View of /trunk/NationalProblemLibrary/ASU-topics/setCalculus/stef16_6p4.pg

    1 ## DESCRIPTION
    2 ## Multivariable Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus','normal','gradient')
    6 ## Tagged by cmd6a 3/12/06
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Vector Calculus')
   10 ## DBsection('Parametric Surfaces and Their Areas')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('ASU')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();
   21 loadMacros("PG.pl",
   22            "PGbasicmacros.pl",
   23            "PGchoicemacros.pl",
   24            "PGanswermacros.pl",
   25            "PGauxiliaryFunctions.pl",
   26            "PGgraphmacros.pl",
   27            "Dartmouthmacros.pl");
   28 
   29 
   30 ## Do NOT show partial correct answers
   31 $showPartialCorrectAnswers = 1;
   32 
   33 ## Lots of set up goes here
   34 $pi = acos(-1);
   35 @angles= ($pi/6, $pi/4, $pi/3);
   36 @angles_print= ("\pi/6", "\pi/4", "\pi/3");
   37 
   38 $which_theta = NchooseK(3,1);
   39 $which_phi = NchooseK(3,1);
   40 $theta = $angles[$which_theta];
   41 $theta_print = $angles_print[$which_theta];
   42 $phi =  $angles[$which_phi];
   43 $phi_print =  $angles_print[$which_phi];
   44 $r = random(2,6,1);
   45 
   46 
   47 $x0 = spf($r * cos($theta) * sin($phi),"%7.5f");
   48 $y0 = spf($r * sin($theta) * sin($phi),"%7.5f");
   49 $z0 = spf($r * cos($phi),"%7.5f");
   50 
   51 
   52 ## Ok, we are ready to begin the problem...
   53 ##
   54 TEXT(beginproblem());
   55 
   56 
   57 BEGIN_TEXT
   58 $BR
   59 A sphere of radius $r is centered at the origin.
   60 $BR
   61 It may be viewed as a parametrized surface:
   62 \( \mathbf{r}(\theta,\phi) = ($r \cos\theta \sin\phi, $r \sin\theta \sin\phi, $r \cos\phi)\),
   63 a level surface of the function \( f(x,y,z) = x^2 + y^2 + z^2 \),
   64 or as the graph of the function \(g(x,y) = \sqrt{\{$r**2\} - x^2 - y^2}\).
   65 $BR
   66 Consider the sphere at the point \( ($x0, $y0, $z0) \) (corresponding to
   67 \((\theta, \phi) = ($theta_print, $phi_print)\) ).
   68 $PAR
   69 $BBOLD A)$EBOLD Find the normal vector \( \mathbf{r}_\theta \times \mathbf{r}_\phi\) at the given point: $BR
   70 \((\)\{ ans_rule(25)\}, \{ ans_rule(25)\},\{ ans_rule(25)\} \()\)
   71 $BR
   72 $BBOLD B)$EBOLD Find the gradient of \(f\) at the indicated point:$BR
   73 \((\)\{ ans_rule(25)\}, \{ ans_rule(25)\},\{ ans_rule(25)\} \()\)
   74 $BR
   75 
   76 They should be parallel ....
   77 $PAR
   78 END_TEXT
   79 
   80 ANS(num_cmp(-$r*sin($phi)*$x0));
   81 ANS(num_cmp(-$r*sin($phi)*$y0));
   82 ANS(num_cmp(-$r*sin($phi)*$z0));
   83 ANS(num_cmp(2 * $x0));
   84 ANS(num_cmp(2 * $y0));
   85 ANS(num_cmp(2 * $z0));
   86 
   87 ENDDOCUMENT();
   88 
   89 
   90 
   91