Where next with Wedge
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A short working note by Phil dated 9.14.15, written after eight sections on the wedge product in the main and dual spaces. It works out wedge products such as 2-blade with vector and 2-blade with 2-blade using antisymmetrized sums over permutations. It then writes general multivectors for n = 3 and as sums over multi-indices, asking how eI^eJ behaves and how this relates to Spivak's and Sjamaar's notation.
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Where next with Wedge? PhL 9.14.15
I have now written a total of 8 sections on the wedge product in both the main and dual spaces.
The next step is to define a wedge product for higher order objects in Λ.
I could start with
w ≡ v1 ^ v2
and call it a bivector and ask about its properties. I might want to know about
h = w ^ v
for example. Well. w lies in the space L2 whose general elements are given by
T = Σij Fij ei ^ ej = Σi<j Tij ei ^ ej
Then I guess one could say
T ^ v = (Σij Fij ei ^ ej) ^ (Σk vk ek) L2 ^ L1
= Σijk Fijvk (ei ^ ej ^ ek) = element of L3.
= Σi<j<k [ΣP (-1)P FP(i)P(j)vP(k)] (ei ^ ej ^ ek)
= Σi<j<k Tijk (ei ^ ej ^ ek) Tijk = ΣP (-1)P FP(i)P(j)vP(k
OK, now try a fancier example:
T ^ H = (Σij Fij ei ^ ej)^(Σab Gab ea ^ eb) L2 ^ L2
= Σijab FijGab ( ei ^ ej ^ ea ^ eb )
= Σi<j<a<b [ΣP (-1)P FP(i)P(j)GP(a)P(b)] ( ei ^ ej ^ ea ^ eb )
= Σi<j<a<b Tijab ( ei ^ ej ^ ea ^ eb ) Tijab = ΣP (-1)P FP(i)P(j)GP(a)P(b)
So I think doing the wedge products of pure "blades" is straightforward in this manner.
Question: for a pair of 2-blads as shown above, how do I relate that to Spivak's notation
Where is it that we get a multibladed answer? Maybe that was the Clifford Geometric product. No, it is the product of multivectors.
So how do I write a general multivector let's say for n = 3
F = f + Σi Fi ei + Σij Fij ei ^ ej + Σijk Fijk ei ^ ej^ ek
G = g + Σi Gi ei + Σij Gij ei ^ ej + Σijk Gijk ei ^ ej^ ek
The result seems straightforward.
F^G = ( f + Σi Fi ei + Σij Fij ei ^ ej + Σijk Fijk ei ^ ej^ ek ) ^
( g + Σi Gi ei + Σij Gij ei ^ ej + Σijk Gijk ei ^ ej^ ek)
= [ ( f + Σi Fi ei + Σij Fij ei ^ ej + Σijk Fijk ei ^ ej^ ek ) ^ g
+ ( f + Σi Fi ei + Σij Fij ei ^ ej + Σijk Fijk ei ^ ej^ ek) ^ (Σi Gi ei)
+ etc
There will be terms which are elements of all four blade spaces. Is there some compact way to do all this?
F = ΣIFIeI I = all possible ordinary multiindices having 0,1,2 or up to n pieces
G = ΣJGJeJ I = all possible ordinary multiindices
F^G = ΣIJ FIGJ (eI^eJ)
What do I know about (eI^eJ)? In the following, a and b are generally different integers.
eI^eJ = ( ei ^ ei ^ .... ^ ei) ^ ( ej ^ ej ^ .... ^ ej)
Sjamaar does not use I and J in the above "super general" sense, he might have I for a k-form and J for n l-form.