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

Abstract

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≥2n\ge 2.

Keywords

• generalized cross product
• jacobi identity
• generalized curl
• matrix representation
• vector laplacian
• helmholtz decomposition
• div-curl lemma
• 15a24
• 15a69
• 47a06
• 35j05