The Hodge Star Operator
نویسنده
چکیده
We’ll start out by defining the Hodge star operator as a map from ∧k(R) to ∧n−k(R). Here ∧k(R) denotes the vector space of alternating k-tensors on R. Later on, we will extend this definition to alternating tensors on a finite dimensional vector space that is equipped with an inner product. Let I = (i1, ..., ik) be some increasing multi-index of length k. That is i1 < i2 < i3 < .... Let J = (j1, ..., jn−k) be the complementary increasing multi-intex. For instance, if n = 7 and I = (1, 3, 5) then J = (2, 4, 6, 7). Let K0 denote the full multi-index (1, ..., n). We first define ∗ on the usual basis elements:
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