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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.
In mathematics, a Green's function (sometimes improperly termed a Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if is a linear differential operator, then
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 .
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 ...
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.
The title page to Green's original essay on what is now known as Green's theorem. In 1828, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, which is the essay he is most famous for today. It was published privately at the author's expense, because he thought it would be ...
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 ...
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) [9] named after Alexis Clairaut and Hermann Schwarz, states that for a function : defined on a set , if is a point such that some neighborhood of is contained in and has continuous second partial derivatives on that neighborhood of , then for all i ...