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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 one dimension, it is equivalent to the fundamental theorem of calculus.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. The Green's function as used in physics is usually defined with the opposite sign, instead.
By Green's theorem, the area of a region enclosed by a smooth, closed, positively oriented curve is given by the integral ¯. This fact is used, for example, in the proof of the area theorem . Quantum mechanics
In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.
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 ...
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)
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 .
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 ...