Instructions.

Drag the slider to rotate the direction \(\mathbf u = \langle\cos\varphi, \sin\varphi\rangle\) at the point \((1,1)\text{.}\) The vertical plane through \((1,1)\) in the direction \(\mathbf u\) cuts the surface \(z = 1 + x^2 + y^2\) in a curve, and the tangent line to this curve at \((1,1,3)\) has slope \(D_{\mathbf u} f = \left(\nabla f\right)_{(1,1)}\cdot\mathbf u\text{,}\) displayed above the figure. Use the buttons to snap \(\mathbf u\) to the direction of \(\nabla f\text{,}\) its opposite, or a direction of zero change. Drag the figure to view it from a different angle, or press the “Rotate” button to spin it automatically.