Hi everyone I am reading Sean Carrol's lecture notes on general relativity.(adsbygoogle = window.adsbygoogle || []).push({});

link to lecture : https://arxiv.org/abs/gr-qc/9712019

In his lecture he introduced dx^{μ}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 dx^{μ}. I have several confusion with the 1 form basis.

1. Is the 1 form basis dx^{μ}really have a meaning for infinitesimal change in x^{μ}?

2. How to convince ourselves that dx^{μ}(dx^{ν})=δ^{μ}_{ν}?

3. Am I correct to understand the formalism of 1 form like this :

Given that a. ) df = ∂_{μ}f dx^{μ}from vector calculus,

b.) ∂_{μ}f is identified to be component of 1 form because it transforms covariently.

Therefore we realised df is a 1-form with dx^{μ}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 dx^{μ}. I also appreciate anyone to explain why dx^{μ}can be a 1 form basis.

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# I Why denote 1 form as dx?

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