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The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
The metric tensor (,) induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ♭ and sharp ♯.A section () corresponds to the unique one-form ♭ such that for all sections (), we have:
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.
MATLAB was first released as a commercial product in 1984 at the Automatic Control ... Versions of the MATLAB product family; Name of release ... 7.3 6.5 R2007a 7.4 6 ...
In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
MATLAB, a technical computing application (MATrix LABoratory) GNU Octave, a high-level language, primarily intended for numerical computations. Scilab, an open source application for numerical computation. Maple, a proprietary application for symbolic calculations. Mathematica, a proprietary application primarily intended for symbolic calculations.
As originally formulated, the Lawson criterion gives a minimum required value for the product of the plasma (electron) density n e and the "energy confinement time" that leads to net energy output. Later analysis suggested that a more useful figure of merit is the triple product of density, confinement time, and plasma temperature T .
The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to φ(h) = id N for all h in H. Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles.