- #1
Silviu
- 624
- 11
Hello! I am reading something about differential geometry and I have that for a manifold M and a point ##p \in M## we denote ##\Omega_p^r(M)## the vector space of r-forms at p. Then they say that any ##\omega \in \Omega_p^r(M)## can be expanded in terms of wedge products of one-forms at p i.e. ##\omega =\frac{1}{r!}\omega_{\mu_1 ...\mu_r}dx^\mu_1 \wedge dx^\mu_2 ... \wedge dx^{\mu_r}##, with ##\omega_{\mu_1 ...\mu_r}## completely antisymmetric. I am not sure why. If I have a 2-form, that would be ##\omega=\omega_{\mu\nu}dx^\mu dx^\nu##, but the ##\omega_{\mu\nu}## and ##\omega_{\nu\mu}## don't need to me in any relationship (equal or opposite), so how can I use the wedge product which would basically contain ##dx^\mu dx^\nu-dx^\nu dx^\mu## to obtain it? Thank you!