In mathematics, the interior product is a degree −1 derivation on the exterior algebra of differential forms on a smooth manifold. It is defined to be the contraction of a differential form with a vector field. Thus if v is a vector field on the manifold M, then Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Contraction can mean: Contraction (childbirth), a contraction during childbirth; Contraction (linguistics), a new word formed from two or more individual words; Contraction (science), one that can occur to solid matter as it cools; Contraction mapping, in mathematics, a type of function on a metric space; Muscle contraction, one that occurs... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

is the map which sends a p-form ω to the (p−1)-form i_vω defined by the property that

for any vector fields u₁,...u_p−1.

The interior product is also called interior or inner multiplication, or the inner derivative or derivation.

Properties

By antisymmetry,

i_ui_vω = − i_vi_uω,

i_vi_vω = 0.

See also

• Tensor contraction