Inner derivative
Encyclopedia
In mathematics
, the interior product is a degree
−1 antiderivation on the exterior algebra
of differential form
s on a smooth manifold. 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 iXω defined by the property that
for any vector fields X1,..., Xp−1.
The interior product, named in opposition to the exterior product, is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an inner product. The interior product ιX ω is sometimes written as X ω; this character is U+2A3C in Unicode
and looks like
.
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
and so
. The interior product relates the exterior derivative
and Lie derivative
of differential forms by Cartan's identity:
This identity is important in symplectic geometry: see moment map
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the interior product is a degree
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
−1 antiderivation on the exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s on a smooth manifold. It is defined to be the contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...
of a differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
with a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
. 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 iXω defined by the property that
for any vector fields X1,..., Xp−1.
The interior product, named in opposition to the exterior product, is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an inner product. The interior product ιX ω is sometimes written as X ω; this character is U+2A3C in Unicode
Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
and looks like
.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
Properties
By antisymmetry,and so
. The interior product relates the exterior derivativeExterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
and Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
of differential forms by Cartan's identity:
This identity is important in symplectic geometry: see moment map
Moment map
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
.