View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/17_Fundamental_Theorems_of_Vector_Analysis/17.3_Divergence_Theorem/17.3.17.pg

    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 
   16 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   17 loadMacros("Parser.pl");
   18 loadMacros("freemanMacros.pl");
   19 loadMacros("PGauxiliaryFunctions.pl");
   20 loadMacros("PGgraphmacros.pl");
   21 loadMacros("PGchoicemacros.pl");
   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();