Understanding Curl in 3D: Using Vector Operators & Components

[snowstorm69]

Can anyone explain to me how to expand this expression for curl which I find in the GR book I'm reading (by Hobson, Efstathiou and Lasenby, page 71)? In a section entitled Vector Operators in Component Form they state the curl as a "rank-2 antisymmetric tensor with components":

(curl)ab = DELaVb - DELbVa where a, b are subscripts, V is a vector(?, tensor?), and I used the admittedly poor notation "DEL" to indicate the nabla or del operator used to denote the grad of a scalar usually or the div of a vector.

I tried to write the components of this out with a going from 1 to 3 while keeping b at 1, then a going 1 to 3 with b at 2, and finally a going 1 to 3 with be at 3. In this way I got 9 pairs of terms, 18 terms in all (I don't have it in front of me but visualizing I think that's what I got). And ALL the terms cancel each other out. I believe my mistake is not taking into account the basis vectors and that if I had the terms would not all cancel out. They would yield the "curl" in 3-D.

What I get instead is for example the 1-3/3-1 pair of terms for the curl in 3-D, and then the same thing a bit later in the expansion, to cancel each other out.

Can you help me? I want to work the case out in 3-D so that I can next work it out in higher D. Obviously there are some things I don't understand about the basis vectors or perhaps the meaning of the notation itself.

Thanks.

 

[Mentz114]

I don't follow how terms 'cancel out'. In 3 dimensions, you get 9 components, each of the form

[tex]\partial^aV^b - \partial^bV^a[/tex].

I'm assuming V is a vector with components Va ( a=1,2,3 etc)

Obviously the tensor is antsymmetric since swapping a and b changes the sign.

I think you are doing some unnecessary summation.

Are you doing a generalised curl ?

[tex]\epsilon^{\alpha\mu\nu}\partial_{\nu}V_{\mu}[/tex]

This answer may be off-beam because I don't have the book.

 

[snowstorm69]

you are right and...

Firstly, Thank you for your response.
Yes, I'm getting 9 components just like that. You are right I am summing them and it sounds like I should not be. Also, yes, this is supposed to be a generalized curl and I'm trying to see if it works out to the curl I know in 3-D (so I'll speak about 3-D in this) so I can then write it out for 4-D and 5-D. I've seen curl written like that with the Levi-civita symbol. But this book writes the curl in this way that I tried to describe which you nicely pasted higher up in your response; I've also seen it in an article that writes maxwell's equations in higher dimensions - as part of the differential form of Faraday's law in higher dimensions. (Once I get comfy with how to write that in higher dimensions, I want to take the curl of the curl of E in the differential form of Faraday's law (curl of both sides of that law) so I can derive the Helmholtz wave equation for electromagnetics in higher dimensions).
If I should not be summing these components, then it sounds like I need to make a matrix of them, which is what the left hand side of the book's equation suggests, namely it says "(curl)ab" where a and b are subscripts. But if this is a matrix or tensor, then if I try to write it out for 3-D to see if it matches Cartesian or better yet a generalized orthogonal 3-D situation, then which are the x components (if in cartesian) and which are the y, and the z? Or rather.. I know (or think) that the terms with derivatives NOT involving x should make up the x-component, and terms NOT involving y the y component, z similarly, and the terms where a=b (3 of my 9 term-pairs of that form you pasted nicely) will be zero, and I'm left with 6 non-zero term-pairs out of the original 9 (in Cartesian). But then that's 2 term-pairs for each coordinate x, y, z. And.. it just doesn't look like the 3-D curl to me. It WOULD looke like it if I ended up with only 3 term-pairs of that form you pasted.
I'm sorry I'm describing this in words instead of just somehow pasting you what I'm getting. If you knew of a place where the curl terms are written (or partially written) out for 3-D or any D really, please let me know. Don't know if this post will make much sense or give you a headache. My apologies if it does, not because it's complicated, but because my description is inadequate.
Thanks.

 

[snowstorm69]

by the way

by the way, it sounds like I should not sum; that I should leave them as a tensor (antisymmetric). If so I'm not sure how I would write the terms out if I wanted to write them equation style.

 

[snowstorm69]

here's a link to the page in the book

here's a link to the page in the book from google books (amazon also has a link to it). Don't worry this link is legal it's from google books as their way of advertising the book to sell it. Only if you have time... it's page 71. This link allows you to preview that page.

http://books.google.com/books?hl=en...Nfd&sig=lsFcukfcT54duv2PRhjWr0CRn6g#PPA71,M1"


http://books.google.com/books?hl=en...6Nfd&sig=lsFcukfcT54duv2PRhjWr0CRn6g#PPA71,M1

 

[Mentz114]

Hi Snowstorm, I see from page 71 that this is the ordinary curl I've written above. The diagonal elements are zero, and there are six remaining independent terms.

This thing is mostly seen in the EM field tensor which may be written -

[tex]F^{\mu\nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}[/tex]

where A is the four-potential.

