I am trying to self-study some concepts in differential geometry to try to update my knowledge from the old-style index games to something more meaningful. I know that there are many threads that have in some way addressed this, but I am still not understanding it completely. I'm new to this, and I could be way off base so please bear with me as I am feeling my way in the dark. I'm looking for more conceptual help so rigor is not that important.(adsbygoogle = window.adsbygoogle || []).push({});

Consider a tangent space of a manifold M at point p. For simplicity, assume T_{p}M is Euclidean and of dimension 2. I understand that {∂_{i}} forms a basis of T_{p}M for some coordinate chart around p.

Question 1:What does the dot product between two vectors in this space look like when the basis vectors are partial derivatives?

Question 2:How can I show that the ∂_{i}form a basis?

Question 3:According to what I have read, the dx_{i}form a basis of T_{p}*M, the dual of T_{p}M. The Euclidean metric (for the two dimensional case) is expressed differentially as ds^{2}=dx^{2}+dy^{2}.

This looks like length is formed from covectors in T_{p}*M and not vectors. Why is this?

Question 4:Consider the following definition:

dx_{i}(∂_{j}) = ∂_{j}x_{i}

I realize that this is a definition, but is there any background to this definition? Where did it come from? (I have a suspicion that the directional derivative has something to do with it, but I'm not there yet.)

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# Yet more elementary questions about the tangent space

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