Distance vs Displacement, 2

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The physical reason of “the magnitude of the gradient vector represents the spatial rate of change” of a scalar field is that represents the spatial rate of change of a scalar field along the direction.

Directional derivative has exactly the same meaning except that its direction may not be along any one of the coordinate axes.

— Me@2016-02-06 07:23:32 AM

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Assume that represents a displacement from point 1 to point 2 along the direction and represents a displacement from point 2 to point 3 along the direction.

Denote “the value of the vector field” as “height”. Then

the height difference between point 3 and point 1

= the height difference between point 2 and point 1

+ the height difference between point 3 and point 2

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That is the exact reason that the change of the due to the displacement is

The “height difference” does not care about the cause or process that introduces that height change.

— Me@2016-04-21 11:16:06 PM

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2016.05.01 Sunday (c) All rights reserved by ACHK