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Chapter 23: Differential Forms and Matrix-Valued Differential Forms

https://doi.org/10.1142/9789813275386_0023Cited by:0 (Source: Crossref)

Abstract:

We define differential p-forms of class C^∞ on an open set Ω of ℝⁿ to be the expressions

$ω : = \sum_{j_{1} < j_{2} < \dots < j_{p}}^{n} c_{j_{1} j_{2} \dots j_{p}} (X) d x_{j_{1}} \land d x_{j_{2}} \land \dots \land d x_{j_{p}}$

where the functions c_{j₁j₂…j_p} ∊ C_∞1(Ω) and the integers j₁, …, j_p lie between 1 and n. Two such differential forms may be added componentwise. One defines the Grabmann product (also called exterior product or wedge product) of a p-form and a q-form as follows: For any permutation σ of the indices j₁, …, j_p,

$d x_{σ (j_{1})} \land d x_{σ (j_{2})} \land \dots \land d x_{σ (j_{p})} = sgn (σ) d x_{j_{1}} \land d x_{j_{2}} \land \dots \land d x_{j_{p}}$

where sgn(σ) denotes the sign of the permutation σ. Let

$ω^{'} = \sum_{k_{1} < k_{2} < \dots < k_{q}}^{n} b_{k_{1} k_{2} \dots k_{q}} (X) d x_{k_{1}} \land d x_{k 2} \land \dots \land d x_{k q} .$

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Chapter 23: Differential Forms and Matrix-Valued Differential Forms

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