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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
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.
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 ...
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 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 ...
Ahead, we’ve rounded up 50 holy grail hyperbole examples — some are as sweet as sugar, and some will make you laugh out loud. 50 common hyperbole examples I’m so hungry, I could eat a horse.
Great orthogonality theorem (group theory) Green–Tao theorem (number theory) Green's theorem (vector calculus) Grinberg's theorem (graph theory) Gromov's compactness theorem (Riemannian geometry) Gromov's compactness theorem (symplectic topology) Gromov's theorem on groups of polynomial growth (geometric group theory)