Green's theorem en · PROPN
Pronunciation
- (US) audio
Etymology
Named after the mathematician George Green.
Meanings
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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(∂Q/∂x-∂P/∂y)dx,dy=∮_(∂R)P,dx+Q,dy.
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Letting ⃑G=(P,Q) be a vector field, and d⃑l=(dx,dy) this can be restated as
where ∧ is the wedge product, or equivalently, as ∬_R∇·⃑Gdx,dy=∮_(∂R)⃑G∧d⃑l,
∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l
with the earlier formula resembling Stokes' theorem, and the latter resembling the divergence theorem.
Translations (7)
fr théorème de Green (theorem) · pt teorema de Green (theorem) · gl teorema de Green (theorem) · es teorema de Green (theorem) · ast teorema de Green (theorem) · it teorema di Green (theorem) · ca teorema de Green (theorem)