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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.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
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}} ).
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,
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.
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After a month-long break, she returned to her YouTube channel in August 2012. [25] Green won a 2016 Streamy Award for Science or Education. [26] In May 2017, Green had a series of dialogs on Twitter, in her own videos, and in the videos of other YouTubers, with critics of identity politics, gender identity, and modern feminism. She said that ...