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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.
Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George Green. Green's treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point: to this function Green gave the name ...
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.
Download as PDF; Printable version; In other projects ... In mathematics, Green formula may refer to: Green's theorem in integral calculus; Green's identities in ...
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 ...
is the derivative of the Green's function along the inward-pointing unit normal vector ^. The integration is performed on the boundary, with measure d s {\displaystyle ds} . The function ν ( s ) {\displaystyle \nu (s)} is given by the unique solution to the Fredholm integral equation of the second kind,
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 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.