Relation Between Interior Product, Inner Product, Exterior Product, Outer Product..

I'm not sure what you're reading, but it certainly contains a lot of terms talked about in Clifford algebra (or "geometric algebra").Clifford algebras were born of a synthesis of inner product spaces and Grassmann's exterior algebras, both of which have geometric applications.A Clifford algebra is constructed from an inner product space $(V,Q)$ by generating an associative algebra (whose product is a descendant of the tensor product in the tensor algebra for $V$). These are compatible in a sense made clear in the Wiki.Suppose we are in a real inner product space, and let's denote the algebra multiplication with $otimes$ and the inner product with $cdot$. One identity that holds for parallel vectors $v,w$ is $votimes wvcdot w$. Part of the reason this is true is that $votimes wwotimes v$ when $v,w$ are parallel.When they are not parallel, then there is a skew component to $votimes w$. This can be retrieved explicitly by computing $frac12(votimes w-wotimes v):vwedge w$. With this notation, $votimes wvcdot wvwedge w$ for all vectors $v,w$. This $wedge$ is called the exterior or outer product in this algebra. They key thing to know is that this inner and this outer product don't uniquely extend to the whole algebra (well the outer product has a slightly more natural extension than the inner product). They are not "natural" to the algebra really: they are mostly just a notational convenience with some algebraic properties that make them easy to view as products.The cross product arises too in the discussion of Clifford algebras, but it seems to be regarded as second class to the outer product. I've seen this attributed to the cross-product somehow not carrying the correct information for physical applications.

I went and read a little and found out that "interior product" looks a lot like an important extension of the inner product in a Clifford algebra which is sometimes called the contraction.In summary, I think these various products might become less mysterious if you read some of these linked articles and see where they are used. It's going to be especially enlightening when you find out where they are all defined (in the vector space $V$ or in an algebra containing $V$.

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Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot Product, I was wondering if the simple map that I drew below is at all an accurate representation of the links between these different products? Vertical lines denote generalisation-specification, horizontal lines denote "in opposition to". I'm just trying to get a quick overview before I dive in. Thanks

·OTHER ANSWER:

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot Product, I was wondering if the simple map that I drew below is at all an accurate representation of the links between these different products? Vertical lines denote generalisation-specification, horizontal lines denote "in opposition to". I'm just trying to get a quick overview before I dive in. Thanks

Relation Between Interior Product, Inner Product, Exterior Product, Outer Product.. 1

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