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

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 y is not defined.
1:14	We need to declare y as a symbol.
1:17	We do it using the var function.
1:19	So type var within brackets and single quotes y.
1:28	Now if you type sin within brackets y, simply returns the expression.
1:32	So type sine y.
1:37	Now, sage treats sin of y 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, x, y, alpha and beta and have defined a symbolic expression using them.
2:25	Here is an expression in theta
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 n 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 n by doing n<tab>, You will see what all arguments it takes and what it returns.
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 sin, cos, log, factorial, gamma, exp, arctan 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 artan 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 Piecewise 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 simplify. 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 find_root 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') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) 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	A = within brackets and again brackets x,0,1 then again brackets y,1,0 again bracket z,0,y
21:18	The solution is on your screen
21:20	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
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 var.
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 use <Tab> to see the documentation of a function.
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 substitute 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 f(x) = sin(x^2)+exp(x^3)
22:50	So, the answers,
22:53	1. We define a symbol using the function var.
22:57	In this case it will be var('y')
23:02	2. The value of pi upto precision 5 digits is given as, 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, diff(f(x),x,3)
23:24	Hope you have enjoyed this tutorial and found it useful.

Contributors and Content Editors

Gaurav, Jyotisolanki, Minal, PoojaMoolya, Pravin1389