TITLE: Does $-\frac{ \nabla f }{ || \nabla f || }$ point in the direction of steepest descent of $f$?
QUESTION [0 upvotes]: One way to interpret $\nabla f$ geometrically is that it yields a vector field who's direction is in the direction of the steepest ascent.
So, we can find the unit vector along this vector:
$$\frac{ \nabla f}{ || \nabla f || }$$
If we multiply it by the scaler $-1$:
$$ -\frac{ \nabla f }{ || \nabla f || } $$
Would this give me the unit vector along the steepest descent?
I needed the unit vector along steepest descent. After thinking a bit I came up with this but I am not sure if this is correct. In case it is correct, is there any proofy way to explain it so I can be more confident about it?

REPLY [0 votes]: Given that you already agree that $\nabla f$ points in the direction of steepest ascent:
If we define the new function $g= -f$, then $g$ ascends precisely when $f$ descends, and so the direction of steepest ascent of $g$ is the direction of steepest descent of $f$. So
$$\nabla g = \nabla (-f) = - \nabla f$$
is the direction of steepest descent of $f$.