Well maybe. There must be some personality clashed when I am told there is an algebra, yet I am informed before that ''The cross product generalises in N dimensions, but no longer forms an algebra. '' By AN no less.
Discussions move on, it isn't necessary for every post to address the original post, it's okay to address later ones. And saying things along the lines of "You don't know enough, this is an advanced topic. You need to learn the basics" is not attacking, it's being honest. If you have further comments about moderation in this forum use the thread in the open government forum. If you wish this thread to continue I suggest you stick to my questions about algebras etc and we can go from there. Anything else will get edited, moved or just deleted. You've changed what you said. I originally said the cross product is an algebra on vectors only in 3d. You then claimed I'd said the Hodge star, rather than cross product, which I corrected you on. Now you're back to cross product. You say you know what an algebra is. Tell me why the cross product is an algebra. Then tell me why the cross product in 4d is not. /edit And Magneto, as Prom said, you're very lucky not to have been given a holiday. People have been suspended for such things in the past. But I tried to demonstrate I'm not desperately looking to ban you, your claims provide entertainment after all. Perhaps you could help Mister with what I asked him, seeing as you've just written an entire book on vector related stuff? Alternatively, if you really want to be banned just stick to the pointless posting.
I changed nothing, do not accuse me of ridiculous accusations. You said ''The cross product generalises in N dimensions, but no longer forms an algebra. ''
from here vs Hence why I corrected you, the Hodge star or a Hodge dual is not an algebra. That statement doesn't even make sense. Seeing as you just don't get it I'll spell it out for you, else we'll be here for a long time with you just misquoting and misunderstanding me. An algebra, among other things, includes a binary operator. Something which takes in 2 elements and spits out a third in the same space. The cross product in 3D does this, taking in two 3d vectors and spitting out a third orthogonal to both inputs. The cross product's action in terms of components of the vectors can be expressed using the Hodge star and p-forms. You construct a pair of 1-forms with components the same as the 2 vectors in question. You take their wedge product, to give a 2-form. The Hodge star then maps the 2-form to an N-2 form when in N dimensions. N=3 gives a 1-form. Thus a pair of 1-forms in, a 1 form out, whose components then define the cross product's components. In 4d to get a 1-form out of the Hodge star you must put a 3-form in. To make a 3-form from 1-forms you need 3 of them. Thus you'd need not a binary operator but a trinary one. That isn't an algebra. You can extend your considerations from just 1-forms to the ring of forms on the space, so objects are sums of different p-forms. Then you get an algebra under the wedge product and the Hodge star is then a map from the algebra to itself. The Hodge star is not an algebra because it's an operator. It's like saying addition is a set, it's meaningless. The Hodge star is a map which takes in a p-form and outputs an (N-p)- form. Unless N is even and p=N/2 the output form space is not the same as the input form space.
HENCE why you corrected me nothing. On one moment you say it does not form an algebra, yet when I say it did not, you scrutanize me for it. Disingenuous much? Just admit you messed up your sentance, instead of trying to make it out to be something else!!!!
Here's the explicit coordinate method. Vector V has components \(v^{i}\) and W has components \(w^{j}\). This and the Euclidean metric gives us 1-forms \(v_{i}\textrm{d}x^{i}\) and \(w_{j}\textrm{d}x^{j}\). Their wedge product is \(v_{i}w_{j}\textrm{d}x^{i} \wedge \textrm{d}x^{j}\). The Hodge star works as \(x^{i} \wedge \textrm{d}x^{j} \to g^{ia}g^{jb}\epsilon_{abc}\textrm{d}x^{c}\). Here we see the Hodge star requires a metric. Remember how I asked you to give an overview of what the Hodge star needs? That's what I was referring to. Simple enough if you've understood the definition. Thus we now have the 1-form \(v_{i}w_{j}\epsilon^{ij}_{\phantom{ij}c}\textrm{d}x^{c}\) (/edit Apparently LaTeX here doesn't support \phantom{}). Since we're using the Euclidean metric the up-down convention isn't necessary (but it very certainly is in general relativity!) and so we have \(v_{i}w_{j}\epsilon^{ijk}\textrm{d}x^{k}\) so the components of the cross product between V and W are \(\epsilon_{ijk}v_{j}w_{k}\). Precisely as you'd get using the usual matrix determinant method (which is closely linked to p-forms for precisely this reason). Did you even read what I said? I originally said the cross product doesn't form an algebra in dimensions higher than 3. You then misrepresented what I said by changing it to the Hodge dual not forming an algebra. I then corrected you by pointing out I said cross product, not Hodge dual. It's right there in the quotes. I didn't screw up any sentence, I was correcting your mistake! In fact neither form an algebra in general but at least the cross product does in a specific case, 3D. Hence why I said what I said. How are you not getting that? It's right there in the quotes. I said one thing, you said another, I then pointed out I didn't say as you said I said. The mistake is yours.
