14.6 Videos Guide

14.6a

• The directional derivative of $f(x,\ y)$ in the direction of $\mathbf{u}=\left\langle a,\ b\right\rangle$, a unit vector:
  $D_{\mathbf{u}}f(x,\ y)=f_{x}(x,\ y)a+f_{y}(x,\ y)b$

• In $\mathbb{R}^{3}$, for $f(x,\ y,\ z)$ and $\mathbf{u}=\left\langle a,\ b,\ c\right\rangle$, a unit vector:
  $D_{\mathbf{u}}f(x,\ y,\ z)=f_{x}(x,\ y,\ z)a+f_{y}(x,\ y,\ z)b+f_{z}(x,\ y,\ z)c$

14.6b

• The gradient vector
  $\mathbf{\nabla}f=\left\langle f_{x},\ f_{y}\right\rangle$ or $\mathbf{\nabla}f=\left\langle f_{x},\ f_{y},f_{z}\right\rangle$

• The del operator $\mathbf{\nabla}$

• Dot product representation of the directional derivative
  $D_{\mathbf{u}}f=\mathbf{\nabla}f\cdot\mathbf{u}$

Exercises:

• Find the directional derivative of $f(x,\ y)=xy^{3}-x^{2}$ at the point $(1,\ 2)$ in the direction $\theta=\frac{\pi}{3}$.

14.6c

• Find the directional derivative of the function $f(x,\ y,\ z)=xy^{2}\tan^{-1}z$ at the point $(2,\ 1,\ 1)$ in the direction $\mathbf{v}=\left\langle 1,\ 1,\ 1\right\rangle$.

• Characteristics of the gradient vector

  • $\mathbf{\nabla}f(a,\ b)$ points in the direction of maximum change of $f$

  • The maximum rate of change of $f$ at $(a,\ b)$ is $\left|\mathbf{\nabla}f(a,\ b)\right|$

14.6d

• Tangent plane and normal line to a level surface $S:F(x,\ y,\ z)=k$ at the point $(x_{0},\ y_{0},z_{0})$

  • Tangent plane:
    $F_{x}\left(x_{0},y_{0},\ z_{0}\right)\left(x-x_{0}\right)+F_{y}\left(x_{0},\ y_{0},\ z_{0}\right)\left(y-y_{0}\right)+F_{z}\left(x_{0},y_{0},\ z_{0}\right)\left(z-z_{0}\right)=0$

  • Normal line: $\frac{x-x_{0}}{F_{x}\left(x_{0},y_{0},\ z_{0}\right)}=\frac{y-y_{0}}{F_{y}\left(x_{0},y_{0},\ z_{0}\right)}=\frac{z-z_{0}}{F_{z}\left(x_{0},y_{0},\ z_{0}\right)}$

Exercise:

• Find equations of (a) the tangent plane and (b) the normal line to the surface
  $x=y^{2}+z^{2}+1$ at the point $(3,\ 1,\ -1)$.