Python/C2/Getting-started-with-symbolics/English-timed
From Script | Spoken-Tutorial
| 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,
|
| 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,Sage 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 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 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. |
