Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Prove Property 4 in Table 4.5.1.
Property 4: Where ∇f≠0, it necessarily points in the direction of maximal increase in f.
Let P be a point where the gradient does not vanish. Then, the directional derivative of f in the direction u is given by
DufP= ∇f·u = ∇f u = ∇f 1 cosθ
where θ is the angle between ∇f and the unit vector u.
The product of the positive quantities ∇f and |cosθ| is maximal when θ=0 so that cosθ=1. Hence, when u is along ∇f, the rate of change in f is maximal. The gradient direction is therefore the direction of maximal increase in f.
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