Vector Calculus: Understanding Divergence

Physical Intuition

Divergence (div) is “flux density”—the amount offluxentering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is asourceof bananas. You’ve hit the Donkey Kong jackpot.

记住，按照惯例，当它离开一个封闭曲面时，通量是正的。Imagine you were your normal self, and could talk to points inside a vector field, asking what they saw:

If the point saw fluxentering, he’d scream that everything was closing in on him. This is anegativedivergence, and the point is capturing flux, like water going down a sink.
If the point saw fluxleaving, he’d sniff his armpits and say all flux was existing. This is apositivedivergence, and the point is a source of flux, like a hose.

So, divergence is just the net flux per unit volume, or “flux density”, just like regular density is mass per unit volume (of course, we don’t know about “negative” density). Imagine a tiny cube—flux can be coming in on some sides, leaving on others, and we combine all effects to figure out if the total flux is entering or leaving.

通量密度越大(正通量或负通量)，通量源或汇就越强。div的值为零，表示该区域一侧没有净通量变化。In plain english:

$\displaystyle{\text{ Divergence } = \frac{\text{Flux}}{\text{Volume}}}$

Math Intuition

Now that we have an intuitive explanation, how do we turn that sucker into an equation? The usual calculus way: take a tiny unit of volume and measure the flux going through it. We need to add up the total flux passing through the x, y and z dimensions.

Imagine a cube at the point we want to measure, with sides of length dx, dy and dz. To get the net flux, we see how much the X component of flux changes in the X direction, add that to the Y component’s change in the Y direction, and the Z component’s change in the Z direction. If there are no changes, then we’ll get 0 + 0 + 0, which means no net flux.

If thereissome change in the field, we get something like 1 -2 +5 (flux increases in X and Z direction, decreases in Y) which gives us the divergence at that point.

In pseudo-math:

Total flux change = (field change in X direction) + (field change in Y direction) + (field change in Z direction)

Or in more formal math:

$\displaystyle{\text{Divergence} = \lim_{\text{Vol} \to 0}\frac{\text{Flux}}{\text{Vol}}}$

$\displaystyle{\text{Divergence} = \frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}}$

(Assuming $F_x$ is the field in the x-direction.)

A few remarks:

The symbol for divergence is the upside down triangle forgradient(调用del)带一个点[$\triangledown \cdot$]。The gradient gives us the partial derivatives $(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$, and thedot productwith our vector $(F_x, F_y, F_z)$ gives the divergence formula above.
Divergence is a single number, like density.
散度和通量密切相关-如果一个体积包含一个正散度(通量的来源)，它将有正通量。
"Diverge" means to move away from, which may help you remember that divergence is the rate of flux expansion (positive div) or contraction (negative div).

一旦你对通量有了直观的理解，散度就不是太糟糕了。它对理解高斯定律等定理非常有用。

Physical Intuition

Math Intuition

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