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Levi-Civita symbol. In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and ...
Not to be confused with the Dirac delta function, nor with the Kronecker symbol. In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, because , whereas ...
Commutation matrix. In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec (A) into vec (AT): K(m,n ...
Triple product. In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar -valued scalar triple product and, less often, the vector -valued vector triple product.
where is the Kronecker delta or identity matrix. Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature (,).
The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions: if , otherwise . unless , one of is zero and the other one is negative. unless , one of is zero and the other one has odd part (definition below) congruent to . For , we have whenever If additionally have the same sign, the same also holds for .
The other two Kronecker delta's state that the row and column indices must be equal (= ′ and = ′) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem. Every group has an identity representation (all group elements mapped to 1).
The derivation of the Navier–Stokes equations as well as their application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such ...