Green's theorem

Etymology

Definitions

1. 1

   A generalization of the fundamental theorem of calculus to the two-dimensional plane,…

   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.
2. 2

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

   • ∬_R∇∧⃑Gdx,dy=∮_(∂R)⃑G·d⃑l
   • 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.

The neighborhood