Divergence — Physica

Equivalent forms

\nabla \cdot \mathbf{F} = \lim_{V\to 0} \frac{1}{V}\oint_{S} \mathbf{F}\cdot d\mathbf{A}

\operatorname{div}\mathbf{F} = \partial_i F_i

∇·F turns the global question 'how much escapes this surface?' into a local number you can evaluate point by point.

Symbol	Name	SI unit	Dimension	Range
$\nabla\cdot F$	Divergenceoutput Net outward flux per unit volume	1/s	T^-1	-5 – 5
$k$	Source strength Coefficient of the radial field F = k·r	1/s	T^-1	-3 – 3
$r$	Radius Distance from the origin	m	L	0 – 5

Unit systems

Symbol	SI	CGS	Imperial
$r$	m	cm	ft
$k$	1/s	1/s	1/s
$\nabla\cdot F$	1/s	1/s	1/s

Where it holds

Any differentiable vector field F on R^n.

Dimensional analysis

\nabla \cdot F \to [F]/[L]

\partial F_x/\partial x \to [F]/[L]

Divergence divides the field's units by length. For

F = k\cdot r

with

[k] = 1/s

,

\nabla \cdot F

has units 1/s.

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Discovery

Carl Friedrich Gauss / Josiah Willard Gibbs · 1881

Gauss used flux integrals in his 1813 study of gravitation; the local 'div' operator and the ∇· notation were standardized by Gibbs and Heaviside in the 1880s as electromagnetism demanded a compact language for sources of fields.

Try this

Is this point a faucet or a drain? The divergence reads the flow and tells you how much field is being created or swallowed right where you stand.

Tune a radial flow from sink to source and watch ∇·F flip sign as field lines either gush outward or pour inward.

Research status: stable

Real-world applications

Gauss's law for electricity, $\nabla \cdot E = \rho /\varepsilon _{0}$
Continuity equation / conservation of mass in fluids
Charge and source localization in field theory
Compressibility diagnostics in CFD solvers

Common misconceptions

Divergence is a vector — it is a scalar.
A field that spreads out always has nonzero divergence — a $1/r^{2}$ field has zero divergence away from its source.
Divergence measures how big the field is — it measures how fast the field spreads.

Experimental verification

Confirmed wherever Maxwell'

s \nabla \cdot E = \rho /\varepsilon _{0} and \nabla \cdot B = 0

are tested, and in fluid dynamics where

\nabla \cdot v = 0

encodes incompressibility.

Derivation

Compute the flux of F out of a tiny box of sides dx,dy,dz.

The x-faces contribute

[F_x(x+dx) - F_x(x)]dy dz \approx (\partial F_x/\partial x)dV

.

Summing all three pairs and dividing by dV gives

\partial F_x/\partial x + \partial F_y/\partial y + \partial F_z/\partial z

— the divergence.

Limiting cases

k > 0⟶ Source

(\nabla \cdot F

> 0)Field lines emanate outward; the point creates flux.

k < 0⟶ Sink

(\nabla \cdot F

< 0)Field lines converge; the point absorbs flux.

Solenoidal field⟶

\nabla \cdot F = 0

Incompressible / source-free flow, e.g. the magnetic field B.

What if…

What

if \nabla \cdot F = 0

everywhere?

F is solenoidal — it can be written as the curl of another field, $F = \nabla \times A$ .

What if the field is a point charge'

s 1/r^{2}

?

$\nabla \cdot F = 0$ away from the origin but a delta-function spike sits at the charge — all the flux comes from that point.

1

Radial source

Given ·

F:: (x, y, z)

Find ·

\nabla \cdot F

Steps

$\partial F_x/\partial x = 1$
$\partial F_y/\partial y = 1$
$\partial F_z/\partial z = 1$
$Sum = 3$ (a uniform 3D source)

Result ·

\nabla \cdot F = 1 + 1 + 1 = 3

2

Source-free swirl

Given ·

F:: $(-y$ , x, 0)

Find ·

\nabla \cdot F

Steps

$\partial (-y)/\partial x = 0$
$\partial (x)/\partial y = 0$
Pure rotation has no divergence

Result ·

\nabla \cdot F = 0

Gradient→Curl→Divergence (Gauss) Theorem→Laplacian→