Bilinear forms and wedge products

  • Thread starter homology
  • Start date
306
1
Hey folks,

I'm reading "Symmetry in Mechanics" by S. Singer and I'm stuck on an exercise. It asks to find an antisymmetric bilinear form on [tex]R^4[/tex] that cannot be written as a wedge product of two covectors.

Here are my thoughts thus far: on [tex]R^2[/tex] its trivial to show that every antisymmetric bilinear form (ABF) is the product of two covectors. I just let F be an ABF and put in two vectors, v & w and expanded them in terms of a basis [tex]e_i[/tex]. I get something like:

[tex]F(e_1,e_2)(v_1w_2-v_2w_1)[/tex] which easily maps into [tex]F(e_1,e_2)dx\wedge dy[/tex] and you can split up the wedge anyway you'd like to, to make two covectors whose wedge gives the same result as F.

Now on [tex]R^3[/tex] are a little more complicated and I have a question about that as well. If I look at [tex]F(v,w)[/tex] where [tex]v,w\in R^3[/tex] and expand it in a basis [tex]e_i[/tex] I get what you'd expect, a cross product looking expression. Now if I also take the wedge product of two 1-forms on [tex]R^3[/tex], say [tex]\alpha,\beta[/tex] where [tex]\alpha = \alpha_idx^i[/tex] and [tex]\beta=\beta_idx^i[/tex] I get a 2-form with the coefficients you'd expect (they look like the components from a cross product).

Now the expansion of the ABF, F, using the basis [tex]e_i[/tex]gives me that cross product looking thing, but with some additional constants which I denote as:

[tex]F(e_i,e_j)=F_{ij}[/tex]. If I wish to take that ABF and associate it to a product of covectors I thought I could just equate coeffcients and solve. If I do that I get the following system:

[tex]\alpha_1\beta_2-\alpha_2\beta_1=F_{12}[/tex]
[tex]\alpha_2\beta_3-\alpha_3\beta_2=F_{23}[/tex]
[tex]\alpha_1\beta_3-\alpha_3\beta_1=F_{13}[/tex]
If i suppose that I pick my [tex]\alpha_k[/tex] first I get a matrix that's inconsistent. But of course I'm doing something stupid here because you're supposed to be able to associate ABF with wedge products of covectors in [tex]R^3[/tex].

So as you can see, I'm lost. I'd appreciate some direction.

Many thanks,

Kevin
 

mathwonk

Science Advisor
Homework Helper
10,733
912
this is discussed in the l;ittle book once read communaly here by david bachman, possibly still available free or at least the running commentray here on all the exercieses. ask tom mattson.
 
306
1
Thanks Math Wonk. I've read it and see now an example of a 2-form which is not a wedge of two covectors and I also see how I had gone awry in my method of associating 2-forms to wedges.

One last point, it seems as though 2-forms are "identical" to antisymmetric bilinear forms?

Thanks for your help,

Kevin
 
28
2
Thanks Math Wonk. I've read it and see now an example of a 2-form which is not a wedge of two covectors and I also see how I had gone awry in my method of associating 2-forms to wedges.

One last point, it seems as though 2-forms are "identical" to antisymmetric bilinear forms?

Thanks for your help,

Kevin
Yes, 2-forms means the same as antisymmetric bilinear forms; 3-forms means antisymmetric trilinear forms, etc.

As for the question of representing a 2-form as a wedge product of two 1-forms. As far as I understand, one can make a general statement that it is sufficient to write n/2 wedge products to represent a 2-form in n-dimensional space. So in four dimensions any 2-form can be written as [tex]a\wedge b+c\wedge d[/tex]. But I can't remember how this is proved. I seem to remember that this is called "Darboux theory" but I'm not sure any more.
 

mathwonk

Science Advisor
Homework Helper
10,733
912
the phrase "n-form" has more than one meaning depending on the author.

to me it means a field of antisymmetric multilinear functions, but as defined above and in bachman, the usage is for just one of them.
 

Related Threads for: Bilinear forms and wedge products

  • Posted
Replies
2
Views
811
Replies
6
Views
4K
  • Posted
Replies
8
Views
3K
  • Posted
2
Replies
32
Views
8K
  • Posted
Replies
4
Views
625
  • Posted
Replies
11
Views
5K
Replies
3
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top