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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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