this script.

|-
|4:36
| Also we can define the number of digits we wish to have in the constants.

|-
|4:40
| For this we have to pass an argument -- digits.

|-
|4:46
|So you can type n within brackets pi comma space digits is equal to 10.

|-
| 5:01
| Apart from the constants Sage also has a lot of built-in functions like <tt>sin</tt>, <tt>cos</tt>, <tt>log</tt>, <tt>factorial</tt>, <tt>gamma</tt>, <tt>exp</tt>, <tt>arctan</tt> which stands for arctangent etc ...

|-
|5:16
|So let us try some of them out on the Sage notebook.

|-
|5:21
|so you can type sine within brackets pi by 2 then arctan oo then log within brackets

|-
|5:44
|so when you type artan , there is an error in arc so we have to type arctan.

|-
|5:54
|Then type log e comma e

|-
| 6:03
| Pause the video here, try out the following exercise and resume the video.

|-
| 6:06
| Find the values of the following constants upto 6 digits precision

|-
|6:14
|First option is pi charat 2

|-
|6:18
|then euler underscore gamma charat 2

|-
|6:23
|Find the value of the following.

|-
|6:26
|1. sin of pi divided by 4

|-
|6:28
|Next one is . ln of 23.

|-
| 6:32
| The solutions are on your screen.

|-
|6:36
|that is n into within brackets pi squared comma digits equal to 6,next one is n into within brackets sin pi by 4 and then third one is n into within brackets log 23 comma e

|-
| 7:05
| Given that we have defined variables like x, y etc., we can define an arbitrary function with desired name in the following way.

|-
|7:14
|So you can type var within brackets and single quotes x and then next line function within brackets and single quotes f comma x

|-
| 7:33
| Here f is the name of the function and x is the independent variable .

|-
| 7:37
| Now we can define f of x

|-
|7:40
|that is f of x within brackets x is equal to x by 2 plus sin x.

|-
| 7:53
| Evaluating this function f for the value x=pi returns pi by 2.

|-
|8:01
|So type f within brackets pi

|-
|8:07
|so we will get the answer as 1 by 2 into pi.

|-
| 8:12
| We can also define functions that are not continuous but defined piecewise.

|-
|8:18
| Let us define a function which is a parabola between 0 to 1 and a constant from 1 to 2 .

|-
|8:24
|We shall use the function <tt>Piecewise</tt> which returns a piecewise function from a list of pairs.

|-
|8:31
| We can type the following

|-
|8:35
|var within brackets in single quotes x

|-
|8:41
|then f of x is equal to x charat 2

|-
|8:52
|then v of x is equal to 1

|-
|8:58
|then next line we can type f is equal to piecewise within brackets 0 comma 1 then comma x then another bracket x then next square bracket it is 1,2, g of x comma x then type f

|-
| 9:21
| We can also define functions convergent series and other series.

|-
|9:26
| We first define a function f(n) in the way discussed before.

|-
|9:29
|So we can type var within brackets n in single quotes

|-
|9:39
|then type function within brackets f,n

|-
| 9:53
|var('n')
function('f', n)
To sum the function for a range of discrete values of n, we use the sage function sum.

|-
|10:03
|For a convergent series , f(n)=1 by n raised to 2 we can say by typing

var('n')

function('f', n)

f(n) = 1/n^2

sum(f(n), n, 1, oo)

|-
| 10:55
| Let us now try another series
f(n) = (-1)^(n-1)*1/(2*n - 1)
sum(f(n), n, 1, oo)

|-
| 11:33
| This series converges to pi by 4.

|-
|11:40
|Pause the video here, try out the following exercise and resume the video.

|-
| 11:46
| Define the piecewise function

|-
| 11:47
| f of x is equal to 3x plus 2 when x is in the closed interval 0 to 4.

|-
|11:55
| f of x is equal to 4x squared between 4 to 6.

|-
|12:03
|Sum of 1 by within brackets n squared -1 where n ranges from 1 to infinity.

|-
| 12:11
| The solution is on your screen

|-
|12:13
|var('x') ,h(x) = 3 into x plus 2 g(x) is equal to 4 into x squared , f = Piecewise within brackets again square brackets and square brackets again and within closing brackets 0,4,h(x),(4,6),g(x),x
|-
|12:40
|Next step you have to type var('n') f = 1/(n squared minus 1) sum(f(n), n, 1, oo)

|-
| 13:00
| Moving on let us see how to perform simple calculus operations using Sage

|-
|13:05
| For example lets try an expression first

|-
| 13:18
| The diff function differentiates an expression or a function.

|-
| 13:27
| It's first argument is expression or function and second argument is the independent variable.

