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Interior Product


The interior product is a dual notion of the wedge product in an exterior algebra LambdaV, where V is a vector space. Given an orthonormal basis {e_i} of V, the forms

 {e_(i_1) ^ ... ^ e_(i_p)}_(i_1<...<i_p)
(1)

are an orthonormal basis for Lambda^pV. They define a metric on the exterior algebra, <alpha,beta>. The interior product with a form gamma is the adjoint of the wedge product with gamma. That is,

 <alpha⌟gamma,beta>=<alpha,beta ^ gamma>
(2)

for all beta. For example,

 e_1 ^ e_2⌟e_3=0
(3)

and

 e_1 ^ e_2 ^ e_3 ^ e_4⌟e_1 ^ e_4=e_2 ^ e_3,
(4)

where the e_i are orthonormal, are two interior products.

An inner product on V gives an isomorphism e:V=V^* with the dual vector space V^*. The interior product is the composition of this isomorphism with tensor contraction.


See also

Exterior Algebra, Inner Product, Tensor Contraction, Wedge Product

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Interior Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InteriorProduct.html

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