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

Sat Jun 3 14:35:45 2006 UTC (6 years, 11 months ago) by gage
File size: 2272 byte(s)
 Cleaned code with convert-functions.pl script


    1 ## DESCRIPTION
2 ## Multivariable Calculus
3 ## ENDDESCRIPTION
4
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();
22            "PGbasicmacros.pl",
23            "PGchoicemacros.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