I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" [Broken]document and the definition directly after. My first question is, is yi(t) many different paths on the manifold or is each i a component of one path on the manifold? Also in the definition after the example I think yj are the ambient components of the actual manifold, not the path, and xi are the local components of the manifold, not the path. Is this correct? And if so how do you take the partials if the number of components of the ambient coordinates are usually one more then the number of components of the local coordinates? If my questions do not make sense, just ask and I will try to make them more clear, and if this is not the right place for this just direct me where to go. Also if anyone had any better suggestions on places to learn this stuff I am happy to hear it.