The gradient collects the partial derivatives into a direction of greatest increase.
For a unit direction u, the rate is ∇f·u. The dot product is largest when u points along a nonzero gradient, giving the maximum rate |∇f|. At a zero gradient there is no unique direction of steepest first-order increase.
The gradient lives in the input plane; it is not a vector pointing vertically off the surface.
Drag to rotate, or use the view sliders. The surface and level curve are sampled; all point values and derivatives use exact formulas.