steepest ascent

The gradient's direction is the direction $f$ increases fastest, and its length is that fastest rate.

$D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}$ is a dot product between a fixed vector and a rotating unit vector, maximized exactly when the two point the same way — so the direction of steepest ascent at a point is $\nabla f$ itself, normalized, and the steepest rate of increase is $|\nabla f|$ . This is a purely local statement about which way to step next from a given point, not a claim about where the highest point on the whole surface is.

Examples

For $f(x,y)=\ln(x^2+y^2)$ at $(3,4)$ , the steepest-ascent direction is $\langle3/5,4/5\rangle$ with rate $2/5$ .