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#+TITLE:   
#+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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* 
#+begin_latex
\begin{center}
\vspace{12pt}
\textcolor{blue}{\huge Getting started with Symbolics}
\end{center}
\vspace{18pt}
\begin{center}
\vspace{10pt}
\includegraphics[scale=0.95]{../images/fossee-logo.png}\\
\vspace{5pt}
\scriptsize Developed by FOSSEE Team, IIT-Bombay. \\ 
\scriptsize Funded by National Mission on Education through ICT\\
\scriptsize  MHRD,Govt. of India\\
\includegraphics[scale=0.30]{../images/iitb-logo.png}\\
\end{center}
#+end_latex
* Objectives
 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.  
 - Simplify0and solve symbolic expressions and functions.

* Pre-requisite
  Spoken tutorial on -
  - Getting started with Sage Notebook.
* Exercise 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
* Exercise 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
* Exercise 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  

* Exercise 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

* Exercise 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
In this tutorial, we have learnt to,

- Define symbolic expression and functions using the method ``var``.
- Use built-in constants like pi,e,oo and functions like 
   sum,sin,cos,log,exp and many more.  
- Use <Tab> to see the documentation of a function. 
- Do simple calculus using functions like --
   - diff()--to find a differential of a function
   - integral()--to integrate an expression
   - simplify--to simplify complicated expression. 
- Substitute values in expressions using ``substitute`` function.
- 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.
* Evaluation
1. How do you define a name 'y' as a symbol?

2. Get the value of pi upto precision 5 digits using sage?

3. Find third order differential function of

   f(x)=sin(x^2)+exp(x^3)
* Solutions
  1. var('y')

  2. n(pi,5)

  3. diff(f(x),x,3)
* 
#+begin_latex
  \begin{block}{}
  \begin{center}
  \textcolor{blue}{\Large THANK YOU!} 
  \end{center}
  \end{block}
\begin{block}{}
  \begin{center}
    For more Information, visit our website\\
    \url{http://fossee.in/}
  \end{center}  
  \end{block}
#+end_latex