|-
|13:33
| We have already tried an expression now lets try a function

|-
|13:41
|f = exp(x^2) + arcsin(x)
diff(f(x),x)

|-
| 14:00
|To get a higher order differential we need to add an extra third argument for order so you can type
diff(f(x),x,3)

|-
|14:35
| in this case it is 3.

|-
|14:38
|Just like differentiation of expression you can also integrate them
x = var('x')
s = integral(1/(1 + (tan(x))**2),x)

|-
| 15:18
| 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)

|-
| 15:46
| One can simplify complicated expression by using the function <tt>simplify</tt>.
f.simplify_full()

|-
| 16:06
| This simplifies the expression fully.

|-
|16:07
| We can also do simplification of just the algebraic part and the trigonometric part
f.simplify_exp()
|-
|16:24
| f.simplify_trig()

|-
|16:33
|One can also find roots of an equation by using <tt>find_root</tt> function
phi = var('phi')
find_root(cos(phi) == sin(phi),0,pi/2)

|-
| 17:07
|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)

|-
| 17:55
|As we can see when we substitute the value the answer is almost = 0 showing the solution we got was correct.

|-
|18:04
|Pause the video here, try out the following exercise and resume the video.

|-
|18:10
|Differentiate the following.

|-
|18:12
|1. sin(x cubed) plus log(3x) , degree=2

|-
|18:24
|2. x raised to 5 into log x raised to 7 , degree=4

|-
|18:32
|Integrate the given expression

|-
|18:33
|sin(x squared)+exp(x cubed)

|-
|18:44
|'''Find x'''

|-
|18:45
|cos(x squared)-log(x)=0

|-
|18:50
|Does the equation have a root between 1,2.

|-
|18:55
|The solution is on your screen

|-
|18:56
|For the first one we have to type var('x') f(x)= x raised to 5 into log of x raised to 7 diff(f(x),x,5)

|-
|19:15
|Next line we have to type var('x')then second line integral(x*sin(x^2),x)

|-
|19:33
|For the third one we have to type var('x') then f=cos(x^2)-log(x) find_root(f(x)==0,1,2)

|-
| 19:53
|So let us now try some matrix algebra symbolically
var('a,b,c,d')
<nowiki>A=matrix([[a,1,0],[0,b,0],[0,c,d]])</nowiki>
A
|-
| 20:29
|Now lets do some of the matrix operations on this matrix
A.det()
A.inverse()

|-
|20:46
|As we can see, we got the determinant and the inverse of the matrix respectively.

|-
|20:50
|So pause the video here, try out the following exercise and resume the video.

|-
|20:57
|Find the determinant and inverse of

|-
|20:59
|<nowiki>A = within brackets and again brackets x,0,1 then again brackets y,1,0 again bracket z,0,y</nowiki>

|-
|21:18
|The solution is on your screen

|-
|21:20
|<nowiki>var('x,y,z') A = matrix([[x,0,1],[y,1,0],[z,0,y]])then third line you have to type A dot det function and next line you have to type A dot inverse function</nowiki>

|-
|21:44
|This brings us to the end of this tutorial.

|-
|21:48
|In this tutorial, we have learnt to,

|-
|21:49
|Define symbolic expression and functions using the method <tt>var</tt>.

|-
|21:53
|Then use built-in constants like pi,e,oo and functions like sum,sin,cos,log,exp and many more.

|-
|22:00
|Then <nowiki>use <Tab> to see the documentation of a function.</nowiki>

|-
|22:03
|4. Do simple calculus using functions - diff()--to find a differential of a function - integral()--to integrate an expression - simplify--to simplify complicated expression.

|-
|22:16
|5. Substitute values in expressions using <tt>substitute</tt> function.

|-
|22:19
|Then 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.

|-
|22:29
|Here are some self assessment questions for you to solve

|-
|22:32
|1. How do you define a name 'y' as a symbol?

|-
|22:37
|2. Get the value of pi upto precision 5 digits using sage?

|-
|22:41
|3. Find third order differential function of<br/> f(x) = sin(x^2)+exp(x^3)

|-
|22:50
|So, the answers,

|-
|22:53
|1. We define a symbol using the function <tt>var</tt>.

|-
|22:57
|In this case it will be<br/> var('y')

|-
|23:02
|2. The value of pi upto precision 5 digits is given as,<br/> n(pi,5)

|-
|23:11
|3. The third order differential function can be found out by adding the third argument which states the order.

|-
|23:18
|The syntax will be,<br/> diff(f(x),x,3)

|-
|23:24
|Hope you have enjoyed this tutorial and found it useful.

