Python/C2/Using-Sage/Gujarati

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Timing Narration
0:00 'સેજના ઉપયોગ' પરના આ ટ્યુટોરીયલમાં સ્વાગત છે.
0:02 આ ટ્યુટોરીયલના અંતે, તમે
  1. વસ્તુઓની શ્રેણી શીખશો જે માટે સેજનો ઉપયોગ કરી શકાય છે.
  2. સેજ માં કેલક્યુલસ માટે વપરાતા ફન્કશન જાણશો.
  3. સેજની મદદથી ગ્રાફ થિયરી અને નંબર થીયરી વિશે શીખશો.
0:16 આ ટ્યુટોરીયલ શરૂ કરો તે પહેલાં, અમારી સલાહ છે કે તમે "Getting started with Sage" પરનું ટ્યુટોરીયલ જુઓ.
0:22 ચાલો કેલક્યુલસ સાથે શરૂ કરીએ.
0:24 આપણે સીમાઓ, વિકલન, સંકલન, અને ટેયલરનો પોલિનોમિયલ વિષે જોઈશું.
0:30 આપણી સેજ નોટબુક ચાલી રહેલ છે.
0:32 જો તે ચાલી રહેલ નથી, તો તે કિસ્સામાં આ આદેશ વાપરી તે શરુ કરો, sage --notebook
0:39 તો sage ટાઇપ કરો અને નોટબુક સ્પષ્ટ કરો.
0:45 તો x =0 પાસે, ફન્કશન x ની સીમા sin(1/x) માં શોધવા માટે, આપણે ટાઇપ કરીશું, lim(x*sin(1/x),x=0)
1:07 ધાર્યા પ્રમાણે, આપણને સીમા 0 મળે છે.
1:11 એક દિશાથી બિંદુ મર્યાદિત કરવું શક્ય છે. ઉદાહરણ તરીકે, ચાલો 1/x at x=0 ની સીમા હકારાત્મક બાજુથી આગમન કરતી વખતે શોધીએ.
1:23 lim કૌશ અંદર (1/x, x=0, dir='right')


1:32 નકારાત્મક બાજુથી સીમા શોધવા માટે, આપણે કહીશું,
1:36 lim(1/x, x=0, dir='left')
1:45 ચાલો સેજ ઉપયોગ કરીને, વિકલન કેવી રીતે શોધવું તે જોઈએ.
1:51 આપણે exp(sin(x squared)) by x એક્ષપ્રેશનનું વિકલન x ના સંદર્ભમાં શોધીશું.
2:11 તે માટે, આપણે પ્રથમ એક્ષપ્રેશન વ્યાખ્યાયિત કરીશું, અને પછી એક્સપ્રેશન નું વિકલન શોધવા માટે diff ફન્કશનનો ઉપયોગ કરીશું.
2:21 તો આપણે ટાઇપ કરીશું, ('x)
f=exp of (sin x squared)/x અને પછી ત્રીજી લાઈન માં ટાઇપ કરો,
diff(f,x)
2:44 આપણે કોઈ એક વેરિયેબલના સંદર્ભમાં એક્ષપ્રેશનનું આંશિક વિકલન પણ મેળવી શકીએ છીએ.
2:51 ચાલો એક્ષપ્રેશનનું વિકલન કરીએ, x અને y ના સંદર્ભમાં, exp(sin (y - x squared))/x
3:07 એટલે કે x અને y ના સંદર્ભમાં
3:10 તો ટાઇપ કરો, var('x y')
3:15 બીજી લાઈનમાં ટાઇપ કરો, f=exp(sin(y - x squared))by x
3:26 પછી ટાઇપ કરો, diff(f,x) પછી આગામી લાઈનમાં ટાઇપ કરો, diff(f,y)
3:43 તો આપણને આપણું આંશિક વિકલન મળે છે.
3:51 હવે, ચાલો સંકલન માટે જોઈએ.
3:53 We shall use the expression obtained from the differentiation that we did before, diff(f, y)which gave us the expression ---e^(sin(-x squared + y)) multiplied by cos(-x squared plus y) by x
4:15 The integrate 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 sin 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 got 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 (x + 1) raised to n 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 taylor() function.
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 Ax = v, where A is the matrix matrix ([[1,2], [3,4]]) and v is the vector vector ([1,2]).
6:19 So, to solve the equation,Ax = v 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, xA = v 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 A, 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 Graph() 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()
8:24 Similarly, we can obtain a directed graph using the DiGraph 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 graph.tab.
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 Complete Graph(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 prime_range gives primes in the range 100 to 200.
9:46 So you can type there prime_range within brackets 100,200.
9:58 is_prime 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 factor(2001) 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 Permutations() gives the permutations of [1, 2, 3, 4]
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 Combinations() gives all the combinations of [1, 2, 3, 4]
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 So 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 graph and digraph 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 x/sin(x) as x tends to 0 from the negative side.
12:32 2. 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
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 A = Matrix of within brackets([[1, -2, 3] comma [2, 3, -1] comma [1, 2, 4]])
13:48 b = vector within brackets([7, 5, 9])
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!

Contributors and Content Editors

Krupali