## DBsubject('Calculus') ## DBchapter('Fundamental Theorems of Vector Analysis') ## DBsection('Greens Theorem') ## KEYWORDS('calculus') ## TitleText1('Calculus: Early Transcendentals') ## EditionText1('2') ## AuthorText1('Rogawski') ## Section1('17.1') ## Problem1('25') ## Author('JustAsk - Kobi Fonarov') ## Institution('W.H.Freeman') ## UsesAuxiliaryFiles('image_17_1_27.png') DOCUMENT(); loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); loadMacros("PGauxiliaryFunctions.pl"); loadMacros("PGgraphmacros.pl"); loadMacros("PGchoicemacros.pl"); TEXT(beginproblem()); $c2=random(2,9);$c3=random(2,9); $curl=random(2,9);$c23=$c2+$c3; $answer=Real(($c23+$curl*23)*$PI); $curve="\mathcal{C}";$domain="\mathcal{D}"; $FF="\mathbf{F}"; TEXT('') if$displayMode eq 'HTML_jsMath'; Context()->texStrings; BEGIN_TEXT \{ textbook_ref_exact("Rogawski ET 2e", "17.1","25") \} $PAR Referring to Figure 11, suppose that $\oint_{{curve}_2} FF \cdot d\mathbf{s} = c2\pi,\qquad \oint_{{curve}_3} FF \cdot d\mathbf{s} = c3\pi$ Use Green's Theorem to determine the circulation of $$FF$$ around $${curve}_1$$, assuming that curl$$_z(FF)=curl$$ on the shaded region.$PAR \{image("image_17_1_27.png", width=>146, height=>186)\} $PAR $$\int_{{curve}_1} FF \cdot d\mathbf{s} =$$ \{ans_rule()\}$PAR END_TEXT ANS($answer->cmp); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION'));$PAR $SOL We must calculate $$\int_{{curve}_1} FF \cdot d\mathbf{s}$$.$BR We use Green's Theorem for the region $$domain$$ between the three circles $${curve}_1$$, $${curve}_2$$, and $${curve}_3$$. $PAR Because of orientation, the line integrals $$\int_{-{curve}_2} FF \cdot d\mathbf{s} =-\int_{{curve}_2} FF \cdot d\mathbf{s}$$ and $$\int_{-{curve}_3} FF \cdot d\mathbf{s} =-\int_{{curve}_3} FF \cdot d\mathbf{s}$$$BR must be used in applying Green's Theorem. That is, $\int_{{curve}_1} FF \cdot d\mathbf{s} -\int_{{curve}_2} FF \cdot d\mathbf{s} -\int_{{curve}_3} FF \cdot d\mathbf{s} =\iint_{domain} \text{curl} (FF) \,dA$ We substitute the given information to obtain $\int_{{curve}_1} FF \cdot d\mathbf{s} -c2\pi -c3\pi =\iint_{domain} curl \,dA=$$curl\iint_{domain} 1\cdot \,dA=curl \,\mathrm{Area} (domain) \quad \mathbf{(1)}$ The area of $$domain$$ is computed as the difference of areas of discs. That is, $\mathrm{Area} (domain)=\pi \cdot 5^2-\pi \cdot 1^2-\pi \cdot 1^2=23\pi$ We substitute in $$\mathbf{(1)}$$ and compute the desired circulation: $\int_{{curve}_1} FF \cdot d\mathbf{s} -c23\pi =curl\cdot 23\pi$ or $\int_{{curve}_1} FF \cdot d\mathbf{s} =\{c23+curl*23\}\pi.$ END_SOLUTION ENDDOCUMENT();