## Proof That Grad f is in the Direction of Greatest Rate of Increase of f

TheoremA funvtion increasing most rapidly in the direction of

\[\nabla f\]

, The gretest rate of increase of \[\nabla f\]

at a point is \[ |\nabla f|\]

at that point.Proof

Suppose

\[f\]

is a parametrised in term of the distance \[s\]

along the curve.\[\frac{df}{ds} = \frac{\partial f}{\partial x} \frac{dx}{ds}+\frac{\partial f}{\partial y} \frac{dy}{ds}+\frac{\partial f}{\partial z} \frac{dz}{ds}=\nabla f \cdot \frac{d \mathbf{r}}{ds}=\nabla f \cdot \mathbf{v}\]

since

\[\mathbf{v}=\frac{d \mathbf{r}}{ds}\]

is a unit vector.But

\[\frac{df}{ds}=\nabla f \cdot \mathbf{v}=|\nabla f| \cos \theta <=|\nabla f|\]

since \[\cos \theta <=1\]

.