Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 12 / d / Solution

For case (i),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),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 $h$.