Curl — Physica

Equivalent forms

(\nabla\times\mathbf{F})_i = \epsilon_{ijk}\,\partial_j F_k

\hat{\mathbf{n}}\cdot(\nabla\times\mathbf{F}) = \lim_{A\to 0}\frac{1}{A}\oint_C \mathbf{F}\cdot d\mathbf{r}

The whole content of Faraday's and Ampère's laws — how changing fields wind around each other — lives in a single cross product with ∇.

Symbol	Name	SI unit	Dimension	Range
$(\nabla\times F)_z$	Curl (z-component)output Circulation per unit area about the z-axis	1/s	T^-1	-6 – 6
$\omega$	Swirl rate Coefficient of the rotational part F = ω(−y, x)	1/s	T^-1	-3 – 3
$a$	Strain rate Coefficient of the irrotational stretch part F = a(x, −y)	1/s	T^-1	0 – 3

Unit systems

Symbol	SI	CGS	Imperial
$\omega$	1/s	1/s	1/s
$a$	1/s	1/s	1/s
$(\nabla\times F)_z$	1/s	1/s	1/s

Where it holds

Any differentiable vector field F on R^3 (the z-component formula holds in 2D).

Dimensional analysis

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

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

Same units as divergence: field units per length. For

F = \omega \cdot r

-type fields with

[\omega ] = 1/s

, curl has units 1/s.

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Discovery

James Clerk Maxwell / Josiah Willard Gibbs · 1873

Maxwell wrote of the 'curl' (he briefly considered 'rotation' and 'twirl') to describe how electric and magnetic fields wind around their sources. The compact ∇× notation came with Gibbs's vector analysis a decade later.

Try this

Float a tiny paddle wheel in a stream. If it spins, the field has curl — and the curl vector points along the axle.

Mix a swirling field with a stretching one and watch a test paddle wheel spin up or freeze as ∇×F changes.

Research status: stable

Real-world applications

Electromagnetic induction (Faraday's law)
Vorticity in fluid dynamics and aerodynamics
Detecting conservative vs non-conservative force fields
Curl-based feature detection in vector-field visualization

Common misconceptions

Curl measures whether field lines are curved — a field can be curved yet curl-free (e.g. 1/r azimuthal field outside the axis).
Curl is a scalar — in 3D it is a vector (pseudovector).
Diverging fields can't have curl — divergence and curl are independent.

Experimental verification

Embodied in Faraday'

s law \nabla \times E = -\partial B/\partial t

(transformers, generators) and Ampère–Maxwell

\nabla \times B = \mu _{0}J + \mu _{0}\varepsilon _{0}\partial E/\partial t

, both verified to extraordinary precision.

Derivation

Take the circulation

\oint F\cdot dr

around a small rectangle in the xy-plane.

The two x-edges contribute

[F_x(y) - F_x(y+dy)]dx \approx -(\partial F_x/\partial y)dA

and the y-edges

(\partial F_y/\partial x)dA

.

Dividing by the area dA gives

(\partial F_y/\partial x - \partial F_x/\partial y)

, the z-component of the curl.

The full vector follows by symmetry.

Limiting cases

Pure rotation

F = \omega (-y

, x)⟶

(\nabla \times F)_z = 2\omega

Rigid swirl gives uniform curl everywhere.

Conservative field

F = \nabla \varphi

⟶

\nabla \times F = 0

Gradients are always curl-free

(\nabla \times \nabla \varphi = 0)

.

Pure strain

F = a(x

,

-y)

⟶

\nabla \times F = 0

Stretching without rotation has no curl, even though field lines diverge.

What if…

What

if \nabla \times F = 0

in a simply-connected region?

F is conservative there — it has a scalar potential, $F = \nabla \varphi$ , $and \oint F\cdot dr = 0$ on every loop.

What if I'm in 2D?

Curl collapses to the single scalar $\partial F_y/\partial x - \partial F_x/\partial y$ , which Green's theorem integrates over an area.

1

Solid-body swirl

Given ·

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

Find ·

\nabla \times F

Steps

$(\nabla \times F)_z = \partial (x)/\partial x - \partial (-y)/\partial y = 1 - (-1) = 2$
x and y components vanish
Uniform curl 2 ẑ — the field rotates rigidly

Result ·

\nabla \times F = (0

, 0, 2)

2

A gradient has no curl

Given ·

F:: $\nabla (x^{2} + y^{2}) = (2x$ , 2y, 0)

Find ·

\nabla \times F

Steps

$(\nabla \times F)_z = \partial (2y)/\partial x - \partial (2x)/\partial y = 0 - 0 = 0$
Confirms $\nabla \times \nabla \varphi = 0$ for any potential $\varphi$

Result ·

\nabla \times F = 0

Gradient→Divergence→Stokes' Theorem→Green's Theorem→