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 * Question 1

    41   - Define the following expression as symbolic

    42     expression in sage.

    43 

    44     - x^2+y^2

    45     - y^2-4ax

    46   

    47 * Solution 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 * Question 2

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

    57    

    58     - pi^2

    59     - euler_gamma^2

    60    

    61       

    62   - Find the value of the following.

    63 

    64    - sin(pi/4)

    65    - ln(23)  

    66 

    67 * Solution 2

    68 #+begin_src python

    69   n(pi^2,digits=6)

    70   n(sin(pi/4))

    71   n(log(23,e))

    72 #+end_src python

    73 * Question 3

    74   - Define the piecewise function. 

    75    f(x)=3x+2 

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

    77    f(x)=4x^2

    78    between 4 to 6. 

    79    

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

    81 

    82 * Solution 3

    83 #+begin_src python

    84   var('x') 

    85   h(x)=3*x+2 

    86   g(x)= 4*x^2

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

    88   f

    89 #+end_src python

    90 

    91 #+begin_src python  

    92   var('n')

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

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

    95 #+end_src python  

    96 

    97 * Question 4

    98   - Differentiate the following. 

    99       

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

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

   102 

   103   - Integrate the given expression 

   104       

   105     - x*sin(x^2) 

   106 

   107   - Find x

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

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

   110 

   111 * Solution 4

   112 #+begin_src python

   113   var('x')

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

   115   diff(f(x),x,5)

   116 

   117   var('x')

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

   119 

   120   var('x')

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

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

   123 #+end_src

   124 

   125 * Question 5

   126   - Find the determinant and inverse of :

   127 

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

   129 

   130 * Solution 5

   131 #+begin_src python  

   132   var('x,y,z')

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

   134   A.det()

   135   A.inverse()

   136 #+end_src

   137 * Summary

   138  - We learnt about defining symbolic expression and functions.

   139  - Using built-in constants and functions.

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

   141  - Simple calculus operations .

   142  - Substituting values in expression using substitute function.

   143  - Creating symbolic matrices and performing operation on them .

   144 * Thank you!

   145 #+begin_latex

   146   \begin{block}{}

   147   \begin{center}

   148   This spoken tutorial has been produced by the

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

   150   \end{center}

   151   \begin{center}

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

   153       Information \& Communication Technology \\ 

   154       MHRD, Govt. of India}.

   155   \end{center}  

   156   \end{block}

   157 #+end_latex

   158 

   159 

   160