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In the other book it is defined as all darivations on all ****tions from the manifold at the point to R. Then they define the tangent of a curve in the manifold to be and operator. And then they define the tangent to the ith coordinate curve of gamma, and proof a theorem about that these coordinate tangents at a point m on the manifold is a basis for all tangents.

The proof are not that hard, but my problem is that i can't see that the two definitions are the same when i choose one of the "manifolds" we worked on in do carmo, fx. the unit sphere in R^3. I now the one is a operator and the other are vectors, but there most be a clear connection?

I've seen an example where they use R^2 as there manifolds, and then state that the directional derivative corospones to the operators. And then it is clear that given and directional derivative in the direction v lets call it Dv, then i have the operator and the if i define a function f(Dv)=v then i get all tangent vectors to all points in R^2.

But somehow i can't find the connection when my surface gets more complicated, like the unit sphere. I can see that the operator is connected to the tangent of curves through m, but it is not so clear to me, in my mind there most be a function (possible a bijection) on the "tangent operators" such that i get all tanget vectors, when my manifold is so simple like the unit sphere that a "normal" tangent vector/plane makes sence. My point is that if it makes sence to define the tangents like operaors, I most have just as much information as if I had defined it the "usual" way (of cause the new defintion makes more sence when i have obscure manifolds where tangentplanes are not clear, but in R^3 there most be a clear connection in my head).

I hope someone can understand what my problem is, anbd have the commitment to read the whole post (it got a bit long, sorry). And by the way, i'm sorry for the bad englsih.