Green's theorem

E620783

result in vector calculus theorem two-dimensional case of Stokes' theorem

Green's theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve in the plane to a double integral over the region it encloses.

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All labels observed (3)

Label	Occurrences
Green's theorem canonical	9
Green’s theorem	1
line integral theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T6788315 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Green's theorem
Context triple: [Stokes' theorem, generalizes, Green's theorem]

A. Stokes' theorem
Stokes' theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the surface’s boundary.
B. Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
C. Fubini's theorem
Fubini's theorem is a fundamental result in measure theory that allows the evaluation of double integrals as iterated integrals under suitable integrability conditions.
D. Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
E. Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Green's theorem
Target entity description: Green's theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve in the plane to a double integral over the region it encloses.

A. Stokes' theorem
Stokes' theorem is a fundamental result in vector calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the surface’s boundary.
B. Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
C. Fubini's theorem
Fubini's theorem is a fundamental result in measure theory that allows the evaluation of double integrals as iterated integrals under suitable integrability conditions.
D. Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
E. Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	result in vector calculus ⓘ theorem ⓘ two-dimensional case of Stokes' theorem ⓘ
appliesTo	region enclosed by a simple closed curve ⓘ simple closed curve in the plane ⓘ
assumes	L and M have continuous partial derivatives on an open region containing D ⓘ
category	integral theorem of vector calculus ⓘ line integral theorem ⓘ linked to: Green's theorem
connects	boundary of a region ⓘ interior of a region ⓘ
dimension	2D ⓘ
expresses	equivalence between circulation integral and area integral ⓘ
field	mathematical analysis ⓘ multivariable calculus ⓘ vector calculus ⓘ
hasFormulation	∮_C (L dx + M dy) = ∬_D (∂M/∂x − ∂L/∂y) dA ⓘ
hasVariant	circulation form of Green's theorem ⓘ flux form of Green's theorem ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	conservative vector fields have path-independent line integrals in simply connected planar regions ⓘ
namedAfter	George Green ⓘ
orientationConvention	counterclockwise orientation is positive ⓘ
relatedTo	Cauchy integral theorem (by analogy) ⓘ linked to: Cauchy integral theorem divergence theorem ⓘ linked to: Gauss’s law fundamental theorem of calculus ⓘ
relates	double integral ⓘ line integral ⓘ
relatesConcept	circulation ⓘ curl ⓘ divergence in the plane ⓘ flux ⓘ
requiresCondition	piecewise smooth boundary curve ⓘ positively oriented boundary curve ⓘ region must be simply connected (for standard form) ⓘ
specialCaseOf	Stokes' theorem ⓘ
topicIn	introductory vector analysis courses ⓘ undergraduate calculus courses ⓘ
typeOfIntegral	double integral over a planar region ⓘ line integral around a closed curve ⓘ
usedIn	complex analysis via relation to Cauchy’s theorem ⓘ computing area via line integrals ⓘ electromagnetism ⓘ planar fluid flow analysis ⓘ potential theory ⓘ proofs of planar versions of the divergence theorem ⓘ
usedToDerive	area formulas using line integrals ⓘ

How these facts were elicited

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

Stokes' theorem → generalizes → Green's theorem ⓘ

Cauchy integral theorem → relatedTo → Green's theorem ⓘ

George Green → knownFor → Green's theorem ⓘ

George Green → notableConcept → Green's theorem ⓘ

Vector Analysis (with Edwin Bidwell Wilson) → subject → Green's theorem ⓘ

Rayleigh–Sommerfeld diffraction theory → derivedFrom → Green’s theorem ⓘ

linked to: Green's theorem

Bendixson–Dulac criterion → proofTechnique → Green's theorem ⓘ

Green's theorem → category → line integral theorem ⓘ

linked to: Green's theorem

Advanced Calculus → coversTopic → Green's theorem ⓘ

Schaum's Outline of Advanced Calculus → topic → Green's theorem ⓘ

Schaum's Outline of Vector Analysis → topic → Green's theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (46)

How these facts were elicited Show

Referenced by (11)

How this entity was disambiguated

How these facts were elicited