Interior product – Wikipedia
From Wikipedia, the free encyclopedia
Mapping from p forms to p-1 forms
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product
is sometimes written as
[1]
Definition[edit]
The interior product is defined to be the contraction of a differential form with a vector field. Thus if
is a vector field on the manifold
then
is the map which sends a
-form
to the
-form
defined by the property that
for any vector fields
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms
where
is the duality pairing between
and the vector
Explicitly, if
is a
-form and
is a
-form, then
The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.
Properties[edit]
If in local coordinates
the vector field
is described by functions
, then the interior product is given by
where
is the form obtained by omitting
from
.
By antisymmetry of forms,
and so
This may be compared to the exterior derivative
which has the property
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula[2] or Cartan magic formula):
This identity defines a duality between the exterior and interior derivatives. Cartan’s identity is important in symplectic geometry and general relativity: see moment map.[3] The Cartan homotopy formula is named after Élie Cartan.[4]
The interior product with respect to the commutator of two vector fields
satisfies the identity
See also[edit]
- Cap product – method of adjoining a chain of with a cochain
- Inner product – Generalization of the dot product; used to define Hilbert spaces
- Tensor contraction – in mathematics and physics, an operation on tensors
References[edit]
- Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
- Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6
Recent Comments