The tangent space at a point 
in an abstract manifold 
can be described without the use of
embeddings or coordinate charts. The elements
of the tangent space are called tangent vectors, and the collection of tangent spaces
forms the tangent bundle.
One description is to put an equivalence relation on smooth paths through the point 
.
More precisely, consider all smooth maps 
where 
and 
. We say that two maps 
and 
are equivalent if they agree to first order. That is, in any
coordinate chart around 
, 
. If they are similar in one chart then they are
similar in any other chart, by the chain rule. The
notion of agreeing to first order depends on coordinate charts, but this cannot be
completely eliminated since that is how manifolds are defined.
Another way is to first define a vector field as a derivation of the ring of smooth functions 
.
Then a tangent vector at a point 
is an equivalence class of vector fields which agree at 
.
That is, 
if 
for every smooth function 
. Of course, the tangent space at 
is the vector space of tangent vectors at 
. The only drawback to this version is that a coordinate
chart is required to show that the tangent space is an 
-dimensional vector space.
