Interior Product: Find from Exterior Product

[joebohr]

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?

 

[Bacle2]

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?

Sorry, seems a little vague: is this a product of vectors, or the exterior product on a vector
space, i.e., the exterior algebra? Besies, if I understood well, the inner-product takes a pair
of vectors and outputs a number, but the exterior product takes two vectors and gives you another vector. Would you clarify?

 

[joebohr]

I am looking for the general expression for the interior product (not the inner product) of two vectors (x and y) defined on a vector space M. I think this is given in terms of the exterior product and the hodge star, but I'm not sure about the exact expression.

 

[quasar987]

You mean this?? http://en.wikipedia.org/wiki/Interior_product

 

[joebohr]

Yes, that's the interior product I was referring to, but wikipedia doesn't give enough info about actually finding an expression for the interior product of two vectors on a vector space and I thought someone here might be able to help.

 

[quasar987]

But the interior product discussed in the link I provided above is between a multilinear map and a vector, not between 2 vectors.

 

[joebohr]

Ok, if there is no interior product for vectors, than what is the inverse of the exterior product? Also, what does the interior product you mentioned represent if it is not this inverse?

 

[quasar987]

Firsty, what do you mean by exterior product of vectors? Usually, given a vector space, the exterior product is a product defined on the exterior algebra of V that assigns to an element of degree p and an element of degree q one of degree p+q. So assuming this is also what you mean by exterior product, then what do you mean by its "inverse"? What properties should such an entity have?

Ah, *ding!*, perhaps you mean so ask somethign like: "given a vector space V, there is the exterior product V x V --> V $\wedge$ V. Given v in V, what is v^-1?"

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.

 

[joebohr]

Ah, *ding!*, perhaps you mean so ask somethign like: "given a vector space V, there is the exterior product V x V --> V $\wedge$ V. Given v in V, what is v^-1?"

Exactly, that's what I'm asking. Sorry if I wasn't clear.

 

[Jamma]

Well, if you are looking for the inverse image of an element of degree two of the canonical map V x V --> V ∧ V, say x∧y, it's just going to be (x,y), (-x,-y), (-y,x) and (y,-x), isn't it? Sorry, this question is still far too vague, can you rephrase it?

 

[joebohr]

Let the exterior product of vectors X and Y be related to the cross product by
X$\times$Y=$\ast$(X$\wedge$Y). Then what is the inverse of the mapping $\wedge$:X$\wedge$Y$\rightarrow$C where C is a vector and X, Y, and C belong to the vector space M, say in Euclidean space? Also, using this same logic, what is the inverse of the cross product?
Rephrased, the question is to find the mappings:

$\wedge$$^{-1}$:X$\wedge$$^{-1}$(X$\wedge$Y)$\rightarrow$X
and
$\times$$^{-1}$:X$\times$$^{-1}$(X$\times$Y)$\rightarrow$X

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.

Since we are talking about binary operators for vectors, vv^-1 would equal the identity vector for the space M.