- #1

Rasalhague

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http://en.wikipedia.org/wiki/Interior_product

I can't yet follow the drift of the Wolfram Mathworld page on the same subject:

http://mathworld.wolfram.com/InteriorProduct.html

But I was struck by their final remark: "An inner product on V gives an isomorphism [itex]e:V \simeq V^*[/itex] with the dual space [itex]V^*[/itex]. The interior product is the composition of this isomorphism with tensor contraction."

This seems more like a description of the inner product or metric tensor than what Wikipedia calls the interior product. Wikipedia's interior product seems to be just a specific contraction, namely inputting a tangent vector into the first argument slot of a covariant alternating tensor. As far as I can see Wikipedia's definition doesn't make use of the isomorphism Mathworld refers to. Is Mathworld actually talking about (what the Wikipedia writer would call) the inner product in that final paragraph, or is Mathworld using a different definition of interior product from Wikipedia's (perhaps even one in which interior and inner products are in some sense the same thing)?

(Aside: Although Wikipedia refers to the C.A.T. as a "differential form", I first came across the interior product in the context of a tangent vector contracted with a volume element, which I gather is generally not the exterior derivative of anything, in spite of the conventional notation.)