Hi everyone I am reading Sean Carrol's lecture notes on general relativity. link to lecture : https://arxiv.org/abs/gr-qc/9712019 In his lecture he introduced dxμ as the coordinate basis of 1 form and ∂μ as the basis of vectors. I understand why ∂μ could be the basis of the vectors but not for the dxμ. I have several confusion with the 1 form basis. 1. Is the 1 form basis dxμ really have a meaning for infinitesimal change in xμ? 2. How to convince ourselves that dxμ(dxν)=δμν? 3. Am I correct to understand the formalism of 1 form like this : Given that a. ) df = ∂μf dxμ from vector calculus, b.) ∂μf is identified to be component of 1 form because it transforms covariently. Therefore we realised df is a 1-form with dxμ as its basis. 4. (more general open question though) The most puzzling part for me is to understand the formalism of 1 form basis. I follows well in realising the basis vector as ∂μ not but for dxμ. I also appreciate anyone to explain why dxμ can be a 1 form basis.