jeo23

I have decided to attempt to pick up some differential geometry on my own, and I am trying to get some traction on the subject which I do by trying to reduce it to familiar and simple cases.

This is a trivial case, I know, but it will go a long way in advancing my understanding. Suppose the manifold of interest is the surface of a 2d-sphere (embedded in a 3d Euclidean space) and consider the tangent plane at point p. According to elementary differential geometry, a basis for this tangent space is [itex]\partial[/itex]_{[itex]\mu[/itex]}.

Now according to elementary linear algebra, one basis for this space would be:

B= { [1 0], [0 1] }. These are written as row vectors because I don't know how to write them as columns here.

My question: How do I get the B basis from the partial derivative basis from differential geometry?

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Another question that I have comes about when I get confused because the differential geometry formalism is introduced talking about a paramaterized curve in a manifold. What is the relationship between this paramaterized curve and the coordinate curves?