MATH 20C Lecture 18

14.5 - The Gradient and Partial Derivatives

Directional Derivative

The directional derivative of at in the direction of the unit vector is

Theorem

If is differentiable at and is a unit vector then

(Recall )
This formula holds for functions with two or three variables
Pf:

Properties of the directional derivative

Suppose and are given. Then it depends on

Theorem

Assume . Let be a unit vector. Then

where is the angle between and

Properties of the directional derivative continued...

• is maximal when
  • In other words, and point in the same direction.
  • points in the direction of the fastest rate of increase of at , and this maximal rate is
    
• is minimal when .
  • In this case, and point in the opposite direction
  • points in the direction of the fastest rate of decrease and the maximal rate of decrease is
• if
  • is orthogonal to the level curve or surface of at .

Tangent plane revisited

Previously, we have an equation for the tangent plane of the graph of (explicit function)
In general, , with as a constant. We say is an implicit function of .

Explicit functions are also implicit functions:
The set is the level surface of in
Note is perpendicular to the tangent plane.

Theorem

Let on the level surface
Then is a normal vector to the tangent plane to the surface at . Moreover, the tangent plane has the equation:

Where