Assume represents the position of an object and is a scalar field on the – plane. Then represents the change of per unit length along the positive direction. In other words, it is the spatial rate of change of along the direction.

Similarly, derivative represents the spatial rate of change of along the direction.

For an arbitrary direction, due to the nature of displacement, the change of is when the object has finished moving in direction and then in direction.

Then, the spatial rate of change of is

.

For simplicity, denote the resultant displacement as :

and define as

Then, the change of the due to the displacement is

.

So the spatial rate of change along the direction of the vector is

is called ** directional derivative**.

— Me@2016-02-06 09:49:22 PM

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This is the reason that is in the steepest direction.

If is chosen to be parallel to , the directional derivative would be maximized.

— Me@2021-08-20 05:20:02 PM

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2016.02.21 Sunday (c) All rights reserved by ACHK