DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
  "PGstandard.pl",
  "PGunion.pl",
  "Parser.pl",
  "parserVectorUtils.pl",
  "PGcourse.pl"
);


TEXT(beginproblem());
BEGIN_PROBLEM();

##############################################
#  Setup

Context("Vector");

#
#  The function
#
$nx = random(2,5,1);
$ny = random(2,5,1);
$nz = random(2,5,1);
$m = $nx*$ny*$nz;

$f = Formula("(x^$nx y^$ny z^$nz)/($nx*$ny*$nz)");

#
#  The point
#
($x,$y,$z) = (non_zero_random(-3,3,1),
              non_zero_random(-3,3,1),
              non_zero_random(-3,3,1));

#
#  The vector
#
$U = Vector((2*non_zero_random(-1,1,1),
             2*non_zero_random(-1,1,1),
             non_zero_random(-1,1,1))[shuffle(3)]);

#
#  The derivatives
#
$fx = Formula("x^($nx-1) y^$ny z^$nz/($ny*$nz)");
$fy = Formula("x^$nx y^($ny-1) z^$nz/($nx*$nz)");
$fz = Formula("x^$nx y^$ny z^($nz-1)/($nx*$ny)");

#
#  Directional derivative
#

$gradf = Vector($fx,$fy,$fz);
$Duf = $gradf->eval(x=>$x,y=>$y,z=>$z) . unit($U);

##############################################
#  Main text

$u = BoldMath('u');

Context()->texStrings;
BEGIN_TEXT

Let \(f(x,y,z) = $DISPLAY $f\).
$PAR

Then \($GRAD\!f\) = \{ans_rule(60)\}, and \(D_{$u} f($x,$y,$z)\) 
in the direction of \($U\) is \{ans_rule(40)\}.

END_TEXT
Context()->normalStrings;

##################################################
#  Answers

ANS($gradf->cmp);
ANS($Duf->cmp);

$showPartialCorrectAnswers = 1;

##################################################

END_PROBLEM();
ENDDOCUMENT();        # This should be the last executable line in the problem.