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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 R 3 {\displaystyle \mathbb {R} ^{3}} ).

  3. Green's identities - Wikipedia

    en.wikipedia.org/wiki/Green's_identities

    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 .

  4. Green's function - Wikipedia

    en.wikipedia.org/wiki/Green's_function

    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.

  5. Symmetry of second derivatives - Wikipedia

    en.wikipedia.org/wiki/Symmetry_of_second_derivatives

    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 ...

  6. Green's function for the three-variable Laplace equation

    en.wikipedia.org/wiki/Green's_function_for_the...

    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 .

  7. 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 ...

  8. Musk v. Altman judge says it is a 'stretch' for Musk to claim ...

    www.aol.com/musk-v-altman-judge-says-232539064.html

    A California federal judge on Tuesday expressed doubt over Elon Musk's "irreparable harm" claims in his lawsuit against Sam Altman.

  9. Divergence theorem - Wikipedia

    en.wikipedia.org/wiki/Divergence_theorem

    A special case of this is =, in which case the theorem is the basis for Green's identities. With F → F × G {\displaystyle \mathbf {F} \rightarrow \mathbf {F} \times \mathbf {G} } for two vector fields F and G , where × {\displaystyle \times } denotes a cross product,