[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/interior-product-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/interior-product-wikipedia\/","headline":"Interior product &#8211; Wikipedia","name":"Interior product &#8211; Wikipedia","description":"From Wikipedia, the free encyclopedia Mapping from p forms to p-1 forms In mathematics, the interior product (also known as","datePublished":"2018-01-05","dateModified":"2018-01-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/56184eeaf256ee60fa3d6dc813bd58f04c41573e","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/56184eeaf256ee60fa3d6dc813bd58f04c41573e","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/interior-product-wikipedia\/","about":["Wiki"],"wordCount":5430,"articleBody":"From Wikipedia, the free encyclopediaMapping from p forms to p-1 formsIn mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree \u22121 (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 \u03b9X\u03c9{displaystyle iota _{X}omega } is sometimes written as X\u231f\u03c9.{displaystyle Xmathbin {lrcorner } omega .}[1]Table of ContentsDefinition[edit]Properties[edit]See also[edit]References[edit]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\u03b9X:\u03a9p(M)\u2192\u03a9p\u22121(M){displaystyle iota _{X}:Omega ^{p}(M)to Omega ^{p-1}(M)}is the map which sends a p{displaystyle p}-form \u03c9{displaystyle omega } to the (p\u22121){displaystyle (p-1)}-form \u03b9X\u03c9{displaystyle iota _{X}omega } defined by the property that(\u03b9X\u03c9)(X1,\u2026,Xp\u22121)=\u03c9(X,X1,\u2026,Xp\u22121){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 X1,\u2026,Xp\u22121.{displaystyle X_{1},ldots ,X_{p-1}.}The interior product is the unique antiderivation of degree \u22121 on the exterior algebra such that on one-forms \u03b1{displaystyle alpha }\u03b9X\u03b1=\u03b1(X)=\u27e8\u03b1,X\u27e9,{displaystyle displaystyle iota _{X}alpha =alpha (X)=langle alpha ,Xrangle ,}where \u27e8\u22c5,\u22c5\u27e9{displaystyle langle ,cdot ,cdot ,rangle } is the duality pairing between \u03b1{displaystyle alpha } and the vector X.{displaystyle X.}  Explicitly, if \u03b2{displaystyle beta } is a p{displaystyle p}-form and \u03b3{displaystyle gamma } is a q{displaystyle q}-form, then\u03b9X(\u03b2\u2227\u03b3)=(\u03b9X\u03b2)\u2227\u03b3+(\u22121)p\u03b2\u2227(\u03b9X\u03b3).{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 (x1,...,xn){displaystyle (x_{1},&#8230;,x_{n})} the vector field X{displaystyle X} is described by functions f1,...,fn{displaystyle f_{1},&#8230;,f_{n}}, then the interior product is given by\u03b9X(dx1\u2227...\u2227dxn)=\u2211r=1n(\u22121)r\u22121frdx1\u2227...\u2227dxr^\u2227...\u2227dxn,{displaystyle iota _{X}(dx_{1}wedge &#8230;wedge dx_{n})=sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}wedge &#8230;wedge {widehat {dx_{r}}}wedge &#8230;wedge dx_{n},}where dx1\u2227...\u2227dxr^\u2227...\u2227dxn{displaystyle dx_{1}wedge &#8230;wedge {widehat {dx_{r}}}wedge &#8230;wedge dx_{n}} is the form obtained by omitting dxr{displaystyle dx_{r}} from dx1\u2227...\u2227dxn{displaystyle dx_{1}wedge &#8230;wedge dx_{n}}.By antisymmetry of forms,\u03b9X\u03b9Y\u03c9=\u2212\u03b9Y\u03b9X\u03c9,{displaystyle iota _{X}iota _{Y}omega =-iota _{Y}iota _{X}omega ,}and so \u03b9X\u2218\u03b9X=0.{displaystyle iota _{X}circ iota _{X}=0.} This may be compared to the exterior derivative d,{displaystyle d,} which has the property d\u2218d=0.{displaystyle dcirc 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):LX\u03c9=d(\u03b9X\u03c9)+\u03b9Xd\u03c9={d,\u03b9X}\u03c9.{displaystyle {mathcal {L}}_{X}omega =d(iota _{X}omega )+iota _{X}domega =left{d,iota _{X}right}omega .}This identity defines a duality between the exterior and interior derivatives. Cartan&#8217;s identity is important in symplectic geometry and general relativity: see moment map.[3] The Cartan homotopy formula is named after \u00c9lie Cartan.[4]The interior product with respect to the commutator of two vector fields X,{displaystyle X,} Y{displaystyle Y} satisfies the identity\u03b9[X,Y]=[LX,\u03b9Y].{displaystyle iota _{[X,Y]}=left[{mathcal {L}}_{X},iota _{Y}right].}See also[edit]Cap product\u00a0\u2013 method of adjoining a chain of with a cochainPages displaying wikidata descriptions as a fallbackInner product\u00a0\u2013 Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targetsTensor contraction\u00a0\u2013 in mathematics and physics, an operation on tensorsPages displaying wikidata descriptions as a fallbackReferences[edit]Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007\/978-1-4419-7400-6  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/interior-product-wikipedia\/#breadcrumbitem","name":"Interior product &#8211; Wikipedia"}}]}]