Hodge Dual - Formal Definition of The Hodge Star of k-vectors

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 (nk)-vectors where n = dim V, for 0 ≤ kn. 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:

    I will not let him stir
    Till I have used the approvèd means I have,
    With wholesome syrups, drugs, and holy prayers,
    To make of him a formal man again.
    William Shakespeare (1564–1616)

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    A rocket is an experiment; a star is an observation.
    José Bergamín (1895–1983)