Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 13 / b / iii / Solution

The horizontal direction of steepest descent on a graph surface is parallel to $-\nabla z$. On the line $y=1$, the component $z_y=X(1-y^2)H$ vanishes identically, so that line is invariant under the downhill path. Starting at $X=0$, $z_x=(1-X^2)e^{-(X^2+1)/2}>0$ for $-1<X<0$, so $x$ decreases towards zero. The path reaches the minimumAt the minimum no strictly downhill direction remains. A constant-speed parametrization reaches it after the finite arc length divided by $u$; the unnormalized gradient flow approaches the same endpoint only asymptotically in its own time parameter.