Directional Derivatives
We did "North" and "East". What about "North-East"?
Introduction
The directional derivative tells us the rate of change of in the direction of a unit vector .
Formula:
Note: MUST be a unit vector!
The Gradient Vector
We define the gradient vector ("del f") as:
Using the dot product, the directional derivative becomes:
Max Rate
Occurs when (direction of ).
Value: .
Min Rate
Occurs when (opposite to ).
Value: .
Zero Rate
Occurs when (orthogonal to gradient).
Change is zero (Contour Line).
Visualizing Steepest Ascent
Interactive: Gradient on Surface
Worked Examples
Example 1: Finding Directional Derivative
Find for at in direction of .
1. Find Gradient:
.
.
.
2. Find Unit Vector:
.
.
3. Dot Product:
.
Example 2: Max/Min Rate of Change
Find the maximum rate of change of at and its direction.
1. Find Gradient:
.
.
.
2. Analyze:
- Max Rate = .
- Direction = .
Example 3: Tangent to Level Curve
Find a vector tangent to the level curve of at .
The gradient is normal (perpendicular) to the level curve.
So any vector perpendicular to is tangent.
1. .
2. Find such that .
.
Let .
Tangent Vector: (or any scalar multiple).
Practice Quiz
Loading...