Rate of Change and Gradient

Recall the directional derivative of f in the direction of a unit vector u, Du(f)=gradient(f).u.
This gives the rate of change of f in the direction of the vector u.
Recall that a.b=||a|| ||b|| cos(theta) where theta is the angle between the vectors a and b. Using this with the directional derivative, we see that
Du(f)=gradient(f).u=||gradient(f)|| ||u|| cos(theta) which implies (since ||u||=1 and cos(theta) is between -1 and 1)
-||gradient(f)||<=Du(f)<=||gradient(f)||

Now if u is in the direction of gradient(f), then theta=0 and we have Du(f)=||gradient(f)||
If u points in the opposite direction of gradient(f), then theta=Pi and we have
Du(f)=-||gradient(f)||

What does this say?

A (differentiable) function increases most rapidly in the direction of the gradient (u=gradient(f)) and the rate of change in this direction is ||gradient(f)||.

A (differentiable) function decreases most rapidly in the opposite direction of the gradient (u=-gradient(f)) and the rate of change in this direction is -||gradient(f)||.

Example

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