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:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Far star that tickles for me my sensitive plate
And fries a couple of ebon atoms white....”
—Robert Frost (1874–1963)