Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in
) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in
Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then
[1][2] In physics, Green's theorem finds many applications.
One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area.
In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
The following is a proof of half of the theorem for the simplified area D, a type I region where C1 and C3 are curves connected by vertical lines (possibly of zero length).
A similar proof exists for the other half of the theorem when D is a type II region where C2 and C4 are curves connected by horizontal lines (again, possibly of zero length).
Green's theorem then follows for regions of type III.
A similar treatment yields (2) for regions of type II.
be a rectifiable, positively oriented Jordan curve in
is a rectifiable, positively oriented Jordan curve in the plane and let
denote the collection of squares in the plane bounded by the lines
into a finite number of non-overlapping subregions in such a manner that Lemma 2 — Let
The outer Jordan content of this set satisfies
is a positively oriented square, for which Green's formula holds.
Every point of a border region is at a distance no greater than
The remark in the beginning of this proof implies that the oscillations of
The hypothesis of the last theorem are not the only ones under which Green's formula is true.
Now, analyzing the sums used to define the complex contour integral in question, it is easy to realize that
These remarks allow us to apply Green's Theorem to each one of these line integrals, finishing the proof.
is the outward-pointing unit normal vector on the boundary.
Green's theorem can be used to compute area by line integral.
It is named after George Green, who stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.
In 1846, Augustin-Louis Cauchy published a paper stating Green's theorem as the penultimate sentence.
This is in fact the first printed version of Green's theorem in the form appearing in modern textbooks.
George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: T. Wheelhouse, 1828).
Green did not actually derive the form of "Green's theorem" which appears in this article; rather, he derived a form of the "divergence theorem", which appears on pages 10–12 of his Essay.
In 1846, the form of "Green's theorem" which appears in this article was first published, without proof, in an article by Augustin Cauchy: A. Cauchy (1846) "Sur les intégrales qui s'étendent à tous les points d'une courbe fermée" (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251–255.
(The equation appears at the bottom of page 254, where (S) denotes the line integral of a function k along the curve s that encloses the area S.) A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his inaugural dissertation: Bernhard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867); see pages 8–9.
