Definition
It is defined to be the contraction of a differential form with a vector field. Thus if X is a vector field on the manifold M, then
is the map which sends a p-form ω to the (p−1)-form ιXω defined by the property that
for any vector fields X1,..., Xp−1.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α
- ,
the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then
The above relation says that the interior product obeys a graded Leibniz rule. An operation equipped with linearity and a Leibniz rule is often called a derivative. The interior product is also known as the interior derivative.
Read more about this topic: Interior Product
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“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)