Chapter 4: Partial Differentiation

Section 4.5: Gradient Vector

Example 4.5.5

$$ Prove Property 3 in Table 4.5.1. $$ Solution $$ Property 3: At any point where $\nabla f\ne \mathbf{0}$, the gradient $\nabla f$ points in the direction of increasing values of $f$. $$ •  Let P be a point where the gradient does not vanish. Then, the directional derivative of $f$ in the direction $\mathbf{u}\= \nabla f\/∥\nabla f∥$ is given by   ${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)\= \nabla f·\mathbf{u}\= \nabla f·\(\nabla f\/∥\nabla f∥\)$ = ${\∥\nabla f\∥}^2\/∥\nabla f∥$ = $∥\nabla f∥\>0$    •  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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