Of couse I read what you said. I have asked you to admit that you could have worded what you said more precise, because in anyones language, what I extracted from you is what many would have. PERIOD
I appreciate now only in light of a badly written sentance, you are now spewing math out left right and center.
No, I was quite clear. Besides, if you understand the cross product and even the smallest bit of what we're talking about here you should have understood what I said. The problem is your lack of understanding. Ha! If you think that's spewing algebra you should see my whiteboard at work! My sentence was clear and I posted the algebra because clearly you were not going to answer my questions, which should have been enough to prompt you into thinking. Instead you've reduced to just throwing out little snipes now. I gave you the opportunity to provide that, rather than me provide it. If you understood the notation you yourself used in your posts you'd have been able to do that yourself. Now you're complaining I'm 'spewing' out mathematics. All I've done is fill in the gaps properly, I've not go any more complicated than what a simple discussion on the Hodge star would require. Besides, if you want to talk about a topic most don't meet until postgrad its hardly surprising some mathematics might be involved. Isn't that what you wanted, to discuss the components etc? Isn't that what is in your original post? Why are you now complaining when I actually provide something you should have found useful, if you understood it? The basic index notation is 1st year undergrad, even if notions like a metric and p-forms are more advanced. I remember proving stuff about the cross product in my 1st year exams, so its hardly rocket science. If you don't want to discuss relevant responses, which you claimed you wanted, then I'll close this thread. Is that what you want or do you actually want to engage in something resembling an honest discussion? I'll check back in the morning....
''no I was quite clear'' YEAH Other to a laymans... well done alphanumeric. You've not only contradicted a statement of yours to a laymen, but you've managed to make a complete disinterest for me on my own thread. No doubt this is what you wanted. Please, close it. I bet your fingers are wet at the idea!
let me for one last time state the obvious of what you said ''Ha! If you think that's spewing algebra you should see my whiteboard at work! You disagreed with me when I said ''Is is an algebra,'' more or less you said before this ''The cross product generalises in N dimensions, but no longer forms an algebra. '' How is that not mistaking something innocently? If that was QH he would have been pardoned immediately as he has said arguably, worst questions in history.
Mister: I have some physics qualifications, but I've never used a Hodge star in my life and I have no idea what it is. Perhaps you can help me get up to speed. I don't understand this explanation, so can you break it down for me? Here are my questions: * What is a functional? * What property does a functional have to have to be linear? * How does a functional for differentials differ from one for other things? What other kinds of functionals are there? * Please give a couple of examples of differentials and functionals of differentials. * What is a k-form? * What is an n+k form, and how does it relate to a k-form? * What is w in the above expression? * what are the v's? Hope you can help.
What gives you the impression I can expand any futher on even some of these. I mean, we should all know what a differential function is. We should even know what some of the math implies. But if someone says they don't know what a Hodge Dual is, should you not be asking someone who is prancing about as if they know the answer? Coming to me and asking these questions, seems backwards in my opinion, gathering the evidence?
Well if you can't expand on any of those points, why would you think yourself qualified to speak on the subject at all?
What makes you think I can't? I can expand on many things given the right direction. does a cat know what a litter tray is, before it is shown how to use it???
Because this statement of your implies you can't. Is this the kind of shit that has gotten you banned from every forum you've appeared in reiku?
You are just a bunch of words, and if you keep it up, derailing my threads, (which you've done to each one for the last half hour, and I can prove) I will report you. If you know how to answer the OP, then please go right ahead. If not, in the kindest respects, piss off.
Quoting your own words in reply to a question you ask is derailing 'your' threads? Report me? :bravo:
Indeed I will, just look at your last thread. In fact, anyone can trail your amount of condescending garbage to the beginning of time, to when I even began my own sockpuppets here. You don't contribute to science.You prance about as if you know shit, which you obviously don't, attack people in the most active area of your pleasure which would be the psuedoscience related area's, and call it in the name of science, without providing one ounce of mathematical back-up. You want me to be more honest than this? I am surprised no mathematician or physics PhD-holder here has actually brought you up on your snide remarks. Even when someone has something to offer, all you can produce is mere remedial words, a superfluous backwash of confrontation which no one can really match. You are a troll at it's highest. Measuring your claims with what you have produced over the years has told me this. If it has gone past anyone elses notice then I am indeed surprised.
