Qestion: Vector field and (n-1)-form representation of current density

[mma]

Question: Vector field and (n-1)-form representation of current density

Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the 2-form. More generally, in an n-dimensional space, between a vector field and an (n-1)-form, which results the same value when integrated on a (n-1) dimensional surface. Is the latter the Hodge-dual of the vector field regarded as an 1-form ? If yes, how can one see this?

 

[robphy]

Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the 2-form. More generally, in an n-dimensional space, between a vector field and an (n-1)-form, which results the same value when integrated on a (n-1) dimensional surface. Is the latter the Hodge-dual of the vector field regarded as an 1-form ? If yes, how can one see this?

In n-dim, Hodge-duality relates a k-form with an (n-k)-multivector.
In 3-D, $$j^a=\epsilon^{abc}J_{bc}$$.
You may enjoy the URLs in these earlier threads:
https://www.physicsforums.com/showthread.php?t=92472
https://www.physicsforums.com/showthread.php?t=89419
https://www.physicsforums.com/showthread.php?t=18963
http://www.ee.byu.edu/forms/

 

[mma]

Dear Robphy, thank you very much.
Are you sure that I will find the answer for this question in these references? And before I read all you showed, could you tell me the answer? Yes, or no? Maybe it's trivial for you but not for me, in spite that I know the definition of the Hodge dual.
mma

 

[robphy]

The tensorial relation I gave shows how the current-vector is associated with a current 2-form, in 3-D space. I literally took the Hodge-dual.

As I said, "In n-dim, Hodge-duality relates a k-form with an (n-k)-multivector. " So, the "Hodge-dual of the vector field regarded as an 1-form" is incorrect... it is an (n-1)-form. I hope I understood your question.

In the old threads I listed,
the links to Burke's pages describes electromagnetism using differential forms... in fact, in a rather unique pictorial way... a sort of visual tensor algebra.
the links to Bossavit's Applied differential geometry discusses forms in more detail than Burke (Bossavit's goal is electromagnetism).
the BYU link describes a course in electrodynamics using differential forms.

The subject is not trivial [to me]... I still struggle with developing my intuition and understanding of them. I have offered a set of references that have helped me get to where I am now.

 

[mma]

So, the "Hodge-dual of the vector field regarded as an 1-form" is incorrect... it is an (n-1)-form. I hope I understood your question.

Perhaps my English is too poor, sorry. I wanted to say that vector fields do not have Hodge dual, because only forms have it, so we have to regard this vector field as an 1-form (really, this 1-form is the dual of the vector field, but I wanted to avoid the usage of word "dual" in one sentence with two different meaning). The Hodge-dual of this 1-form is of course an (n-1)-form.
Thank you again the references, I will study them.

 

[robphy]

Perhaps my English is too poor, sorry. I wanted to say that vector fields do not have Hodge dual, because only forms have it, so we have to regard this vector field as an 1-form (really, this 1-form is the dual of the vector field, but I wanted to avoid the usage of word "dual" in one sentence with two different meaning). The Hodge-dual of this 1-form is of course an (n-1)-form.
Thank you again the references, I will study them.

In my understanding, Hodge duality operates on totally-antisymmetric tensors, which include vectors v^a and 1-forms w_a. Based on http://en.wikipedia.org/wiki/Hodge_star_operator , a metric is needed for the Hodge-dual to relate a k-form to a (n-k)-form... or a k-multivector to a (n-k)-multivector. [To distinguish the two duals, you could say metric-dual and Hodge-dual.]

Your restriction may depend on your starting definitions and specification of the available structures [e.g. volume form, metric, etc...], which you may wish to present here.

 

[mma]

Hodge duality operates on totally-antisymmetric tensors, which include totally-antisymmetric tensors, which include vectors v^a and 1-forms w_a.

Of course, v^a is an 1-form, but it acts on the dual space. That's why we must use for our purpose the metric-dual v_a of it (as long as I am right that the 2-form representing the current density is really a 2-form, and not a 2-multivector) . Certainly, this duality requires the same metric as the Hodge-dual requires.