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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?

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up vote 3 down vote accepted

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.

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will you write it down in equations? – webster Dec 29 '11 at 23:32
If $v$ is a unit vector, the directional derivative of the function $f$ is $$ v \cdot \nabla f = |\nabla f| \cos \theta, $$ where $\theta$ is the angle between $v$ and $\nabla f.$ – Will Jagy Dec 30 '11 at 4:27

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