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joebohr
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If the interior product is defined as the inverse of the exterior product, then how would I find the interior product of a space given its exterior product?
joebohr said:If the interior product is defined as the inverse of the exterior product, then how would I find the interior product of a space given its exterior product?
quasar987 said:Ah, *ding!*, perhaps you mean so ask somethign like: "given a vector space V, there is the exterior product V x V --> V [itex]\wedge[/itex] V. Given v in V, what is v-1?"
quasar987 said:Usually, by v-1 we mean an element such that vv-1=1 in some sense or another. Here, I see no obvious candidate for what the equation vv-1=1 could mean.
The interior product and exterior product are both operations in multilinear algebra. The interior product is a contraction of two vector fields, resulting in a scalar function, while the exterior product is a wedge product of two vector fields, resulting in a bivector field.
To find the interior product of two vector fields, you first take the exterior product of the two vector fields. Then, you contract the resulting bivector field with the metric tensor to obtain a scalar function.
Yes, the interior product can be extended to higher dimensions. In three-dimensional space, the interior product of two vector fields results in a scalar function. In four-dimensional space, the interior product of two vector fields results in a vector field, and so on.
The interior product is closely related to differential forms. It is used to define the Lie derivative of a differential form along a vector field. The interior product also allows for the integration of differential forms over manifolds.
The interior product has various practical applications in physics and engineering. It is used in the study of electromagnetic fields, general relativity, and fluid mechanics. It is also used in computer graphics and geometric modeling.