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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.
A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space , at a point s, is any solution of. (1) where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form.
In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory.
Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the -plane. We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector -valued function . Start with the left side of Green's theorem:
Advanced. Specialized. Miscellanea. v. t. e. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface ...
In Einstein notation, the vector field has curl given by: where = ±1 or 0 is the Levi-Civita parity symbol. For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then ...
For elements a and b of S, Green's relations L, R and J are defined by . a L b if and only if S 1 a = S 1 b.; a R b if and only if a S 1 = b S 1.; a J b if and only if S 1 a S 1 = S 1 b S 1.; That is, a and b are L-related if they generate the same left ideal; R-related if they generate the same right ideal; and J-related if they generate the same two-sided ideal.
List of logarithmic identities. MacWilliams identity. Matrix determinant lemma. Newton's identity. Parseval's identity. Pfister's sixteen-square identity. Sherman–Morrison formula. Sophie Germain identity. Sun's curious identity.