## Problem on gradient, directional derivative and level curves

Here are equispaced level curves of a function $f(x,y)$.
a) Where is $\nabla f$ biggest in magnitude?
b) Where is $\nabla f$ smallest in magnitude?

c) Where is $\partial_x f =0$?

d) Where is the directional derivative $D_{(\mathbf{i}/\sqrt{2} + \mathbf{j}/\sqrt{2})} f = 0$?
• ## Solution

(a) Because the only information we have is a sketch of the level curves of $f$, we need to relate these level curves to the magnitude of $\nabla f$.
Recall that
We now relate the level curves to directional derivatives of $f$.
Because the level curves correspond to equally spaced values of $f$, the directional derivative is largest where where the level curves are closest together.
Thus, the maximal magnitude $|\nabla f|$ is located as shown:
(b) The magnitude of $\nabla f$ is small at points where $f$ isn't very steep in any direction.
The smallest possible value $|\nabla f|$ can take is 0. It occurs if the function is locally flat.
The function $f$ must be flat (at its maximal or minimal value) somewhere inside the innermost level set. Hence $|\nabla f|$ is smallest at a point such as:
(c) We need to relate the partial derivative of $f$ to the level sets of $f$.
Some points where the level curves are parallel to $\mathbf{i}$ are as shown:
(d) We need to relate the directional derivative of $f$ to the level sets of $f$.
Some points where the level curves are parallel to $\mathbf{i} + \mathbf{j}$ are as shown: