= Solution
For case (i),
$$
\boxed{\nabla h=(e^x-1,e^y-1)^T}.
$$
Each component has the sign of its coordinate, so the <gradient> points outward from the <minimum>. Near the origin it is $(x,y)^T+O(x^2+y^2)$.
For case (ii),
$$
\boxed{\nabla h=(e^x-1,1-e^y)^T}.
$$
Its $x$ component has the sign of $x$, while its $y$ component has the opposite sign to $y$. Thus arrows point outward on the $x$-axis and inward on the $y$-axis, with local form $(x,-y)^T+O(x^2+y^2)$. These directions are included in the contour figure; in both cases the <gradient> is perpendicular to the <level curves> and points toward increasing $h$.
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