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.
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