getting-started-with-symbolics/slides.org
author anand
Thu, 11 Nov 2010 00:03:57 +0530
changeset 472 fcdec2d28c9a
parent 442 a9b71932cbfa
child 458 9a1c5d134feb
permissions -rw-r--r--
Checklist OK for `The other kinds of plots`
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     1
#+LaTeX_CLASS: beamer
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     2
#+LaTeX_CLASS_OPTIONS: [presentation]
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     3
#+BEAMER_FRAME_LEVEL: 1
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     4
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
     5
#+BEAMER_HEADER_EXTRA: \usetheme{Warsaw}\usecolortheme{default}\useoutertheme{infolines}\setbeamercovered{transparent}
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     6
#+COLUMNS: %45ITEM %10BEAMER_env(Env) %10BEAMER_envargs(Env Args) %4BEAMER_col(Col) %8BEAMER_extra(Extra)
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
     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
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
     8
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
     9
#+LaTeX_CLASS: beamer
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    10
#+LaTeX_CLASS_OPTIONS: [presentation]
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    11
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    12
#+LaTeX_HEADER: \usepackage[english]{babel} \usepackage{ae,aecompl}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    13
#+LaTeX_HEADER: \usepackage{mathpazo,courier,euler} \usepackage[scaled=.95]{helvet}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    14
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    15
#+LaTeX_HEADER: \usepackage{listings}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    16
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    17
#+LaTeX_HEADER:\lstset{language=Python, basicstyle=\ttfamily\bfseries,
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    18
#+LaTeX_HEADER:  commentstyle=\color{red}\itshape, stringstyle=\color{darkgreen},
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    19
#+LaTeX_HEADER:  showstringspaces=false, keywordstyle=\color{blue}\bfseries}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    20
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    21
#+TITLE:   Getting started with symbolics
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    22
#+AUTHOR:    FOSSEE
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    23
#+EMAIL:     
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    24
#+DATE:    
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    25
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    26
#+DESCRIPTION: 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    27
#+KEYWORDS: 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    28
#+LANGUAGE:  en
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    29
#+OPTIONS:   H:3 num:nil toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    30
#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:not-in-toc
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    31
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    32
* Outline
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    33
  - Defining symbolic expressions in sage.  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    34
  - Using built-in constants and functions.   
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    35
  - Performing Integration, differentiation using sage. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    36
  - Defining matrices. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    37
  - Defining Symbolic functions.  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    38
  - Simplifying and solving symbolic expressions and functions.
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    39
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    40
* Questions 1
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    41
  - Define the following expression as symbolic
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    42
    expression in sage.
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
    43
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    44
    - x^2+y^2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    45
    - y^2-4ax
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    46
  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    47
* Solutions 1
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    48
#+begin_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    49
  var('x,y')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    50
  x^2+y^2
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
    51
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    52
  var('a,x,y')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    53
  y^2-4*a*x
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    54
#+end_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    55
* Questions 2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    56
  - Find the values of the following constants upto 6 digits  precision 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    57
   
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    58
    - pi^2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    59
   
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    60
      
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    61
  - Find the value of the following.
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    62
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    63
   - sin(pi/4)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    64
   - ln(23)  
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
    65
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    66
* Solutions 2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    67
#+begin_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    68
  n(pi^2,digits=6)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    69
  n(sin(pi/4))
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    70
  n(log(23,e))
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    71
#+end_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    72
* Question 2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    73
  - Define the piecewise function. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    74
   f(x)=3x+2 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    75
   when x is in the closed interval 0 to 4.
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    76
   f(x)=4x^2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    77
   between 4 to 6. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    78
   
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    79
  - Sum  of 1/(n^2-1) where n ranges from 1 to infinity. 
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
    80
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    81
* Solution Q1
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    82
#+begin_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    83
  var('x') 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    84
  h(x)=3*x+2 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    85
  g(x)= 4*x^2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    86
  f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    87
  f
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    88
#+end_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    89
* Solution Q2
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    90
#+begin_src python  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    91
  var('n')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    92
  f=1/(n^2-1) 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    93
  sum(f(n), n, 1, oo)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    94
#+end_src python  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    95
 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    96
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    97
* Questions 3
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    98
  - Differentiate the following. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
    99
      
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   100
    - x^5*log(x^7)  , degree=4 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   101
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   102
  - Integrate the given expression 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   103
      
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   104
    - x*sin(x^2) 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   105
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   106
  - Find x
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   107
    - cos(x^2)-log(x)=0
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   108
    - Does the equation have a root between 1,2. 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   109
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   110
* Solutions 3
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   111
#+begin_src python
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   112
  var('x')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   113
  f(x)= x^5*log(x^7) 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   114
  diff(f(x),x,5)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   115
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   116
  var('x')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   117
  integral(x*sin(x^2),x) 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   118
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   119
  var('x')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   120
  f=cos(x^2)-log(x)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   121
  find_root(f(x)==0,1,2)
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   122
#+end_src
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   123
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   124
* Question 4
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   125
  - Find the determinant and inverse of :
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   126
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   127
      A=[[x,0,1][y,1,0][z,0,y]]
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   128
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   129
* Solution 4
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   130
#+begin_src python  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   131
  var('x,y,z')
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   132
  A=matrix([[x,0,1],[y,1,0],[z,0,y]])
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   133
  A.det()
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   134
  A.inverse()
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   135
#+end_src
418
8a42b4203f6d "Merging heads"
Amit Sethi
parents:
diff changeset
   136
* Summary
442
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   137
 - We learnt about defining symbolic 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   138
   expression and functions.  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   139
 - Using built-in constants and functions.  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   140
 - Using <Tab>  to see the documentation of a 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   141
   function.  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   142
 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   143
* Summary 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   144
 - Simple calculus operations .  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   145
 - Substituting values in expression 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   146
   using substitute function.
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   147
 - Creating symbolic matrices and 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   148
   performing operation on them .
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   149
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   150
* Thank you!
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   151
#+begin_latex
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   152
  \begin{block}{}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   153
  \begin{center}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   154
  This spoken tutorial has been produced by the
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   155
  \textcolor{blue}{FOSSEE} team, which is funded by the 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   156
  \end{center}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   157
  \begin{center}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   158
    \textcolor{blue}{National Mission on Education through \\
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   159
      Information \& Communication Technology \\ 
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   160
      MHRD, Govt. of India}.
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   161
  \end{center}  
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   162
  \end{block}
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   163
#+end_latex
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   164
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   165
a9b71932cbfa Added exercises and slides to getting started with symbolics
Amit Sethi
parents: 418
diff changeset
   166