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#+TITLE:   Getting started with symbolics

#+AUTHOR:    FOSSEE

#+EMAIL:     

#+DATE:    

#+DESCRIPTION: 

#+KEYWORDS: 

#+LANGUAGE:  en

#+OPTIONS:   H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t

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* Outline

  - Defining symbolic expressions in sage.  

  - Using built-in constants and functions.   

  - Performing Integration, differentiation using sage. 

  - Defining matrices. 

  - Defining Symbolic functions.  

  - Simplifying and solving symbolic expressions and functions.

* Question 1

  - Define the following expression as symbolic

    expression in sage.

    - x^2+y^2

    - y^2-4ax

* Solution 1

#+begin_src python

  var('x,y')

  x^2+y^2

  var('a,x,y')

  y^2-4*a*x

#+end_src python

* Question 2

  - Find the values of the following constants upto 6 digits  precision 

    - pi^2

    - euler_gamma^2

  - Find the value of the following.

   - sin(pi/4)

   - ln(23)  

* Solution 2

#+begin_src python

  n(pi^2,digits=6)

  n(sin(pi/4))

  n(log(23,e))

#+end_src python

* Question 3

  - Define the piecewise function. 

   f(x)=3x+2 

   when x is in the closed interval 0 to 4.

   f(x)=4x^2

   between 4 to 6. 

  - Sum  of 1/(n^2-1) where n ranges from 1 to infinity. 

* Solution 3

#+begin_src python

  var('x') 

  h(x)=3*x+2 

  g(x)= 4*x^2

  f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x)

  f

#+end_src python

#+begin_src python  

  var('n')

  f=1/(n^2-1) 

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

#+end_src python  

* Question 4

  - Differentiate the following. 

    - sin(x^3)+log(3x), to the second order

    - x^5*log(x^7), to the fourth order

  - Integrate the given expression 

    - x*sin(x^2) 

  - Find x

    - cos(x^2)-log(x)=0

    - Does the equation have a root between 1,2. 

* Solution 4

#+begin_src python

  var('x')

  f(x)= x^5*log(x^7) 

  diff(f(x),x,5)

  var('x')

  integral(x*sin(x^2),x) 

  var('x')

  f=cos(x^2)-log(x)

  find_root(f(x)==0,1,2)

#+end_src

* Question 5

  - Find the determinant and inverse of :

      A=[[x,0,1][y,1,0][z,0,y]]

* Solution 5

#+begin_src python  

  var('x,y,z')

  A=matrix([[x,0,1],[y,1,0],[z,0,y]])

  A.det()

  A.inverse()

#+end_src

* Summary

 - We learnt about defining symbolic expression and functions.

 - Using built-in constants and functions.

 - Using <Tab> to see the documentation of a function.

 - Simple calculus operations .

 - Substituting values in expression using substitute function.

 - Creating symbolic matrices and performing operation on them .

* Thank you!

#+begin_latex

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  This spoken tutorial has been produced by the

  \textcolor{blue}{FOSSEE} team, which is funded by the 

  \end{center}

  \begin{center}

    \textcolor{blue}{National Mission on Education through \\

      Information \& Communication Technology \\ 

      MHRD, Govt. of India}.

  \end{center}  

  \end{block}

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