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  2. Green's theorem - Wikipedia

    en.wikipedia.org/wiki/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 ). In one dimension, it is equivalent to the fundamental theorem of calculus.

  3. Green's identities - Wikipedia

    en.wikipedia.org/wiki/Green's_identities

    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.

  4. Green's function - Wikipedia

    en.wikipedia.org/wiki/Green's_function

    A Green's function can also be thought of as a right inverse of L. Aside from the difficulties of finding a Green's function for a particular operator, the integral in equation 3 may be quite difficult to evaluate. However the method gives a theoretically exact result.

  5. Green formula - Wikipedia

    en.wikipedia.org/wiki/Green_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; the Green formula for the Green measure in stochastic analysis

  6. George Green (mathematician) - Wikipedia

    en.wikipedia.org/wiki/George_Green_(mathematician)

    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 ...

  7. Green's function (many-body theory) - Wikipedia

    en.wikipedia.org/wiki/Green's_function_(many-body...

    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 ...

  8. Generalized Stokes theorem - Wikipedia

    en.wikipedia.org/wiki/Generalized_Stokes_theorem

    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.

  9. Dirichlet problem - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_problem

    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,