Instructions.

Press “Play” to watch the vertical plane through \(P_0(1,1)\) turn once around. The gold curve is where the plane cuts the surface \(z = 1 + x^2 + y^2\text{,}\) and the straight line through \((1,1,3)\) is the tangent to that curve; its slope is \(D_{\mathbf u} f = \left(\nabla f\right)_{(1,1)}\cdot\mathbf u\text{,}\) read off at the lower left. The tangent is green and steepest upwards when \(\mathbf u\) points along \(\nabla f\text{,}\) red and steepest downwards along \(-\nabla f\text{,}\) and yellow and level in the two directions tangent to the pink level curve. Stop the animation at any time and drag the \(\varphi\) slider, drag the figure to change the viewpoint, or press “Spin”.