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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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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
orientationConvention	counterclockwise orientation is positive ⓘ
relatedTo	Cauchy integral theorem (by analogy) NERFINISHED ⓘ divergence theorem NERFINISHED ⓘ 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 NERFINISHED ⓘ
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 ⓘ

Referenced by (2)

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 ⓘ