Matrix representation of a cross product and related curl-based differential operators in all space dimensions

Peter Lewintan (1)

1 Universität Duisburg-Essen = University of Duisburg-Essen [Essen]

A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann’s, Jacobi’s and Room’s identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian, which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions n ≥ 2 .

Mots clés: vector Laplacian; matrix representation; generalized curl; Jacobi identity; Generalized cross product; Helmholtz decomposition; div-curl lemma

Domaines: Mathématiques générales [math.GM]; Physique mathématique [math-ph]

Peter Lewintan. Matrix representation of a cross product and related curl-based differential operators in all space dimensions. Open Mathematics, 2021, 19 (1), pp.1330-1348. ⟨10.1515/math-2021-0115⟩. ⟨hal-03233545v2⟩