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
Famous quotes containing the words formal, definition and/or star:
“It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.”
—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“And though in tinsel chain and popcorn rope
My tree, a captive in your window bay,
Has lost its footing on my mountain slope
And lost the stars of heaven, may, oh, may
The symbol star it lifts against your ceiling
Help me accept its fate with Christmas feeling.”
—Robert Frost (1874–1963)