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In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area.
Green's functions may be categorized, by the type of boundary conditions satisfied, by a Green's function number. Also, Green's functions in general are distributions, not necessarily functions of a real variable. Green's functions are also useful tools in solving wave equations and diffusion equations.
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 ...
The divergence theorem is an important result for the mathematics of physics and engineering, ... it is equivalent to Green's theorem ... This equation is also known ...
By Cauchy's theorem, the left-hand integral is zero when () is analytic (satisfying the Cauchy–Riemann equations) for any smooth closed curve L. Correspondingly, by Green's theorem, the right-hand integrals are zero when = ¯ is irrotational (curl-free) and incompressible (divergence-free).
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.
There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density. Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. [ 1 ]
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...