Instructions.

Press “Play” to watch \(\mathbf u\) turn once around \(P_0(1,1)\text{.}\) The reading at the lower left follows \(D_{\mathbf u} f = \nabla f\cdot\mathbf u = \|\nabla f\|\cos\theta\) as it sweeps from \(\|\nabla f\|\) down to \(-\|\nabla f\|\) and back: it is largest along \(\nabla f\) (A), smallest along \(-\nabla f\) (B), and zero in the two directions tangent to the highlighted level curve \(f = 1\) (C), where the second vector \(\tilde{\mathbf u} = -\mathbf u\) appears. Stop the animation at any time and drag \(\mathbf u\) on the picture, or use the slider.