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 ι X ω {\displaystyle \iota _{X}\omega } is sometimes written as X ⌟ ω . {\displaystyle X\mathbin {\lrcorner } \omega .} [1]
Definition [ edit ] The interior product is defined to be the contraction of a differential form with a vector field . Thus if X {\displaystyle X} is a vector field on the manifold M , {\displaystyle M,} then
ι X : Ω p ( M ) → Ω p − 1 ( M ) {\displaystyle \iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)} is the
map which sends a
p {\displaystyle p} -form
ω {\displaystyle \omega } to the
( p − 1 ) {\displaystyle (p-1)} -form
ι X ω {\displaystyle \iota _{X}\omega } defined by the property that
( ι X ω ) ( X 1 , … , X p − 1 ) = ω ( X , X 1 , … , X p − 1 ) {\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)} for any vector fields
X 1 , … , X p − 1 . {\displaystyle X_{1},\ldots ,X_{p-1}.} The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α {\displaystyle \alpha }
ι X α = α ( X ) = ⟨ α , X ⟩ , {\displaystyle \displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,} where
⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is the
duality pairing between
α {\displaystyle \alpha } and the vector
X . {\displaystyle X.} Explicitly, if
β {\displaystyle \beta } is a
p {\displaystyle p} -form and
γ {\displaystyle \gamma } is a
q {\displaystyle q} -form, then
ι X ( β ∧ γ ) = ( ι X β ) ∧ γ + ( − 1 ) p β ∧ ( ι X γ ) . {\displaystyle \iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).} 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 ( x 1 , . . . , x n ) {\displaystyle (x_{1},...,x_{n})} the vector field X {\displaystyle X} is given by
X = f 1 ∂ ∂ x 1 + ⋯ + f n ∂ ∂ x n {\displaystyle X=f_{1}{\frac {\partial }{\partial x_{1}}}+\cdots +f_{n}{\frac {\partial }{\partial x_{n}}}}
then the interior product is given by
ι X ( d x 1 ∧ . . . ∧ d x n ) = ∑ r = 1 n ( − 1 ) r − 1 f r d x 1 ∧ . . . ∧ d x r ^ ∧ . . . ∧ d x n , {\displaystyle \iota _{X}(dx_{1}\wedge ...\wedge dx_{n})=\sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}\wedge ...\wedge {\widehat {dx_{r}}}\wedge ...\wedge dx_{n},} where
d x 1 ∧ . . . ∧ d x r ^ ∧ . . . ∧ d x n {\displaystyle dx_{1}\wedge ...\wedge {\widehat {dx_{r}}}\wedge ...\wedge dx_{n}} is the form obtained by omitting
d x r {\displaystyle dx_{r}} from
d x 1 ∧ . . . ∧ d x n {\displaystyle dx_{1}\wedge ...\wedge dx_{n}} .
By antisymmetry of forms,
ι X ι Y ω = − ι Y ι X ω , {\displaystyle \iota _{X}\iota _{Y}\omega =-\iota _{Y}\iota _{X}\omega ,} and so
ι X ∘ ι X = 0. {\displaystyle \iota _{X}\circ \iota _{X}=0.} This may be compared to the
exterior derivative d , {\displaystyle d,} which has the property
d ∘ d = 0. {\displaystyle d\circ d=0.} 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 ) :
L X ω = d ( ι X ω ) + ι X d ω = { d , ι X } ω . {\displaystyle {\mathcal {L}}_{X}\omega =d(\iota _{X}\omega )+\iota _{X}d\omega =\left\{d,\iota _{X}\right\}\omega .} where the anticommutator was used. 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 X , {\displaystyle X,} Y {\displaystyle Y} satisfies the identity
ι [ X , Y ] = [ L X , ι Y ] . {\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right].} See also [ edit ] Cap product – method of adjoining a chain of with a cochainPages displaying wikidata descriptions as a fallback Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets Tensor contraction – Operation in mathematics and physics 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