Given a dual variable function $$f(x,y)$$ the gradient,$\nabla f$, at a point $(x_1,y_1)$ on the surface defined by this function would give us the vector at that point, pointing in the direction of the greatest change of the function. And given a random vector, $u$ at that same point, the directional derivative of the function along this random vector gives us the slope or the rate of change along the direction pointed by the vector. This directional derivative can also be given as the dot product of (projection of) the gradient vector,$\nabla f$, with this random vector $u$. How is this possible as another function, $$g(x,y)$$ with the same gradient as that of $f(x,y)$, at $(x_1,y_1)$, but different slope in the direction $u$ is also possible.
Is it because the angle between these two vectors on the different surfaces would be different?