For a function of one variables $f(x)$, $f'(x)>1$ implies that the rate of increase of $f(x) $ at $x$ is more than 1. How can we define the analogous notion for a function of two variables $f(x,y)$, i.e. whenever $f(x,y)$ increases, the rate of increase is more than 1?
If the length of the gradient vector is exactly one, then the function increases at the rate of 1 in that direction only. If the gradient vector is longer than 1, then there is a wedge of directions in which the rate of increase of the function is larger than 1, with the gradient vector in the center of the wedge.