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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.
The problem now lies in finding the Green's function G that satisfies equation 1. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator L. Not every operator admits a Green's function. A Green's function can also be thought of as a right inverse of L.
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
The theorem of de Rham shows that this map is actually an isomorphism, a far-reaching generalization of the Poincaré lemma. As suggested by the generalized Stokes' theorem, the exterior derivative is the "dual" of the boundary map on singular simplices.
There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician George Green and are generalizations of the classical Green's function and Green formula to the stochastic case using Dynkin's formula .
In mathematics, Green formula may refer to: Green's theorem in integral calculus; Green's identities in vector calculus; Green's function in differential equations;
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.
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements.