Green's theorem

proper noun

A generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

“∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l”

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Meanings

proper noun
  1. calculusA generalization of the fundamental theorem of calculus to the two-dimensional plane, which states that given two scalar fields P and Q and a simply connected region R, the area integral of derivatives of the fields equals the line integral of the fields, or

    ∬_R(∂Q/∂x-∂P/∂y)dx,dy=∮_(∂R)P,dx+Q,dy.

  2. calculusLetting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as

    ∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l

    2 more examples

    where ∧ is the wedge product, or equivalently, as ∬_R∇·⃑Gdx,dy=∮_(∂R)⃑G∧d⃑l,

    with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.

Origin

Named after the mathematician George Green.

Full word history & related roots →

Translations

Catalan teorema de Green
  • teorema de Greentheorem
French théorème de Green
  • théorème de Greentheorem
Italian teorema di Green
  • teorema di Greentheorem
Portuguese teorema de Green
  • teorema de Greentheorem
Spanish teorema de Green
  • teorema de Greentheorem
Cite this entry
APA

Green's theorem. (n.d.). In WatkinsX Dictionary. Retrieved November 4, 2026, from https://dictionary.watkinsx.com/word/Green's%20theorem

MLA

“Green's theorem.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/Green's%20theorem. Accessed November 4, 2026.

Chicago

WatkinsX Dictionary, s.v. “Green's theorem,” accessed November 4, 2026, https://dictionary.watkinsx.com/word/Green's%20theorem.

Source: Green's theorem on Wiktionary, CC BY-SA 4.0.