Understanding the Equality in Equation 2.96: Volume Element as n-Form?

In summary, the conversation discusses the meaning of equation 2.96 in Carroll's text on Integration and how it relates to the wedge product. It is stated that the formula (1.81) in Carroll's "Lecture Notes on General Relativity" (1997) explains the relationship between the wedge product and the tensor product. The conversation also mentions formula (1.80) in the edition of Carroll's text and how it can be used to derive the equation in question.
  • #1
LAHLH
409
1
Hi,

I'm reading Sean Carroll's text, ch2, and believe I understood most of the discussion on Integration in section 2.10. However in equation 2.96, he states that

[tex] \epsilon\equiv\epsilon_{\mu_1\mu_2...\mu_n}dx^{\mu_1}\otimes dx^{\mu_2}\otimes...\otimes dx^{\mu_n} =\frac{1}{n!}\epsilon_{\mu_1\mu_2...\mu_n}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge...\wedge dx^{\mu_n}[/tex]

I don't quite understand this equality. For example just taking n=2, the LHS is [tex] dx^0\otimes dx^1-dx^1\otimes dx^0[/tex] (which isn't zero because tensor products don't commute). Where on the RHS one would have [tex] \frac{1}{2}\left(dx^0\wedge dx^1-dx^1\wedge dx^0\right)=dx^0\wedge dx^1[/tex], by the antisymmetry of the wedge product.

So I'm at a loss to understand this part of 2.96 despite understand the following lines.

Thanks a lot for any replies.
 
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  • #2
Formula (1.81) in Carroll's "Lecture Notes on General Relativity" (1997) tells you:

[tex]A\wedge B=A\otimes B-B\otimes A[/tex]

So, what is it that causes you the problem?
 
  • #3
Oh I was unaware of this formula.

Do you mean (1.80) in this edition of Carroll? namely [tex] (A\wedge B)_{\mu\nu}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}[/tex]

Which I guess can be written [tex] (A\wedge B)_{\mu\nu} =(A\otimes B)_{\mu\nu}-(B\otimes A)_{\mu\nu}[/tex], leading to the equation you stated: [tex] (A\wedge B) =(A\otimes B)-(B\otimes A)[/tex]
 
  • #4
Yes - that is another way of writing it.
 

1. What is a volume element as n-form?

A volume element as n-form is a mathematical concept used in multivariable calculus to represent the infinitesimal volume of a n-dimensional space. It is essentially a differential form that can be integrated over a region of space to calculate the volume of that region.

2. How is a volume element as n-form related to differential forms?

A volume element as n-form is a type of differential form. Differential forms are mathematical objects that are used to represent various geometric concepts in higher dimensions. Volume elements as n-forms are a specific type of differential form that represents the infinitesimal volume of n-dimensional space.

3. What is the significance of using a volume element as n-form?

Using a volume element as n-form allows for the calculation of volume in spaces with more than three dimensions. It also allows for the integration of more complex functions over these higher-dimensional spaces.

4. How is a volume element as n-form calculated?

A volume element as n-form is calculated by taking the determinant of the Jacobian matrix of the transformation from the n-dimensional space to a coordinate system in which the volume element is a simple product of differentials.

5. Can a volume element as n-form be used in physics and engineering?

Yes, volume elements as n-forms have many applications in physics and engineering. They are used in various fields, such as electromagnetism, fluid mechanics, and general relativity, to calculate volumes and perform integrations in higher-dimensional spaces.

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