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 alot for any replies.
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?
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]