Divergence / Gradient / Laplace Operator

注意这里反复使用了 $\mathrm{\nabla }$，但要注意的是，$\mathrm{\nabla }$ 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

DivergencePermalink

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 $\overrightarrow{F}=[\begin{array}{c}U\\ V\\ W\end{array}]$，它其实是一个 vector
2. 这里 $\mathrm{\nabla }\cdot \mathbf{F}$ 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 $\mathrm{\nabla }f$ 不带这个 dot

Divergence 的物理意义Permalink

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 OperatorPermalink

注意我们把 laplace operator 写作 ${\mathrm{\nabla }}^2$ 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. $\mathrm{lap}f=\mathrm{div}(\mathrm{grad}f)$

注意 $\mathrm{\nabla }\cdot \mathrm{\nabla }$ 仍然不是 dot product，而且我们把这个结果写成 ${\mathrm{\nabla }}^2$ 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 ${\mathrm{\nabla }}^{2}f$ 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