Green's Theorem — Physica

Equivalent forms

\oint \mathbf{F}\cdot d\mathbf{r} = \iint (\nabla\times\mathbf{F})_z\,dA

\text{Area}(D) = \tfrac12\oint (x\,dy - y\,dx)

A boundary walk equals an area survey — the same trade that lets a planimeter measure a field's area by tracing its outline.

Symbol	Name	SI unit	Dimension	Range
$k$	Field strength Amplitude of the rotational field	1	1	-2 – 2
$R$	Loop radius Radius of the circular boundary	1	1	0.5 – 1.8
$\oint$	Circulationoutput Line integral around the boundary	1	1	-30 – 30
$\iint$	Curl integraloutput Area integral of the curl	1	1	-30 – 30

Unit systems

Symbol	SI	CGS	Imperial
$k$	1	1	1

Where it holds

Requires a positively-oriented, piecewise-smooth simple closed curve bounding a region where P, Q have continuous partials. Multiply-connected regions need boundary orientation care.

Dimensional analysis

\oint (Pdx+Qdy) = [F][

length]

∬

(\partial Q/\partial x-\partial P/\partial y)dA = [F]/[

length

]\times [

length

]^{2}

Both sides carry [field

]\times [

length]; the curl loses one length and the area integral restores two.

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Discovery

George Green · 1828

A self-taught miller's son, Green published his essay on electricity and magnetism privately in 1828; it was nearly lost until Lord Kelvin rediscovered it. The plane theorem is the 2D case of Stokes' theorem and the ancestor of the divergence theorem — cornerstones of electromagnetism.

Try this

Want to know how much a field swirls inside a whole region? Green's theorem says don't check every point — just walk the boundary once. The circulation around the edge equals the total swirl inside.

Send a marker around a closed loop while the enclosed area glows with the field's curl; the loop's total push matches the area's total spin.

Research status: stable

Real-world applications

Planimeters measuring area by boundary tracing
Computing 2D flux and circulation in fluid flow
Deriving Ampère's and Faraday's laws in integral form
Shoelace formula for polygon area in computer graphics

Common misconceptions

It needs a circle — any simple closed piecewise-smooth curve works.
Orientation doesn't matter — reversing the boundary flips the sign.
It's unrelated to Stokes/divergence — it's the 2D special case of both.

Experimental verification

Mechanical planimeters compute land area on maps by tracing the boundary — a direct physical realization of

the \tfrac{1}{2}\oint (xdy-ydx)

area formula, accurate to fractions of a percent.

Derivation

Tile D with tiny rectangles.

On each,

\oint (Pdx+Qdy) \approx (\partial Q/\partial x - \partial P/\partial y)dA

by expanding the four edge contributions.

Summing, interior edges are traversed twice in opposite directions and cancel, leaving the outer boundary integral equal to the summed area integral.

Limiting cases

curl

= 0

(conservative)⟶

\oint F\cdot dr = 0

No enclosed swirl — the field is a gradient; path-independent.

F = (-y

, x)/2⟶

\oint

= area of DThe area formula — a planimeter in equation form.

R \to 0

⟶ circulation → curl

\times

areaShrinking the loop recovers the pointwise definition of curl.

What if…

What if the region has a hole?

Add the inner boundary with opposite orientation; the theorem still holds for the multiply-connected region.

What if I lift it into 3D?

It becomes Stokes' theorem — circulation around a loop equals the flux of the curl through any surface it bounds.

1

Circulation of F = (-y, x)

Given ·

F:: (-y, x)
D:: disk radius R

Find ·

\oint F\cdot dr

Steps

$\partial Q/\partial x - \partial P/\partial y = 1-(-1) = 2$
∬ $2\,\mathrm{dA} = 2\cdot (\pi R^{2})$
$= 2\pi R^{2}$

Result ·

2\pi R^{2}

2

Area of a region by boundary

Given ·

formula:: $\tfrac{1}{2}\oint (x dy - y dx)$

Find · area of unit circle

Steps

Parametrize $x=\cos t$ , $y=\sin t$
$\tfrac{1}{2}\oint (\cos ^{2}t+\sin ^{2}t)dt = \tfrac{1}{2}\oint dt$
$= \tfrac{1}{2}\cdot 2\pi = \pi$

Result ·

\pi

Stokes' Theorem→Divergence (Gauss) Theorem→Curl→Fundamental Theorem of Calculus→