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Tue Nov 8 15:17:41 2011 UTC (19 months, 1 week ago) by aubreyja
File size: 4454 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    1 ## DBsubject('Calculus')
2 ## DBchapter('Fundamental Theorems of Vector Analysis')
3 ## DBsection('Stokes Theorem')
4 ## KEYWORDS('calculus')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('17.2')
9 ## Problem1('13')
10 ## Author('JustAsk - Kobi Fonarov')
11 ## Institution('W.H.Freeman')
12 ## UsesAuxiliaryFiles('image_17_2_13.png')
13
14 DOCUMENT();
15
22
23 TEXT(beginproblem());
24
25 $context = Context(); 26$context->variables->add(y=>'Real');
27
28
29 $a=random(2,9); 30$b=random(2,9);
31
32 $fy=Formula("$b*x^($b-1)*e^(x^($b))")->reduce();
33 $az=Formula("$a*e^y - e^(x^($b))")->reduce(); 34 35 36$curve="\mathcal{C}";
37 $domain="\mathcal{D}"; 38$surf="\mathcal{S}";
39 $FF="\mathbf{F}"; 40$curl="\text{curl}";
41 $ii="\mathbf{i}"; 42$jj="\mathbf{j}";
43 $kk="\mathbf{k}"; 44$BB="\mathbf{B}";
45 $AA="\mathbf{A}"; 46$rr="\mathbf{r}";
47 $cc="\mathbf{c}"; 48$GG="\mathbf{G}";
49
50 TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>')
51        if $displayMode eq 'HTML_jsMath'; 52 53 Context()->texStrings; 54 55 BEGIN_TEXT 56 \{ textbook_ref_exact("Rogawski ET 2e", "17.2","13") \} 57$PAR
58 Let $$I$$ be the flux of $$GG = \left<a e^y,fy,0\right>$$
59 through the upper hemisphere $$surf$$ of the unit sphere. $PAR 60$BBOLD (a) $EBOLD Find a vector field $$\mathbf{A}$$ such that $$curl(\mathbf{A})=GG$$. 61$BR
62 $BBOLD (b)$EBOLD Calculate the circulation of $$\mathbf{A}$$ around $$\partial surf$$.
63 $BR 64$BBOLD (c) $EBOLD Compute the flux of $$GG$$ through $$surf$$ 65$PAR $PAR 66$BBOLD (a) $EBOLD $$\mathbf{A}=$$ \{ans_rule()\} $$+ \left<C_1,C_2,C_3\right>$$$BR
67 $BBOLD (b)$EBOLD $$\int_{curve} \mathbf{A}\cdot d\mathbf{s}=$$ \{ans_rule()\} $BR 68$BBOLD (c) $EBOLD $$I =$$ \{ans_rule()\} 69$PAR
70
71 END_TEXT
72
73 Context("Vector");
74 ANS(Vector(Real(0),Real(0),Formula($az))->cmp); 75 Context("Numeric"); 76 ANS(Real(0)->cmp); 77 ANS(Real(0)->cmp); 78 79 80 Context()->texStrings; 81 SOLUTION(EV3(<<'END_SOLUTION')); 82$PAR
83 $SOL$PAR
84 $BBOLD (a)$EBOLD We search for a vector field $$\mathbf{A}$$ so that $$GG=curl (\mathbf{A})$$. That is,
85 $\left< \frac{\partial {\mathbf{A}}_3}{\partial y}-\frac{\partial {\mathbf{A}}_2}{\partial z},\frac{\partial {\mathbf{A}}_1}{\partial z} 86 -\frac{\partial {\mathbf{A}}_3}{\partial x},\frac{\partial {\mathbf{A}}_2}{\partial x}-\frac{\partial {\mathbf{A}}_1}{\partial y} \right> = 87$$88 \left< a e^y ,fy,0 \right>$
89 We note that the third coordinate of this curl vector must be zero;
90 this can be satisfied if $$\mathbf A_1 = 0$$
91 and $$\mathbf A_2 = 0$$. $BR 92 With this in mind, we let $$\mathbf{A}= \left< 0,0, a e^y - e^{x^{b}} \right>$$.$BR
93 The vector field $$\mathbf{A}= \left< 0,0,a e^y - e^{x^{b}} \right>$$ satisfies this equality. Indeed,
94 $\frac{\partial {\mathbf{A}}_3}{\partial y}-\frac{\partial {\mathbf{A}}_2}{\partial z}=a e^y ,\quad 95 \frac{\partial {\mathbf{A}}_1}{\partial z}-\frac{\partial {\mathbf{A}}_3}{\partial x}=fy,$$96 \frac{\partial {\mathbf{A}}_2}{\partial x}-\frac{\partial {\mathbf{A}}_1}{\partial y}=0$
97
98 $BBOLD (b)$EBOLD
99 The boundary $$curve$$ is the circle $$x^2+ y^2=1$$, parametrized by
100 $\gamma (t)=(\cos t, \sin t,0),\quad 0\le t\le 2\pi$
101 \{image("image_17_2_13.png", width=>208, height=>186)\} $BR 102 We compute the following values: 103 $\mathbf{A}\left(\gamma (t)\right)= 104 \left< 0,0,a e^y - e^{x^{b}} \right> \bigg|_{x=\cos t\text{,}y=\sin t}= 105 \left< 0,0,a e^{\sin t}- e^{\cos^{b}t} \right>$ 106 ${\gamma }'(t)=\left< - \sin t, \cos t,0 \right>$ 107 $\mathbf{A}\left(\gamma (t)\right)\cdot {\gamma }'(t) =$$108 \left< 0,0, a e^{\sin t}- e^{{ \cos}^{b}t} \right> \cdot \left< - \sin t, \cos t,0 \right> = 109 0$ 110 Therefore, 111 $\int_{curve} \mathbf{A}\cdot d\mathbf{s} =\int_0^{2\pi } 0 \,dt=0 \quad \mathbf{(1)}$ 112$BBOLD (c) $EBOLD We found that $$GG=curl (\mathbf{A})$$, where $$\mathbf{A}= \left< 0,0,a e^y - e^{x^{b}} \right>$$.$PAR
113 We compute the flux of $$\mathbf{G}$$ through $$surf$$. By Stokes' Theorem,
114 $115 \iint_{surf} GG\cdot d\mathbf{S} = 116 \iint_{surf} curl (\mathbf{A})\cdot d\mathbf{S} = 117 \int_{curve} \mathbf{A}\cdot d\mathbf{s} 118$
119 Combining with $$\mathbf{(1)}$$ we get
120 $\iint_{surf} GG\cdot d\mathbf{S} =0$
121
122
123
124
125 END_SOLUTION
126
127 ENDDOCUMENT();


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