How Does the Alt Operator Apply to a Differential 2-Form in R^4?

[arthurhenry]

Homework Statement

Suppose the standard coordinates in R^4 are x,y,x,w.
We have a differential 2-form a= z dx \wedge dy.
Trying to evaluate Alt(a)

I am trying to see this form as a bilinear form that acts on a pair of vectors so that I can apply the Alt operator formula. I am able understand the formula for the operator that makes a form alternating (i.e. The Alt operator), but I cannot apply it in this case.

Homework Equations

I bet I would be able to approach this problem one way or the other if I had found some source that can explain where these notions originate...originate in the sense: I see them in a math book and I would like to know why people needed this definition and to address what, etc. I tried reading different sources, but I am still here. (I have looked at David Bachmann's notes, they are very good, but I am still missing something)

The Attempt at a Solution

I am trying to see this form as a bilinear form that acts on a pair of vectors so that I can apply the Alt operator formula. I am able understand the formula for the operator that makes a form alternating (i.e. The Alt operator), but I cannot apply it in this case.

Thank you for your time

 

[mark726]

and effort in trying to understand this problem. It is clear that you have put in a lot of effort to try and find a solution. Let me try to explain the concept of alternating forms and the Alt operator in a bit more detail.

Firstly, a differential 2-form is a type of mathematical object that takes in two vectors as inputs and produces a real number as an output. In this case, the 2-form "a" takes in two vectors (dx and dy) and produces a real number (z) as an output.

Now, the Alt operator is a mathematical operation that takes in a differential k-form (in this case, a 2-form) and produces a new differential k-form that is "alternating" in its inputs. This means that if you swap the inputs of the new form, the output will also change sign. In other words, the Alt operator ensures that the form is "antisymmetric" in its inputs.

To apply the Alt operator to the 2-form a= z dx \wedge dy, we need to understand how it acts on our inputs (dx and dy). In this case, the Alt operator acts on the wedge product (denoted by \wedge) of dx and dy, which is a special type of multiplication operation for forms. The result of this action is a new differential 2-form that is alternating in its inputs.

To evaluate Alt(a), we first need to understand the wedge product of dx and dy. This operation produces a new form that is defined as follows:

(dx \wedge dy)(v,w) = dx(v) * dy(w) - dx(w) * dy(v)

where v and w are vectors. So, for example, if we have dx = xdx + ydy + zdz and dy = xdx + wdw, then the wedge product would be:

(dx \wedge dy) = (x^2 - yw) dx \wedge dy \wedge dz

Now, to apply the Alt operator, we need to swap the inputs of this new form. So, if we swap dx and dy, we would get:

(dx \wedge dy)(w,v) = dx(w) * dy(v) - dx(v) * dy(w)

= xdx(w) * xdy(v) + ydy(w) * xdx(v) + zdz(w) * xdy(v) + xdx(w) * wdy(v)