|}

@@ Line 337: / Line 337: @@
 |-
 | 11:46
-| Define the piecewise function
+| Define the piecewise function,f of x is equal to 3x plus 2 when x is in the closed interval 0 to 4.
-−
-|-
-| 11:47
-| f of x is equal to 3x plus 2 when x is in the closed interval 0 to 4.
 |-
@@ Line 461: / Line 457: @@
 |-
 |18:32
-|Integrate the given expression
+|Integrate the given expression,x start sin into x square
-−
-|-
-|18:33
-|x start sin into x square
 |-
 |18:44
-|'''Find x'''
+|'''Find x''',cos(x squared)-log(x)=0
-−
-|-
-|18:45
-|cos(x squared)-log(x)=0
 |-
@@ Line 481: / Line 469: @@
 |-
 |18:55
-|The solution is on your screen
+|The solution is on your screen,For the first one we have to type var('x') f(x)= x raised to 5 into log of x raised to 7 diff(f(x),x,5)
-−
-|-
-|18:56
-|For the first one we have to type var('x') f(x)= x raised to 5 into log of x raised to 7 diff(f(x),x,5)
 |-

@@ Line 418: / Line 418: @@
 |-
 | 16:06
-| This simplifies the expression fully.
+| This simplifies the expression fully. We can also do simplification of just the algebraic part and the trigonometric part
 |-
 |16:24
@@ Line 540: / Line 537: @@
 |-
 |21:48
-|In this tutorial, we have learnt to,
+|In this tutorial, we have learnt to, Define symbolic expression and functions using the method <tt>var</tt>.
-−
-|-
-|21:49
-|Define symbolic expression and functions using the method <tt>var</tt>.
 |-

@@ Line 474: / Line 474: @@
 |-
 |18:33
-|sin(x squared)+exp(x cubed)
+|x start sin into x square
 |-

@@ Line 10: / Line 10: @@
 | 00:07
 | At the end of this tutorial, you will be able to,
 Define symbolic expressions in sage.
-# Define symbolic expressions in sage.
+Use built-in constants and functions.
-# Use built-in constants and functions.
+Perform Integration, differentiation using sage.
-# Perform Integration, differentiation using sage.
+Define matrices.
-# Define matrices.
+Define Symbolic functions.
-# Define Symbolic functions.
+Simplify and solve symbolic expressions and functions.
-# Simplify and solve symbolic expressions and functions.
 |-
@@ Line 317: / Line 316: @@
 |10:03
 |For a convergent series , f(n)=1 by n raised to 2 we can say by typing
 var('n')
 function('f', n)
 f(n) = 1/n^2
 sum(f(n), n, 1, oo)
@@ Line 393: / Line 388: @@
 |-
 |13:41
-|f = exp(x^2) + arcsin(x)
+|f = exp(x^2) + arcsin(x) diff(f(x),x)
 diff(f(x),x)
 |-
@@ Line 462: / Line 456: @@
 |-
 |18:12
-|1. sin(x cubed) plus log(3x) , degree=2
+|sin(x cubed) plus log(3x) , degree=2
 |-
 |18:24
-|2. x raised to 5 into log x raised to 7 , degree=4
+|x raised to 5 into log x raised to 7 , degree=4
 |-

@@ Line 118: / Line 118: @@
 | Define following expressions as symbolic expressions in Sage.
-|0-
+|-
-|3:11
+|03:11
 |that is  x squared plus y squared

@@ Line 283: / Line 283: @@
 |-
 |8:41
-|then f of x is equal to x charat 2
+|then h of x is equal to x charat 2
 |-
 |8:52
-|then v of x is equal to 1
+|then g of x is equal to 1
 |-
 |8:58
-|then next line we can type f is equal to piecewise within brackets 0 comma 1 then comma x then another bracket x then next square bracket it is 1,2, g of x comma x then type f
+|then next line we can type f is equal to piecewise within brackets 0 comma 1 then comma h then another bracket x then next square bracket it is 1,2, g of x comma x then type f
 |-
@@ Line 311: / Line 311: @@
 |-
 |  9:53
-|var('n')
+|To sum the function for a range of discrete values of n, we use the sage function sum.
-function('f', n)
 To sum the function for a range of discrete values of n, we use the sage function sum.
 |-
@@ Line 376: / Line 374: @@
 |13:05
 | For example lets try an expression first
 |-

@@ Line 59: / Line 59: @@
 |-
 |  1:28
-| Now if you type sin within brackets y,Sage simply returns the expression.
+| Now if you type sin within brackets y, simply returns the expression.
 |-
@@ Line 183: / Line 183: @@
 |-
 |5:21
-|so you can type sine within brackets pi by 2 then arctan oo then log within brackets
+|so you can type sine within brackets pi by 2 then artan oo then log within brackets
 |-
@@ Line 307: / Line 307: @@
 |-
 |9:39
-|then type function within brackets f,n
+|then type function within brackets ' f ',n
 |-
@@ Line 478: / Line 478: @@
 |'''Find x'''
 |-
 |18:45
 |cos(x squared)-log(x)=0