If you want a quick refresher on curl, try this

http://mathworld.wolfram.com/Curl.html

by the way, it sounds like I should not sum; that I should leave them as a tensor (antisymmetric). If so I'm not sure how I would write the terms out if I wanted to write them equation style.

Like this -

[tex]F^{01} = \partial^{0}A^{1} - \partial^{1}A^{0}[/tex]

[tex]F^{02} = \partial^{0}A^{2} - \partial^{2}A^{0}[/tex]

etc

 

[snowstorm69]

Thank you!

Hi, Mentz114,
My heartfelt thanks for your reply.
I had to think about it to see if I was going to run into another question - I still might. But I was summing (as you aptly suggested) when I shouldn't have been, and you writing those out (and the link to the curl) helped out immensely. Thanks for your patience with this now seemingly silly question.

 

[snowstorm69]

1. wave eqn vs. particle motion; 2. curl curl in 5-D

I've got 2 questions now.
1. It seems that the GR book I'm reading comes up with a "wave equation" to describe the motion of a charged particle. This equation reminds me of the EM wave equation that you get when you take the curl of both sides of the differential form of Faraday's Law, which I thought described the motion of a propagating EM wave rather than the motion of a charged particle. Are both phenomena described by the same equation? Are both phenomena the same thing physically? Or are the 2 equations really different? Perhaps if one were to take the curl of both sides of Faraday's law in 3+1-D one would get something much more complicated looking than the equation that describes the motion of a charged particle?
2. I'd like to get that wave equation in 4+1-D. How should I do that? If the 2 above are the same equation, then I'll just follow the GR books derivation of this and apply it the way Klein did.
If they are different things then (and even if they aren't I'm still curious to know how to do this) I should perhaps write Maxwell's equations in 4+1-D and then take the curl of both sides of Faraday's Law to derive that good ole EM wave equation that I thought I learned about in undergrad.
To do that I'd need to know how to write the curl of the curl of E. If the 2 above equations mentioned in number 1 of my post now are the same then I'd like to still take the curl of the curl just to see if I get the same equation as the one the GR books come
up with.
I thought of 2 ways to get the curl of the curl. 1. use the levi-cevita symbol with ... 5?.. indices to operate on the expression given in the above post; 2. reapply the above to get the merates/components of the curl curl tensor; in index notation I'm not sure how to write it, but maybe it would be (curl curl E)abd = DELa Ebd - DELb Ead though I don't know if that makes sense index-wise...
Sorry for the gaping holes in my education.
- Snowstorm69

 

[Mentz114]

It seems that the GR book I'm reading comes up with a "wave equation" to describe the motion of a charged particle.

No idea what you are referring to. What equation ?

 

[snowstorm69]

wave equation? maybe not..

Here are 2 links to show the equation. Maybe I need to read more, but here they are:
In the first link page 141 looks like the EM wave equation if I think about the d'Alembertian operator being split into time and spatial derivatives.

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

Then when they talk about gravity here they say this situation is analogous to the EM situation and then on page 474 further below they call this the "wave equation".

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

 

[snowstorm69]

wave equation? maybe not..

Here are 2 links to show the equation. Maybe I need to read more, but here they are:
In the first link page 141 looks like the EM wave equation if I think about the d'Alembertian operator being split into time and spatial derivatives.
(Darn this isn't working now. You'd have to search in the book for the page, it would get you to the index of the book and you'd click on the page number to get there.. but I don't want you to do that.. I tried to screenshot it and put it in a word doc but it won't let me screenshot it... so I'll have to get latex saavy)... the below explanation won't work therefore. I'll post again tomorrow hopefully. thanks.

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

Then when they talk about gravity here on page 472 they say this situation is analogous to the EM situation and then on page 474 further below they call this the "wave equation".

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

https://www.amazon.com/gp/product/0521829518/?tag=pfamazon01-20

 

[snowstorm69]

malarchy

Ok, I'm full of malarchy, my apologies. Ill nevertheless try to explain the above question and then cut to the chase.
So, 2 things, first to explain the above:
1. In the 474 page area they are talking about the linearized gravity field equations an earlier form of which they stated on page 472, as wave equations; and on page 472 they had already said that this is analogous to an EM equation. The EM equation they said this gravity "wave equation" was analagous to is foundon the top most page I tried to link to, and is the D'Alembert operator applied to the vector potential, and equated with mu times the current density; an equation which they describe as a compact way to write the EM field equations.
So I thought this reminded me of either the scalar or the vector helmholtz equation. I was interested in seeing that in an extra dimension, 4+1-D. (The geodesic equation also looks like the wave equation to me by the way).

2. Here's my real question: is there what I'll call a "wave equation" in 4+1 or 5 dimensions? I mean by this an equation analagous to the *Vector* Helmholtz equation of Electromagnetics that describes a propagating EM wave. If so I'd like to know where I can read about it and see it.

-Snowstorm69

 

[Mentz114]

I can't see the pages in question, but I will say that all wave equations in 1,2,3 or 4 dimensions that I've seen are the same in relating the spatial dependence of something to its temporal dependence.

This link has a good summary, and a section on more than 3 dimensions.

http://en.wikipedia.org/wiki/Wave_equation