So I was reviewing some of the old lectures on forms and I realized I kind of glossed over something that caught can cause confusion if you start studying other books so I want to quickly go over the normalization of forms and I think this would be best inserted right after lesson 27 of what is a tensor right so right after that lesson maybe this little quick lesson would best fit but the issue is that forms can be normalized a couple ways and what lesson 27 offered up was the anti symmetries ation operator which 4p forms which all or 4p for rank 0p tensors which I'll call KP this all works for of course as usual this all works for P vectors as well but we'll just do it in terms of P tensors this guy and so the idea was you could take any tensor and run it through the anti symmetry zation operator which was writing 1 over P factorial x multiplied by the sum over all permutations of the parody of the permutation multiplied by the what we were calling the Sigma transpose of the tensor T and we went through this in less than 27 about what this exactly meant the point is is I think I also demonstrated why this P factorial is in there and the reason it's in there is we want this to equal T when T starts anti-symmetric so that's a very reasonable thing so what I want K a P of T to equal T whenever it's always true that P transpose of T equals when the Sigma transpose of T equals the parody of that transposition times T itself right when this is true for T then this has to be true for the anti symmetrized version of T and we went through that proof in less 27 but that explains the origin of this factorial so that's really important place to start and the other point I want to remind people of is that the Sigma transpose of say a collection of seco vectors that are tensor product dead tensor product did together say there's a P of them right that is going to be equal by definition so what I'm doing is I'm doing the Sigma transpose of this guy here right that guy I might call say capital V or something that's a tensor all right so this here is a P a this here is a 0p rank tensor right here and I'm just writing it all out so it's it's P acting on this tensor so I could call that P the Sigma transpose of the tensor V right is is what I mean and that's going to simply equal V of Sigma inverse of 1 tensor product V of Sigma inverse of 2 dot dot dot tensor product V of Sigma inverse of P and we went through this notation a little bit also in lesson 27 of what is a tensor so that's the other interesting fact but it's this last fact which I don't think I really emphasized a lot in the whole process of talking about forms in the what is a tensor serious let me go to this last one which I want to begin to highlight the last one looks like this the last one is the wedge product of a series of one forms that's what this means this is the wedge product of p1 forms so that this guy here is an element of the P text earier power that is whoops I've already screwed up but I'm trying to get across okay that is obviously we know that this guy is going to be an anti-symmetric tensor and we want it to be related to the anti symmetrized part of the same vectors the anti symmetrized at its anti symmetry operator acting on the tensor product of V 1 V 2 and V P but if we just did this if we just took the anti symmetry zation operator and acted on the tensor product of these guys we would end up with a result that I'm not using it's at well it's a convention right there we can always throw a factor in front of this right we can define this wedge product to be the anti symmetrized anti symmetry operator acting on the tensor product times some arbitrary factor and we can choose this arbitrary factor so what I've been working with so far the way I've been presenting it I've been saying things like okay well V one wedge V two I wanted you to think in your mind as that V 1 tensor products the one tensor product V 2 minus V two tensor product V one so that's the wedge product that I've been using for between of the to form tensor product right meaning the tensor product of two one forms I've defined exactly this way and other books do to write Meissner Thorne and wheeler does it this way plenty of books you know it will get you the wedge product this way and if we did the wedge product of three right I mean I'm not out there on a limb with this definition of how the wedge product should look I should just anti symmetries these three guys in the tensor products base like we did and that will be the definition of the wedge product of three-in-one forms as as written here right we want this to equal this the problem is is this guy here if you operate using the definition of of the anti symmetries ation operator if you do that this part here actually is it's a process that goes over all these permutations and you end up with this part giving you what you want here that in other words the sum is this section what I've written down is just this sum for example all
