Friday, March 3, 2017

lecture 17: properties of the gradient vector

We'll finish the example problem from Wednesday and then look at the properties of the gradient vector (12.6). They are:
  1. The range of the directional derivative is $-| \, \vec{\nabla}f \,| \le D_{\hat{u}}f \le | \, \vec{\nabla}f \,|$.
  2.  $\vec{\nabla}f$ is the direction in which $f$ increases most rapidly, aka the direction of maximum increase.
  3. $-\vec{\nabla}f$ is the direction in which $f$ decreases most rapidly, aka the direction of maximum decrease.
  4. $\vec{\nabla}f$ is perpendicular to level curves of $f(x,y)$ in $\mathbb{R}^2$ or level surfaces of $f(x,y,z)$ in $\mathbb{R}^3$.
The fourth property gives us a spiffy way to create tangent planes to surfaces. And, tangent planes are a gateway to linear approximations.