Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Prove Property 5 in Table 4.5.1.
Property 5: The maximal rate of change in f, as measured by the directional derivative, has the value ∇f.
At a point P where ∇f≠0, the directional derivative of f in the direction u= ∇f/∇f, is given by
DufP= ∇f·u= ∇f·(∇f/∇f) = ∥∇f∥2/∇f = ∇f
The value of the maximal rate of increase of f is thus ∇f.
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