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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2990 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('Divergence Theorem')
4 ## KEYWORDS('calculus')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('17.3')
9 ## Problem1('17')
10 ## Author('JustAsk - Kobi Fonarov')
11 ## Institution('W.H.Freeman')
12 ## UsesAuxiliaryFiles('image_17_3_15_a.png','image_17_3_15_b.png')
13
14 DOCUMENT();
15
22
23 TEXT(beginproblem());
24
25 $r=random(2,9); 26$r2=$r**2; 27$a=random(2,9);
28 $b=random(2,9); 29 if ((($a+$b)%2)!=0) {$b=$b+1;}; 30$div=$a+$b;
31
32 $sol=($div/2)*($r**4); 33$answer=Real($sol*$PI);
34
35 $curve="\mathcal{C}"; 36$domain="\mathcal{D}";
37 $region="\mathcal{W}"; 38$surf="\mathcal{S}";
39 $FF="\mathbf{F}"; 40$curl="\text{curl}";
41 $ii="\mathbf{i}"; 42$jj="\mathbf{j}";
43 $kk="\mathbf{k}"; 44$dive="\text{div}";
45 $rec="\mathcal{R}"; 46$nn="\mathbf{n}";
47 $TT="\mathbf{T}"; 48 49 TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>') 50 if$displayMode eq 'HTML_jsMath';
51
52 Context()->texStrings;
53
54 BEGIN_TEXT
55 \{ textbook_ref_exact("Rogawski ET 2e", "17.3","17") \}
56 $PAR 57 Use the Divergence Theorem to evaluate the surface integral $$\iint_{surf} FF\cdot d\mathbf{S}$$ .$PAR
58 $$FF = \left<a x+y, z,b z-x\right>$$, $$surf$$ is the boundary of the
59 region between the paraboloid $$z=r2-x^2-y^2$$ and the $$xy$$-plane.
60 $PAR 61 $$\iint_{surf} FF\cdot d\mathbf{S}=$$ \{ans_rule()\}$PAR
62
63 END_TEXT
64
65 ANS($answer->cmp); 66 67 68 Context()->texStrings; 69 SOLUTION(EV3(<<'END_SOLUTION')); 70$PAR
71 $SOL We compute the divergence of $$FF = \left<a x+y,z,b z-x \right>$$, 72 $73 dive (FF)=\frac{\partial }{\partial x}(a x+y)+\frac{\partial }{\partial y}(z)+\frac{\partial }{\partial z}(b z-x)=$$a+0+b=div\text{.} 74$ 75 \{image("image_17_3_15_a.png", width=>230, height=>190)\} With $$r_0=r$$ 76$PAR
77 Using the Divergence Theorem we have
78 $\iint_{surf} FF \cdot d\mathbf{S} =\iiint_{region} dive (FF) \,dV=\iiint_{region} div \,dV$
79 We compute the triple integral:
80 $\iint_{surf} FF \cdot d\mathbf{S} = 81 \iiint_{region} div \,dV=$$82 \iint_{domain} \int_0^{r2- x^2- y^2}div \,dz \,dx \,dy= 83 \iint_{domain} div z\bigg|_0^{r2- x^2- y^2} \,dx \,dy=$$84 \iint_{domain} div(r2- x^2- y^2) \,dx \,dy 85$
86 \{image("image_17_3_15_b.png", width=>237, height=>234)\} With $$r_0=r$$
87 \$PAR
88 We convert the integral to polar coordinates:
89 $x=r\cos\theta, \quad y=r\sin\theta, \quad 0\le r\le r, \quad 0\le \theta \le 2\pi$
90 $91 \iint_{surf} FF \cdot d\mathbf{S} = 92 \int_0^{2\pi } \int_0^{r} div\left(r2-r^2\right)r \,dr \,d\theta =$$93 \{2*div\}\pi \int_0^{r}(r2 r- r^3) \,dr= 94 \{2*div\}\pi \left(\frac{r2 r^2}{2}-\frac{r^4}{4}\bigg|_0^{r}\right)= 95 sol \pi 96$
97 END_SOLUTION
98
99 ENDDOCUMENT();