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

Theorem
A 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 since
\[\cos \theta .