Chapter 4: Partial Differentiation

Section 4.5: Gradient Vector

Example 4.5.7

$$ Prove Property 5 in Table 4.5.1. $$ Solution $$ Property 5: The maximal rate of change in $f$, as measured by the directional derivative, has the value $∥\nabla f∥$. $$ •  At a point P where $\nabla f\ne 0$, the directional derivative of $f$ in the direction $\mathbf{u}\= \nabla f\/∥\nabla f∥$, is given by   ${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)\= \nabla f·\mathbf{u}\= \nabla f·\(\nabla f\/∥\nabla f∥\)$ = ${\∥\nabla f\∥}^2\/∥\nabla f∥$ = $∥\nabla f∥$  $$ •  The value of the maximal rate of increase of $f$ is thus $∥\nabla f∥$.

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