# Trying to Understand the Hodge Star

Discussion in 'Physics & Math' started by Reiku, Dec 7, 2011.

1. ### ReikuBannedBanned

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Performing the exterior product on two vectors will give you an anti-symmetric operator.

$u \wedge v= A(u \otimes v)$

where $A$ is such an operator. There is a relation between the cross product and the exterior product using the Cevi Levita Symbol.

$(u \times v)_i = \epsilon_{ijk} v_j v_k$

$\frac{1}{2}\epsilon_{ijk} (v_jv_k - v_kv_j)$

which is simple enough. But then comes my problems, the introduction of the Hodge Star, which I am not terribly familiar with, I've only ever heard of it

$\epsilon_{ijk}(u \wedge v)_{jk}$

$(* 1)_{ijk} = \epsilon_{ijk}$

Now as I understand it, this identity arises because the hodge star is the identity element in three dimensions. This is a special case isn't it, because the cross product in three dimensions will give you the ability to express something as a hodge dual?

Is it also right to say a hodge dual is a mapping of permutations of space, or is it more refined than this?

Last edited: Dec 7, 2011
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3. ### ReikuBannedBanned

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My star never showed up... that should be

(*1)_ijk

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5. ### ReikuBannedBanned

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I am under the impression that the hodge star can be used as thus

$\epsilon_{ijk}(u \wedge v)_{jk}$

$=*(u \wedge v)_i$

This is right, right?

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7. ### CptBorkRobbing the Shalebridge CradleValued Senior Member

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Before worrying about Hodge duals, I would advise you to start by learning something more basic, such as functions and vectors. Then calculus, multivariable calculus, vector calculus, some analysis, linear and abstract algebra, differential geometry, etc. etc. Then in 4 years' time we can talk about exterior products. See you then.

8. ### ReikuBannedBanned

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Stop trying to patronize me in front of everyone.

If I have a question, it is the duty of people here to answer my question. This constant dual in trying to put me down is more than tiring.

Answer my question, or leave it for someone who can.

9. ### CptBorkRobbing the Shalebridge CradleValued Senior Member

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No one can answer your question, because the answer would have to include 4 years' worth of background material in order for you to be able to understand it.

10. ### ReikuBannedBanned

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Anyone who knows how to answer the question can. Anyone competent in physics could give a basic overview.

You might like to start with whether my impression of what is going on is basically right.

11. ### Guest254Valued Senior Member

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1,056
But you don't have an impression of what's going on. You simply copy out bits and bobs from different sources, then make out they're the product of your own thoughts.

Are you not bothered by the fact you're called up on this kind of thing every time you make a new account? Come one, be sensible. Use this place to ask genuine questions and I'm sure we'll all get on just fine.

12. ### ReikuBannedBanned

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You must have missed the questions:

Now as I understand it, this identity arises because the hodge star is the identity element in three dimensions. This is a special case isn't it, because the cross product in three dimensions will give you the ability to express something as a hodge dual?

Is it also right to say a hodge dual is a mapping of permutations of space, or is it more refined than this?

13. ### rpennerFully WiredStaff Member

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4,833
No, because you view math and physics jargon as tools of rhetoric, not a language to communicate reliable observations and inferences.

14. ### ReikuBannedBanned

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Try me. If quarkhead could come to this forum and ask questions, why can't I? You are so up your own arses drenched in biased behaviour it is incomprehensible how any could come here and even see sciforums as a true place for people to present their idea's or present a question.

15. ### AlphaNumericFully ionizedModerator

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This is absolutely not true. In fact it isn't even a coherent sentence.

The cross product generalises in N dimensions, but no longer forms an algebra. You can always construct a Hodge dual of something, that's what the Hodge star does.

If you understood the definition of the Hodge star you wouldn't need to ask this. Please give the definition, along with a description in your own words, of the Hodge star.

It isn't patronising, it's completely sound and reasonable advice. If you can't do linear algebra, and you can't, then something like the Hodge star is beyond your comprehension. You need to walk before you can run and you currently can't even stand.

In cases where you ask questions whose answers you cannot understand then Cpt's response is a valid answer. It's easy to ask questions you don't understand the answers of. For example if I asked "What is the proof to Fermat's Last Theorem?" the answer is beyond my comprehension, beyond anyones here. So it would be a waste for someone who understands the proof to explain it to me in explicit detail. Instead a valid reply would be "Perhaps you should learn about Galois theory first?", which is just what Cpt has said to you in regards to the Hodge star.

Stop making threads like this then. We all know you don't understand this stuff, your very questions show it, so you're wasting your own time as well as ours.

Give me your view on the definition, in your own words, of the Hodge star and its applications in maths or physics. Failure to do so will render this thread pointless because discussion isn't occurring and I'll close it.

16. ### ReikuBannedBanned

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whattttt???

