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Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of ...
If is a multi-index, and a is a positive real number, then | | (). Any smooth function f with compact support is in 𝒮(R n).This is clear since any derivative of f is continuous and supported in the support of f, so has a maximum in R n by the extreme value theorem.
By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa: B γ β ⋯ = g γ α A α β ⋯ and A α β ⋯ = g α γ B γ β ⋯ {\displaystyle B^{\gamma }{}_{\beta \cdots }=g^{\gamma \alpha }A_{\alpha \beta \cdots }\quad {\text{and}}\quad A_{\alpha ...
Equivalence class: given an equivalence relation, [] often denotes the equivalence class of the element x. 3. Integral part : if x is a real number , [ x ] {\displaystyle [x]} often denotes the integral part or truncation of x , that is, the integer obtained by removing all digits after the decimal mark .
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...