View of /trunk/rochester_problib/instructiveProblems/NewProblems/ur_vc_12_10N.pg

    1 ##DESCRIPTION
    2 #
    3 # File Created: 6/5/2000
    4 # Last Modified: 6/5/2000
    5 # Problem Author: Joseph Neisendorfer
    6 # WeBWorK Entry: David Etlinger
    7 # Location: University of Rochester
    8 #
    9 # Compute divergence, curl, and div curl
   10 # of a field
   11 #
   12 ##ENDDESCRIPTION
   13 
   14 ##KEYWORDS('Vector','Divergence','Curl','Field')
   15 
   16 DOCUMENT();        # This should be the first executable line in the problem.
   17 
   18 loadMacros(
   19 "PGstandard.pl",
   20 "PGcourse.pl",
   21 "MathObjects.pl",
   22 # "source.pl",
   23 );
   24 
   25 TEXT( beginproblem() );
   26 $showPartialCorrectAnswers = 1;
   27 
   28 ###################
   29 #
   30 #  Setup
   31 
   32 Context("Vector");
   33 
   34 $a = random(1,10,1);
   35 $b = random(1,10,1);
   36 $c = random(1,10,1);
   37 
   38 $ans1 = 0;
   39 
   40 $ans7 = 0;
   41 
   42 $VectI = Formula("$a*y*z");
   43 $VectJ = Formula("$b*x*z");
   44 $VectK = Formula("$c*x*y");
   45 
   46 $VectEqu = Vector("$VectI*i + $VectJ*j + $VectK*k")->reduce;
   47 
   48 $ans1 = ($VectI->D('x'))+($VectJ->D('y'))+($VectK->D('z'));
   49   #   For multi-variable functions, one must specify the variable to
   50   # differentiate with respect to as a parameter of the differentiation
   51   # function, i.e. $Function->D('variable');
   52 
   53 $ans2 = &Curl($VectI,$VectJ,$VectK);
   54   #   Using a subroutine defined below, the Curl is returned as a vector given
   55   # the function -I,J,K components as parameters.
   56 
   57 $ans3 = Real("0");
   58   #   Div of Curl of F = 0.
   59 
   60 ###################
   61 #
   62 #  Text
   63 
   64 Context()->texStrings;
   65 BEGIN_TEXT
   66 Let \( \mathbf{F} = $VectEqu \). Compute the following:
   67   $PAR
   68 A. div \( \mathbf{F} = \) \{ans_rule(40)\}
   69   $PAR
   70 B. curl \( \mathbf{F} = \) \{ans_rule(80)\}
   71   $PAR
   72 C. div curl \( \mathbf{F} = \) \{ans_rule(40)\}
   73   $PAR
   74 Note: Your answers should be expressions of x, y and/or z; e.g. "3xy" or "z" or
   75 "5".  To enter a vector as an answer use either the \( \mathbf{i} -
   76 \mathbf{j}-\mathbf{k} \) notation or \(<x_1,x_2,\ldots,x_n>\) notation; e.g. for
   77 \(5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\) enter either "5i + 3j + 4k" or
   78 "<5,3,4>."  $PAR
   79 END_TEXT
   80 Context()->normalStrings;
   81 
   82 ###################
   83 #
   84 #  Answers
   85 
   86 ANS($ans1->cmp);
   87 ANS($ans2->cmp);
   88 ANS($ans3->cmp);
   89 
   90 ###################
   91 #
   92 #  Subroutine:
   93 #
   94 #    Given i-j-k components of a vector function with respect to x, y,
   95 #  and z, and returns the curl of the function as a vector object.
   96 
   97 sub Curl {
   98 
   99   my ($FnI,$FnJ,$FnK) = @_;  #   Parameters passed into those three
  100                              # local variables.
  101 
  102   my $FnIder = ($FnK->D('y')) - ($FnJ->D('z')),
  103      $FnJder = ($FnI->D('z')) - ($FnK->D('x')),
  104      $FnKder = ($FnJ->D('x')) - ($FnI->D('y'));
  105 
  106   Vector("$FnIder*i + $FnJder*j + $FnKder*k");
  107                              #   The curl of the vector function is returned
  108                              # as a vector object.
  109 
  110 };
  111 
  112 ENDDOCUMENT();        # This should be the last executable line in the problem.