Green's theorem

Green's theorem equates a positively oriented line integral around a simple closed plane curve with a double integral over the region it encloses. It connects circulation or flux along the boundary to derivatives throughout the region.

Circulation form of Green's theorem: The form equating a counterclockwise boundary integral of a vector field's tangential component with the region's integral of its scalar curl. It expresses boundary circulation as the accumulated curl inside the curve.
Line integral: An integral that accumulates a scalar or vector quantity along a parametrized curve. Green's theorem begins with a line integral taken around the region's boundary.
Isaac Newton: An English mathematician and natural philosopher who developed calculus and the laws of motion and gravitation. His calculus established ideas that helped make boundary-to-interior integral relations possible.
Area of a plane region: The two-dimensional measure of the points contained in a plane region. Choosing a suitable vector field makes Green's theorem compute enclosed area from a boundary integral.
Stokes' theorem: A theorem equating the integral of a differential form's exterior derivative over a surface with its integral over the surface boundary. It generalizes Green's theorem from plane regions to oriented surfaces in three-dimensional space.
Flux form of Green's theorem: The form equating outward flux across a closed plane curve with the region's integral of the vector field's divergence. It converts outward boundary flow into total divergence within the enclosed region.
Double integral: An integral that accumulates a function over a two-dimensional region. The theorem's boundary integral is replaced by a double integral over the enclosed region.
Gottfried Wilhelm Leibniz: A German philosopher and mathematician who independently developed calculus and introduced much of its standard notation. His integral calculus is part of the mathematical tradition behind Green's theorem.
Centroid: The geometric center of a region or mass distribution, defined by its first moments. Boundary integrals derived from Green's theorem can calculate a planar region's first moments.
Divergence theorem: A theorem equating outward flux through a closed surface with the volume integral of divergence inside it. It extends the flux relation in Green's theorem to three-dimensional regions.

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