Directional Derivative
The directional derivative of in the direction of the unit vector is , where is the angle between and : the rate of change of in the direction of .
Definition 9.1 (Directional Derivative)
The directional derivative of in the direction of the unit vector is defined as
where is the angle between and .
This is the rate of change of in the direction of .
Properties of the gradient vector
Proposition 9.2 (Properties Of Gradient Vector)
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The direction of is the direction of maximum increase of .
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The magnitude of is the maximum rate of change of , i.e.
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If is parallel to contours of , then
Therefore, is perpendicular to the contours of .
Geometric characterisation
Conversely, the gradient vector can be defined geometrically the other way round, as the vector such that
Knowing the rate of change of in every unit direction therefore determines itself.
Related
Stated in
- Definition 9.1 (Directional Derivative)§9.1 Directional derivatives
- Proposition 9.2 (Properties Of Gradient Vector)§9.1 Directional derivatives