# Volume element as n-form

by LAHLH
Tags: element, nform, volume
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 P: 411 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 $$\epsilon\equiv\epsilon_{\mu_1\mu_2...\mu_n}dx^{\mu_1}\otimes dx^{\mu_2}\otimes...\otimes dx^{\mu_n} =\frac{1}{n!}\epsilon_{\mu_1\mu_2...\mu_n}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge...\wedge dx^{\mu_n}$$ I don't quite understand this equality. For example just taking n=2, the LHS is $$dx^0\otimes dx^1-dx^1\otimes dx^0$$ (which isn't zero because tensor products don't commute). Where on the RHS one would have $$\frac{1}{2}\left(dx^0\wedge dx^1-dx^1\wedge dx^0\right)=dx^0\wedge dx^1$$, 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 alot for any replies.
 P: 1,411 Formula (1.81) in Carroll's "Lecture Notes on General Relativity" (1997) tells you: $$A\wedge B=A\otimes B-B\otimes A$$ So, what is it that causes you the problem?
 P: 411 Oh I was unaware of this formula. Do you mean (1.80) in this edition of Carroll? namely $$(A\wedge B)_{\mu\nu}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$ Which I guess can be written $$(A\wedge B)_{\mu\nu} =(A\otimes B)_{\mu\nu}-(B\otimes A)_{\mu\nu}$$, leading to the equation you stated: $$(A\wedge B) =(A\otimes B)-(B\otimes A)$$
 P: 1,411 Volume element as n-form Yes - that is another way of writing it.

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