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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, 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 .
Green’s theorem is also a special case of Stokes’ formula. Stokes' formula also yields a general version of Cauchy's integral formula . To state and prove it, for the complex variable z = x + i y {\displaystyle z=x+iy} and the conjugate z ¯ {\displaystyle {\bar {z}}} , let us introduce the operators
Titchmarsh (1939) proves it in a straightforward way using Riemann approximating sums corresponding to subdivisions of a rectangle into smaller rectangles. To prove Clairaut's theorem, assume f is a differentiable function on an open set U, for which the mixed second partial derivatives f yx and f xy exist and are continuous.
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;
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.
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.
It uses only the arc length formula, expression for the area of a plane region from Green's theorem, and the Cauchy–Schwarz inequality. For a given closed curve, the isoperimetric quotient is defined as the ratio of its area and that of the circle having the same perimeter.