View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/14_Differentiation_in_Several_Variables/14.5_The_Gradient_and_Directional_Derivatives/14.5.7.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Differentiation in Several Variables')
    3 ## DBsection('The Gradient and Directional Derivatives')
    4 ## KEYWORDS('calculus')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('14.5')
    9 ## Problem1('7')
   10 ## Author('JustAsk - Vladimir Finkelshtein')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 
   15 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 loadMacros("PGauxiliaryFunctions.pl");
   19 loadMacros("PGgraphmacros.pl");
   20 loadMacros("PGchoicemacros.pl");
   21 
   22 TEXT(beginproblem());
   23 
   24 Context()->texStrings;
   25 $a=non_zero_random(-3,3,1);
   26 $b=non_zero_random(-3,3,1);
   27 $c=non_zero_random(-3,3,1);
   28 
   29 $context = Context();
   30 $context->variables->add(y=>'Real');
   31 $context->variables->add(z=>'Real');
   32 
   33 $f=Formula("x^($a)*y^($b)*z^($c)")->reduce();
   34 $fx=Formula("$a*x^($a-1)*y^($b)*z^($c)")->reduce();
   35 $fy=Formula("$b*x^($a)*y^($b-1)*z^($c)")->reduce();
   36 $fz=Formula("$c*x^($a)*y^($b)*z^($c-1)")->reduce();
   37 
   38 BEGIN_TEXT
   39 \{ textbook_ref_exact("Rogawski ET 2e", "14.5","7") \}
   40 $PAR
   41 Calculate the gradient of \(h(x,y,z)=$f\)
   42 $PAR
   43 \( \nabla h=\) \{ans_rule()\}
   44 $BR
   45 END_TEXT
   46 
   47 Context()->normalStrings;
   48 Context("Vector");
   49 ANS(Vector(Formula($fx),Formula($fy),Formula($fz))->cmp);
   50 Context("Numeric");
   51 Context()->texStrings;
   52 
   53 SOLUTION(EV3(<<'END_SOLUTION'));
   54 $PAR
   55 $SOL
   56 $BR
   57 We compute the partial derivatives of \(h(x,y,z)=$f\), obtaining
   58 \[\frac{\partial{h}}{\partial{x}}=$fx \quad \frac{\partial{h}}{\partial{y}}=$fy \quad \frac{\partial{h}}{\partial{z}}=$fz \]
   59 The gradient vector is thus
   60 \[\nabla h=\left<\frac{\partial{h}}{\partial{x}},\frac{\partial{h}}{\partial{y}},\frac{\partial{h}}{\partial{z}}\right>=
   61 \left<$fx, $fy, $fz \right>\]
   62 
   63 $BR
   64 END_SOLUTION
   65 
   66 ENDDOCUMENT();