getting-started-with-symbolics/slides.org
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#+LaTeX_CLASS: beamer
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#+LaTeX_CLASS_OPTIONS: [presentation]
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#+BEAMER_FRAME_LEVEL: 1
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#+BEAMER_HEADER_EXTRA: \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}
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#+COLUMNS: %45ITEM %10BEAMER_env(Env) %10BEAMER_envargs(Env Args) %4BEAMER_col(Col) %8BEAMER_extra(Extra)
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#+PROPERTY: BEAMER_col_ALL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 :ETC
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#+LaTeX_CLASS_OPTIONS: [presentation]
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#+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl}
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#+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}
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#+LaTeX_HEADER: \usepackage{listings}
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#+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries,
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#+LaTeX_HEADER:  commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},
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#+LaTeX_HEADER:  showstringspaces=false, keywordstyle=\color{blue}\bfseries}
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#+TITLE:   Getting started with symbolics
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#+AUTHOR:    FOSSEE
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#+EMAIL:     
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#+DATE:    
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#+DESCRIPTION: 
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#+KEYWORDS: 
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#+LANGUAGE:  en
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#+OPTIONS:   H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
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#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc
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* Outline
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  - Defining symbolic expressions in sage.  
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  - Using built-in constants and functions.   
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  - Performing Integration, differentiation using sage. 
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  - Defining matrices. 
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  - Defining Symbolic functions.  
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  - Simplifying and solving symbolic expressions and functions.
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* Question 1
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  - Define the following expression as symbolic
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    expression in sage.
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    - x^2+y^2
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    - y^2-4ax
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* Solution 1
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#+begin_src python
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  var('x,y')
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  x^2+y^2
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  var('a,x,y')
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  y^2-4*a*x
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#+end_src python
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* Question 2
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  - Find the values of the following constants upto 6 digits  precision 
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    - pi^2
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    - euler_gamma^2
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  - Find the value of the following.
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   - sin(pi/4)
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   - ln(23)  
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* Solution 2
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#+begin_src python
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  n(pi^2,digits=6)
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  n(sin(pi/4))
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  n(log(23,e))
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#+end_src python
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* Question 3
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  - Define the piecewise function. 
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   f(x)=3x+2 
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   when x is in the closed interval 0 to 4.
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   f(x)=4x^2
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   between 4 to 6. 
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  - Sum  of 1/(n^2-1) where n ranges from 1 to infinity. 
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* Solution 3
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#+begin_src python
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  var('x') 
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  h(x)=3*x+2 
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  g(x)= 4*x^2
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  f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x)
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  f
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#+end_src python
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#+begin_src python  
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  var('n')
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  f=1/(n^2-1) 
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  sum(f(n), n, 1, oo)
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#+end_src python  
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* Question 4
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  - Differentiate the following. 
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    - sin(x^3)+log(3x), to the second order
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    - x^5*log(x^7), to the fourth order
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  - Integrate the given expression 
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    - x*sin(x^2) 
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  - Find x
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    - cos(x^2)-log(x)=0
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    - Does the equation have a root between 1,2. 
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* Solution 4
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#+begin_src python
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  var('x')
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  f(x)= x^5*log(x^7) 
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  diff(f(x),x,5)
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  var('x')
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  integral(x*sin(x^2),x) 
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  var('x')
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  f=cos(x^2)-log(x)
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  find_root(f(x)==0,1,2)
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#+end_src
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* Question 5
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  - Find the determinant and inverse of :
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      A=[[x,0,1][y,1,0][z,0,y]]
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* Solution 5
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#+begin_src python  
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  var('x,y,z')
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  A=matrix([[x,0,1],[y,1,0],[z,0,y]])
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  A.det()
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  A.inverse()
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#+end_src
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* Summary
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 - We learnt about defining symbolic expression and functions.
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 - Using built-in constants and functions.
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 - Using <Tab> to see the documentation of a function.
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 - Simple calculus operations .
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 - Substituting values in expression using substitute function.
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 - Creating symbolic matrices and performing operation on them .
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* Thank you!
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#+begin_latex
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  \begin{block}{}
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  \begin{center}
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  This spoken tutorial has been produced by the
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  \textcolor{blue}{FOSSEE} team, which is funded by the 
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  \end{center}
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  \begin{center}
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    \textcolor{blue}{National Mission on Education through \\
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      Information \& Communication Technology \\ 
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      MHRD, Govt. of India}.
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  \end{center}  
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  \end{block}
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#+end_latex
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