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\noindent {\bf Problem statement} Suppose $\bf H$ is a vector field of
the form ${\bf H}(x,y,z)=h(x,y,z)(x{\bf i}+y{\bf j})$, where
$h(x,y,z)$ is a positive scalar function.  If $S$ is the closed
surface which is the boundary of the solid region bounded below by the
paraboloid $z=x^2+y^2$ and above by the plane $z=4$, with positive
(outward) orientation, is the integral $\displaystyle \int\!\int_S
{\bf H}\cdot d{\bf S}$ positive, negative, or zero?










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