View Single Post
jeo23
jeo23 is offline
#1
Jan18-12, 02:16 PM
P: 6
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?

------------------------------------------------------------------------------------------

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?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes