- #1
orion
- 93
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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.
Consider a tangent space of a manifold M at point p. For simplicity, assume TpM is Euclidean and of dimension 2. I understand that {∂i} forms a basis of TpM 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 dxi form a basis of Tp*M, the dual of TpM. The Euclidean metric (for the two dimensional case) is expressed differentially as ds2=dx2+dy2.
This looks like length is formed from covectors in Tp*M and not vectors. Why is this?
Question 4: Consider the following definition:
dxi(∂j) = ∂jxi
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.)
Consider a tangent space of a manifold M at point p. For simplicity, assume TpM is Euclidean and of dimension 2. I understand that {∂i} forms a basis of TpM 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 dxi form a basis of Tp*M, the dual of TpM. The Euclidean metric (for the two dimensional case) is expressed differentially as ds2=dx2+dy2.
This looks like length is formed from covectors in Tp*M and not vectors. Why is this?
Question 4: Consider the following definition:
dxi(∂j) = ∂jxi
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.)