## 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
$\frac{df}{ds}=\nabla f \cdot \mathbf{v}=|\nabla f| \cos \theta <=|\nabla f|$
since
$\cos \theta <=1$
.