Divergence / Gradient / Laplace Operator

Yao Yao on June 11, 2018

注意这里反复使用了 $\nabla$,但要注意的是,$\nabla$ 并不是一个有统一定义的 operator,它只是一个符号而已,在不同的高阶 operator 定义中有不同的解读和记法。具体可以参见 Wikipedia: Del


Divergence

Quote from Wikipedia: Divergence:

Let $x$, $y$, $z$ be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field $F = U \mathbf{i} + V \mathbf{j} + W \mathbf{k}$ is defined as the scalar-valued function:

注意:

  1. 写法。$F = U \mathbf{i} + V \mathbf{j} + W \mathbf{k}$ 其实就是 $\vec F = \icol{U \newline V \newline W}$,它其实是一个 vector
  2. 这里 $\nabla \cdot \mathbf {F}$ 明显不是 dot product,但是计算方法类似,最后的结果是一个 scalar
  3. Gradient 的写法 $\nabla f$ 不带这个 dot

Divergence 的物理意义

Quote from Erik Anson’s answer on Quora:

Imagine a fluid, with the vector field representing the velocity of the fluid at each point in space. Divergence measures the net flow of fluid out of (i.e., diverging from) a given point. If fluid is instead flowing into that point, the divergence will be negative.

A point or region with positive divergence is often referred to as a “source” (of fluid, or whatever the field is describing), while a point or region with negative divergence is a “sink”.

Quote from Better Explained - Vector Calculus: Understanding Divergence:

The bigger the flux density (positive or negative), the stronger the flux source or sink. A div of zero means there’s no net flux change in side the region.

从这个角度来看,divergence 更像是 gradient of vector fields

Laplace Operator

注意我们把 laplace operator 写作 $\nabla^2$ 其实是有原因的。其实你对 gradient field 求 divergence,就有:

which happens to be the $lap$ of $f(x, y, z)$.

I.e. $\operatorname{lap} f = \operatorname{div}(\operatorname{grad} f)$

注意 $\nabla \cdot \nabla$ 仍然不是 dot product,而且我们把这个结果写成 $\nabla^2$ 也是人为规定的,完全是为了简便记法,所以并不什么特别的 operator 叠加法则。

考虑 $\nabla^2 f$ 的物理含义:

  • 如果你在 $f$ 的 local minimum $a$,你周围的 gradient 全部流出 $a$ (任意方向都是 ascending),divergence 是 highly positive,所以 $a$ 是 gradient 的 source
  • 如果你在 $f$ 的 local maximum $b$,你周围的 gradient 全部流入 $b$ (任意方向都是 descending),divergence 是 highly negative,所以 $b$ 是 gradient 的 sink


blog comments powered by Disqus