Gradient Operator

$$\vec{\nabla }f=⟨f_x^{\prime },f_y^{\prime },\dots ⟩$$

Geometric Properties

• $\Vert \vec{\nabla }f(P_0)\Vert$ is the max rate of change of $f$ at $P_0$
  • $\vec{\nabla }f(P_0)$ is the direction on which that maximum occurs
• $-\Vert \vec{\nabla }f(P_0)\Vert$ is the min rate of change of $f$ at $P_0$
  • $-\vec{\nabla }f(P_0)$ is the direction on which that minimum occurs
• $\vec{\nabla }f$ is normal to the level surface (at any point on the surface)