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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.
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Template documentation GreenBook3rd is for referencing the IUPAC recommendations in Quantities, Units and Symbols in Physical Chemistry (3rd edition, 2008), also known as the "Green Book". Parameters are:
The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L u,v: V → V, defined by L u,v (w) = [u, v, w], is a derivation of the triple product.
The Kaya identity was developed by Japanese energy economist Yoichi Kaya. [1] It is the subject of his book Environment, Energy, and Economy: strategies for sustainability co-authored with Keiichi Yokobori as the output of the Conference on Global Environment, Energy, and Economic Development (1993 : Tokyo, Japan).
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In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. [1] [2] It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens ...
An example is the High Commissioner on National Minorities of the Organization for Security and Co-operation in Europe. The Directorate-General for the Environment of the European Commission has as its mission "protecting, preserving and improving the environment for present and future generations, and promoting sustainable development".