Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Prove Property 3 in Table 4.5.1.
Property 3: At any point where ∇f≠0, the gradient ∇f points in the direction of increasing values of f.
Let P be a point where the gradient does not vanish. Then, the directional derivative of f in the direction u= ∇f/∇f is given by
DufP= ∇f·u= ∇f·(∇f/∇f) = ∥∇f∥2/∇f = ∇f>0
Since at P the rate of change of f in the direction of the gradient is positive, the function values of f are indeed increasing in the direction of the gradient.
<< Previous Example Section 4.5
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document