Instructions.

Press “Play” to watch the vertical plane through \(P_0(1,1)\) turn once around. The gold curve is where it cuts the surface \(z = \frac{x^2}{2} + \frac{y^2}{2}\text{,}\) and the straight line through \((1,1,1)\) is the tangent to that curve, whose slope is \(D_{\mathbf u} f\text{:}\) it is green and steepest upwards along \(\nabla f\) (A), red and steepest downwards along \(-\nabla f\) (B), and yellow and level in the two directions tangent to the pink level curve \(f = 1\) (C). Stop the animation at any time and drag the \(\varphi\) slider, drag the figure to change the viewpoint, or press “Spin”.