Why is a function decreasing the fastest in dir of neg of the gradient?

[toofle]

Why is the function decreasing the fastest in the direction of the negative of the gradient?

Just because it increases the fastest in the direction of the positive of the gradient why does this have to mean it has to decrease the fstest in the negative of the gradient?
If you stand facing a steep mountain and ityou have flats behind you and a huge drop on your sides, it would decrease faster to 90 degress than behind you. What am I not understanding with gradients?

 

[bigfooted]

Why is the function decreasing the fastest in the direction of the negative of the gradient?

It doesn't (always), exactly for the reason you are giving. Your mountain and your function are not allowed to have saddle points.
There cannot be valleys where straight ahead and behind you, you go up, and left and right of you, you go down. Symmetric (positive) definiteness is often a requirement to guarantee this.

If this is about steepest descent or conjugate gradient methods, You should read 'Introduction to the conjugate gradient method without the agonizing pain' by Jonathan Shewchuk.

 

[Erland]

This is only claimed to be valid at points where the function surface has a nonhorizontal tangent plane. If there is such a plane, walking at the surface near the point can be approximated with walking at the tangent plane. Wouldn't you then agree that the directions of steepest increase and decrease are opposite?