I HAVE ADMITTED I DON'T KNOW MUCH TO NEXT TO NOTHING ABOUT A HODGE STAR, SO WHY WOULD MY QUESTIONS IMPLY TO YOU, THAT I WOULD NEED TO KNOW WHAT A HODGE DUAL IS? ISN'T THAT THE WHOLE POINT OF ASKING IN THE FIRST PLACE?

''This is absolutely not true. In fact it isn't even a coherent sentence. ''

Ok, I could only ask.

''The cross product generalises in N dimensions, but no longer forms an algebra. You can always construct a Hodge dual of something, that's what the Hodge star does''

I'll come back to this

''Stop making threads like this then. We all know you don't understand this stuff, your very questions show it, so you're wasting your own time as well as ours.''

I'm to stop asking questions period. Would you like me to stop posting in the physics subforum altogther???

''Give me your view on the definition, in your own words, of the Hodge star and its applications in maths or physics. Failure to do so will render this thread pointless because discussion isn't occurring and I'll close it.''

I'll try.

17. ### AlphaNumericFully ionizedModerator

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The problem is you don't know anything about the things the Hodge star is defined in terms of. It's like giving an exam on Shakespeare to someone who can't read, they aren't even going to understand the symbols on the page, never mind the specifics of the story.

No, I didn't say that. How many times are you going to do this (for others, Mister has this problem in PMs a lot), where you completely misconstrue what I say? I didn't say you're to stop asking questions. Asking questions whose answers you know you're incapable of understanding is what I said. You don't know enough linear algebra, you know you don't, so why ask for something more advanced than that? Why are you aiming at complicated topics like gravitons, neutrinos and the Hodge star when you probably would struggle with an A Level exam in physics or maths? It's a waste of your time to ask the questions and a waste of our time to answer. If you did it very occasionally then it might not be so bad but you're producing threads multiple times a week, bouncing from topic to topic you don't understand.

So why didn't you before you clicked 'submit reply'?

18. ### Magneto_1Super PrincipiaRegistered Senior Member

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295
It it true, that this forum is not a place to present new ideas or theories. And this is mainly due to the personality disorder of the moderators!

Mister, you should change your moniker to "Anti-Numeric" because you are the anti-particle of "Alpha Numeric."

You are what "Alpha Numeric" would have become had he not prepared himself mathematically, and learned some physics. But he did prepare himself mathematically, and he even learned some physics rigor along the way.

Now, if you go and prepare yourself in mathematics and in physics, by taking some classes at a college, it might help in your transition into the real world of physics.

You, mentioned that you have been doing physics for nine (9) years in one post. It looks more like four (4) years.

You are doing "Dart Physics". Dart Physics is throwing physics concepts and mathematics at a wall of physicists to see what sticks. Your philosophy is, if it sticks to the wall of physicists then, hey "I am viewed as a Genius"!!

Good luck!!

19. ### ReikuBannedBanned

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I know what you are referring to... no, I said I have studied consciousness in regards to physics for over 9 years. I spent 3 years getting a diploma in physics, which is generally different.

20. ### ReikuBannedBanned

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11,238
''The cross product generalises in N dimensions, but no longer forms an algebra. You can always construct a Hodge dual of something, that's what the Hodge star does''

Right, the reason why I asked my first question, which you said was not true, was because (if my OP was right) is that

(*1)_ijk = ε_ijk

this identity arises because the hodge star is the identity element in three dimensions. So I wondered if that was a special case since it could be given in terms of the Cevi-Levita symbol

''Give me your view on the definition, in your own words, of the Hodge star and its applications in maths or physics. Failure to do so will render this thread pointless because discussion isn't occurring and I'll close it.''

Keep in mind my general ignorance of the Hodge star.

My definition would simply be what I have read on the subject to gain an understanding of it. (How does one define it in a way when you've only read about it??)

Don't be very surprised if it sounds similar to anything you might find on the web. I've been using the web to learn about it since three days ago.

The Hodge star is a linear functional for differentials. It changes the k-forms to n+k forms. The best mathematical description so far I have seen is that if $w$ is a changing $k$ form, then

$w(v_1... v_k) = (*w)(v_{k+1}... v_n)$

You said ''The cross product generalises in N dimensions''. But it is no longer an algebra, what does this mean?

21. ### ReikuBannedBanned

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11,238
And I've never taken mathematics outside of college, so yeah, alphanumeric will know a lot of magical things beyond me. That's one of the reasons I come here to ask questions, because I know some here will be able to answer them. Call it dart physics if you want, I call it using a forum to the best of your ability.

22. ### funkstarratsknufValued Senior Member

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1,390
Pray tell, what kind of diploma?

23. ### ReikuBannedBanned

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11,238
Can't you read?

A college diploma here in scotland mind have different meanings over there in America. It might sound fancy, but its no all that great. It's a pre-requisit though if you want to go and study physics in university, for instance.