I have some questions about the definition of tangent space that arose after reading the book Differential Geometry of Curves and Surfaces of Manfredo do Carmo.
First, what's the best way to define tangent space? Using tangent vector to curves or derivations?
I've researched about the concept of derivation but it seems to require knowledge of abstract algebra, is is possible to understand derivations without knowing anything of abstract algebra?
Can someome recommend some material about this topic? And if it's really needed to understand derivations, can someone recommend also some material about derivations?
Thanks in advance.