Formal Definition of The Hodge Star of k-vectors
The Hodge star operator on a vector space V with a nondegenerate symmetric bilinear form (herein aka inner product) is a linear operator on the exterior algebra of V, mapping k-vectors to (n−k)-vectors where n = dim V, for 0 ≤ k ≤ n. It has the following property, which defines it completely: given two k-vectors α, β
where denotes the inner product on k-vectors and ω is the preferred unit n-vector.
The inner product on k-vectors is extended from that on V by requiring that for any decomposable k-vectors and .
The preferred unit n-vector ω is unique up to a sign. The choice of ω defines an orientation on V.
Read more about this topic: Hodge Dual
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