Divergence — definition, physical meaning (flux density): Worked examples

This page is a drill. The Divergence — definition, physical meaning (flux density) parent note built the idea (flux per tiny volume) and the formula. Here we hit every kind of problem that can appear, so you never meet a case you haven't already seen.

Before we compute anything, one reminder in plain words. A vector field just means: at every point of space, an arrow. The arrow $F = (F_{1}, F_{2}, F_{3})$ has three numbers — how far east ( $F_{1}$ ), how far north ( $F_{2}$ ), how far up ( $F_{3}$ ) it points. Divergence asks: around this point, is arrow-stuff net leaving (a tap) or net arriving (a drain)? We measure it with

$div F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z} .$

The symbol $\frac{\partial F _{1}}{\partial x}$ (read "partial of $F_{1}$ with respect to $x$ ") means: how fast does the east-arrow $F_{1}$ grow as you step east, holding $y, z$ frozen. We only ever take the matching partial — $F_{1}$ with $x$ , $F_{2}$ with $y$ , $F_{3}$ with $z$ . That "diagonal" rule is the whole game.

The scenario matrix

Here is the full list of case-classes this topic can throw at you. Every cell below is covered by a worked example further down.

#	Case class	What's tricky	Example
A	Positive constant divergence (source)	reading a uniform "tap"	Ex 1
B	Negative constant divergence (sink)	sign of a "drain"	Ex 2
C	Zero divergence, big arrows (rotation)	size ≠ spreading	Ex 3
D	Sign-varying divergence (positive here, negative there)	one field, many behaviours	Ex 4
E	Degenerate: uniform field	biggest arrows, div $= 0$	Ex 5
F	Cross-term trap ( $\partial F_{2} / \partial x$ )	which partials count	Ex 6
G	Limiting check: shrink a real box, take $V \to 0$	definition = formula	Ex 7
H	Real-world word problem (fluid tap)	units, interpretation	Ex 8
I	Exam twist: find where $div = 0$	solve, not just compute	Ex 9
J	2D radial with a singularity at the origin	zero everywhere yet a source	Ex 10

Cell A — positive constant divergence (source)

Cell B — negative constant divergence (sink)

Cell C & E — zero divergence with big arrows

Figure — Divergence — definition, physical meaning (flux density)

Cell D — sign-varying divergence

Cell F — the cross-term trap

Worked example Example 6: partials that look relevant but aren't

$F = (3 y, 5 x, 7 z)$ . Compute $div F$ .

Forecast: there's a $y$ in the first slot and an $x$ in the second — tempting to differentiate them. Guess the answer, then check whether those go in.

$F_{1} = 3 y$ : the matching partial is $\partial (3 y) / \partial x = 0$ (no $x$ inside $3 y$ ). Why this step? Divergence uses only $\partial F_{1} / \partial x$ — not $\partial F_{1} / \partial y$ . The cross-partial $\partial (3 y) / \partial y = 3$ belongs to curl, not here.
$F_{2} = 5 x$ : matching partial $\partial (5 x) / \partial y = 0$ .
$F_{3} = 7 z$ : matching partial $\partial (7 z) / \partial z = 7$ .
Sum the diagonal: $0 + 0 + 7 = 7$ .

Meaning: despite lots of cross-coupling, the net spreading comes entirely from the $z$ -part.

Verify: only $F_{3}$ grows along its own axis; $div = 7$ . (If you'd wrongly used the cross-terms $3$ and $5$ , you'd get $15$ — that's a curl-flavoured mistake.) ✓

Cell G — the limiting definition, checked by hand

Worked example Example 7: shrink a real box and recover the formula

$F = (x, 0, 0)$ . The formula says $div = 1$ . Prove it by the geometric definition (flux per volume) on a cube of side $2 h$ centred at the origin.

Forecast: does actual flux ÷ volume give exactly $1$ , independent of $h$ ?

Only the two $x$ -faces matter. On the top/bottom/front/back faces the outward normal is $\pm \overset{y}{^}$ or $\pm \overset{z}{^}$ , and $F = (x, 0, 0)$ has zero $y, z$ parts, so $F \cdot \overset{n}{^} = 0$ there. Why this step? Flux is $F \cdot \overset{n}{^}$ ; a field with no $y, z$ components passes no flux through faces facing $y, z$ .
Right face $x = h$ : $F_{1} = h$ , outward normal $+ \overset{x}{^}$ , area $(2 h)^{2} = 4 h^{2}$ . Flux out $= h \cdot 4 h^{2} = 4 h^{3}$ .
Left face $x = - h$ : $F_{1} = - h$ , outward normal $- \overset{x}{^}$ , so $F \cdot \overset{n}{^} = - (- h) = + h$ . Flux out $= h \cdot 4 h^{2} = 4 h^{3}$ . Why this step? On the left, both the field value and the normal are negative — two minus signs make outflow positive. Fluid genuinely leaves both faces.
Total flux $= 4 h^{3} + 4 h^{3} = 8 h^{3}$ . Volume $= (2 h)^{3} = 8 h^{3}$ .
Flux per volume $= \frac{8 h ^{3}}{8 h ^{3}} = 1$ , for every $h$ . Take $h \to 0$ : still $1$ .

Meaning: the geometric definition and the formula agree exactly — see the balloon shrinking in the figure.

