Gradient 1.2

Distance vs Displacement, 2

.

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

.

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

.

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

.

.

2016.05.01 Sunday (c) All rights reserved by ACHK