= Solution
The <gradient> is
$$
\boxed{\nabla f=E\bigl(v(1-u^2),u(1-v^2)\bigr)^T.}
$$
It is perpendicular to each nonzero <level set> and points towards increasing $f$. Near the two positive <local maxima>, the <Hessian matrix> is $-2e^{-1}I$, so the <gradient> arrows point inward. Near the two negative <local minima>, the <Hessian matrix> is $2e^{-1}I$, so they point outward. These arrows are shown on the figure in part (c).
At the <saddle point>, $\nabla f\approx(v,u)^T$. Along $v=u$, the arrows point away from the centre on both sides; along $v=-u$, they point towards the centre on both sides. On the positive $u$ axis they point up, on the negative $u$ axis down, on the positive $v$ axis right, and on the negative $v$ axis left. This specifies the four intervening sectors of the <gradient> pattern.
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