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 $\displaystyle{\frac{\partial f}{\partial x}}$ represents the spatial rate of change of a scalar field along the $\displaystyle{x}$ 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 $\displaystyle{\delta x}$ represents a displacement from point 1 to point 2 along the $\displaystyle{x}$ direction and $\displaystyle{\delta y}$ represents a displacement from point 2 to point 3 along the $\displaystyle{y}$ 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 $\displaystyle{f}$ due to the displacement $\displaystyle{\mathbf{v}}$ is

\displaystyle{ \begin{aligned} \left(\delta f\right)_{\mathbf{v}} &= \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y \\ &= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}\right) \cdot (\delta x, \delta y) \\ &= \left(\nabla f\right) \cdot \mathbf{v} \\ \end{aligned}}

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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