The intrinsic tangent space at a point 
in a smooth 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, for
a 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.
Equivalently, a tangent vector at 
is a derivation at 
: a real linear functional 
satisfying
 |
Here 
denotes the algebra of smooth
functions 
.
A smooth curve 
through 
defines this operation by 
, and every such derivation
comes from a unique tangent vector (Meinrenken
2002, pp. 9-11).
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.