Verify: $8 h^{3} /8 h^{3} = 1 = \partial x / \partial x$ . Independent of $h$ , as the limit demands. ✓

Cell H — real-world word problem (with units)

Worked example Example 8: a leaky pipe field

Water velocity in a tank is $v = (0.4 x, 0.4 y, - 0.8 z)$ , measured in metres per second, position in metres. Is water being created or destroyed at the origin, and at what rate per unit volume?

Forecast: two positive spreads in $x, y$ and a squeeze in $z$ — will they cancel?

$\frac{\partial ( 0.4 x )}{\partial x} = 0.4 s^{- 1}$ . Why this step? Units: velocity (m/s) differentiated by length (m) gives per-second ( $s^{- 1}$ ) — a rate, exactly what "per unit volume per second" needs.
$\frac{\partial ( 0.4 y )}{\partial y} = 0.4 s^{- 1}$ ; $\frac{\partial ( - 0.8 z )}{\partial z} = - 0.8 s^{- 1}$ .
Sum: $div v = 0.4 + 0.4 - 0.8 = 0 s^{- 1}$ .

Meaning: divergence $0$ everywhere → the flow is incompressible; nothing is created or destroyed. Water spreading sideways is exactly compensated by squeezing downward. This is the Continuity equation with constant density: $\nabla \cdot v = 0$ .

Verify: $0.4 + 0.4 - 0.8 = 0$ ; units all $s^{- 1}$ , consistent. ✓

Cell I — exam twist: find where divergence vanishes

Cell J — 2D radial with a singularity

Worked example Example 10: the field that's source-free everywhere yet acts like a tap

In the plane, let $F = (\frac{x}{x ^{2} + y ^{2}}, \frac{y}{x ^{2} + y ^{2}})$ (arrows point out, but shrink with distance — see figure). Compute its divergence away from the origin, and explain the origin.

Forecast: the arrows point outward, so you'd guess a positive source. But they weaken outward — maybe that weakening exactly cancels the spreading?

Write $r^{2} = x^{2} + y^{2}$ , so $F_{1} = x / r^{2}$ . Use the quotient rule with $\partial r^{2} / \partial x = 2 x$ : $\frac{\partial}{\partial x} (\frac{x}{r ^{2}}) = \frac{1 \cdot r ^{2} - x \cdot 2 x}{r ^{4}} = \frac{r ^{2} - 2 x ^{2}}{r ^{4}} = \frac{y ^{2} - x ^{2}}{r ^{4}} .$ Why this step? $F_{1}$ is a fraction whose top and bottom both hold $x$ ; the quotient rule handles "how fast does a ratio change."
By symmetry ( $x \leftrightarrow y$ ): $\frac{\partial}{\partial y} (\frac{y}{r ^{2}}) = \frac{x ^{2} - y ^{2}}{r ^{4}}$ .
Add: $div F = \frac{y ^{2} - x ^{2}}{r ^{4}} + \frac{x ^{2} - y ^{2}}{r ^{4}} = 0$ (for $r \neq = 0$ ).

Meaning: away from the origin the outward spreading is exactly undone by the arrows shrinking — divergence $0$ everywhere except one point. But total flux through any circle around the origin is a fixed positive number: all the "source" is concentrated at the singular origin, where the formula breaks down (division by zero). This is the flavour behind point sources and the Divergence Theorem (Gauss).

Verify: the two fractions are exact negatives, summing to $0$ for all $r \neq = 0$ ; check at $(1, 0)$ : $(0 - 1) /1 + (1 - 0) /1 = 0$ ✓. At $(1, 2)$ : $(4 - 1) /25 + (1 - 4) /25 = 3/25 - 3/25 = 0$ ✓.

Recall Which cell was hardest for you?

Zero-divergence-with-big-arrows (C/E) and the singular source (J) trip up the most people. Both say the same thing: divergence measures net leaving, not arrow size, and not appearance.

Recall

Field with big arrows can still have zero divergence — true or false?

True; e.g.

(100, 0, 0)

or the swirl

(- y, x, 0)

, both div

= 0

div (2 x, 2 y, 2 z) = ?

2 + 2 + 2 = 6

On what surface does

(x^{2}, y^{2}, z^{2})

have zero divergence?

The plane

x + y + z = 0

Why does

(x / r^{2}, y / r^{2})

have zero divergence away from the origin?

Outward spreading is exactly cancelled by the arrows shrinking as

1/ r

; the source is concentrated at the singular origin.

Which partials enter divergence for

F = (3 y, 5 x, 7 z)

Only the diagonal ones

\partial (3 y) / \partial x, \partial (5 x) / \partial y, \partial (7 z) / \partial z = 0 + 0 + 7 = 7

Connections

Divergence — definition, physical meaning (flux density) (parent — the idea and derivation)
Curl — rotation density (where the cross-terms of Ex 3 & 6 actually live)
Divergence Theorem (Gauss) (turns Ex 7's local flux/volume into global flux; explains Ex 10's origin)
Continuity equation (Ex 8: $\nabla \cdot v = 0$ is incompressibility)
Flux through a surface (the surface integral computed by hand in Ex 7)
Gradient and the del operator (where $\nabla$ comes from)
Laplacian ( $\nabla^{2} f = div (grad f)$ )