1 #+LaTeX_CLASS: beamer

     2 #+LaTeX_CLASS_OPTIONS: [presentation]

     3 #+BEAMER_FRAME_LEVEL: 1

     4 

     5 #+BEAMER_HEADER_EXTRA: \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}

     6 #+COLUMNS: %45ITEM %10BEAMER_env(Env) %10BEAMER_envargs(Env Args) %4BEAMER_col(Col) %8BEAMER_extra(Extra)

     7 #+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

     8 

     9 #+LaTeX_CLASS: beamer

    10 #+LaTeX_CLASS_OPTIONS: [presentation]

    11 

    12 #+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl}

    13 #+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}

    14 

    15 #+LaTeX_HEADER: \usepackage{listings}

    16 

    17 #+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries,

    18 #+LaTeX_HEADER:  commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},

    19 #+LaTeX_HEADER:  showstringspaces=false, keywordstyle=\color{blue}\bfseries}

    20 

    21 #+TITLE:   Getting started with symbolics

    22 #+AUTHOR:    FOSSEE

    23 #+EMAIL:     

    24 #+DATE:    

    25 

    26 #+DESCRIPTION: 

    27 #+KEYWORDS: 

    28 #+LANGUAGE:  en

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

    30 #+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc

    31 

    32 * Outline

    33   - Defining symbolic expressions in sage.  

    34   - Using built-in constants and functions.   

    35   - Performing Integration, differentiation using sage. 

    36   - Defining matrices. 

    37   - Defining Symbolic functions.  

    38   - Simplifying and solving symbolic expressions and functions.

    39 

    40 * Questions 1

    41   - Define the following expression as symbolic

    42     expression in sage.

    43 

    44     - x^2+y^2

    45     - y^2-4ax

    46   

    47 * Solutions 1

    48 #+begin_src python

    49   var('x,y')

    50   x^2+y^2

    51 

    52   var('a,x,y')

    53   y^2-4*a*x

    54 #+end_src python

    55 * Questions 2

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

    57    

    58     - pi^2

    59    

    60       

    61   - Find the value of the following.

    62 

    63    - sin(pi/4)

    64    - ln(23)  

    65 

    66 * Solutions 2

    67 #+begin_src python

    68   n(pi^2,digits=6)

    69   n(sin(pi/4))

    70   n(log(23,e))

    71 #+end_src python

    72 * Question 2

    73   - Define the piecewise function. 

    74    f(x)=3x+2 

    75    when x is in the closed interval 0 to 4.

    76    f(x)=4x^2

    77    between 4 to 6. 

    78    

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

    80 

    81 * Solution Q1

    82 #+begin_src python

    83   var('x') 

    84   h(x)=3*x+2 

    85   g(x)= 4*x^2

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

    87   f

    88 #+end_src python

    89 * Solution Q2

    90 #+begin_src python  

    91   var('n')

    92   f=1/(n^2-1) 

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

    94 #+end_src python  

    95  

    96 

    97 * Questions 3

    98   - Differentiate the following. 

    99       

   100     - x^5*log(x^7)  , degree=4 

   101 

   102   - Integrate the given expression 

   103       

   104     - x*sin(x^2) 

   105 

   106   - Find x

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

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

   109 

   110 * Solutions 3

   111 #+begin_src python

   112   var('x')

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

   114   diff(f(x),x,5)

   115 

   116   var('x')

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

   118 

   119   var('x')

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

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

   122 #+end_src

   123 

   124 * Question 4

   125   - Find the determinant and inverse of :

   126 

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

   128 

   129 * Solution 4

   130 #+begin_src python  

   131   var('x,y,z')

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

   133   A.det()

   134   A.inverse()

   135 #+end_src

   136 * Summary

   137  - We learnt about defining symbolic 

   138    expression and functions.  

   139  - Using built-in constants and functions.  

   140  - Using <Tab>  to see the documentation of a 

   141    function.  

   142  

   143 * Summary 

   144  - Simple calculus operations .  

   145  - Substituting values in expression 

   146    using substitute function.

   147  - Creating symbolic matrices and 

   148    performing operation on them .

   149 

   150 * Thank you!

   151 #+begin_latex

   152   \begin{block}{}

   153   \begin{center}

   154   This spoken tutorial has been produced by the

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

   156   \end{center}

   157   \begin{center}

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

   159       Information \& Communication Technology \\ 

   160       MHRD, Govt. of India}.

   161   \end{center}  

   162   \end{block}

   163 #+end_latex

   164 

   165 

   166