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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 ).
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green , who discovered Green's theorem .
Green's work on the motion of waves in a canal (resulting in what is known as Green's law) anticipates the WKB approximation of quantum mechanics, while his research on light-waves and the properties of the Aether produced what is now known as the Cauchy-Green tensor. Green's theorem and functions were important tools in classical mechanics ...
Plugging this into the divergence theorem produces Green's theorem, = ^. Suppose that the linear differential operator L is the Laplacian , ∇ 2 , and that there is a Green's function G for the Laplacian.
Herein also his remarkable theorem in pure mathematics, since universally known as Green's theorem, and probably the most important instrument of investigation in the whole range of mathematical physics, made its appearance. We are all now able to understand, in a general way at least, the importance of Green's work, and the progress made since ...
In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely ...
Great orthogonality theorem (group theory) Green–Tao theorem (number theory) Green's theorem (vector calculus) Grinberg's theorem (graph theory) Gromov's compactness theorem (Riemannian geometry) Gromov's compactness theorem (symplectic topology) Gromov's theorem on groups of polynomial growth (geometric group theory)
One easy way to establish this theorem (in the case where =, =, and =, which readily entails the result in general) is by applying Green's theorem to the gradient of . An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case ...