I’ve recently learnt what manifolds are to prepare myself for a course in GR. My relevant mathematical background is linear algebra (abstract, proof-ish course) and multivariable/vector calculus (course mostly focussed on computations). I use Knuth and Renteln as my main references.

The idea of an n-dimensional manifold is introduced as a combination of open sets whose union forms the manifold. Each such open set must have a continuous 1-to-1 map to an open set in n-dimensional Euclidean space; that is: each point within these open sets can be described as an n-tuple, just like vectors and points in ”regular” space can. Please correct me if I’m wrong here, I really want to get my definitions right.

Subsequently the tangent space is introduced as a vector space for each point on the manifold, whose elements are differential operators. I recognise that that this vector space is very much usable to describe vector fields on the manifold. 

But as I read the definition of manifolds, I intuitively expected the displacement between points to be a good definition for vectors on my manifold. If our manifold consists of multiple charts, this would not be possible, since the space would not be closed: if we added the displacement between points to itself long enough we would eventually “exit” the open set in which the tuples are defined, which is by definition not possible. I’m still interested in knowing why in general relativity we use tangent spaces instead of conventional classic coordinate tuples.

**So my first question is: if a manifold is describable by a single chart, can we define a vector space simply by taking displacements between points on that chart as n-tuple vectors?** For instance, take the manifold $\mathbf{r} = (x,y,z)^T = (f\_1(u,v), f\_2(u,v), f\_3(u,v))^T$: a 2D-manifold embedded in 3D space. If the functions $f\_i$ are well behaved so that the manifold is smooth and has no sharp edges/crossings/etc, wouldn’t $(u,v)$ tuples form a nice vector space? You could add them, scale them, and fulfil all the properties of a vector space.

My second question is more to manage my own expectations for later: **is this single-coordinate chart situation one that occurs at all in G.R.?** If each/most manifold(s) in G.R. naturally requires two or more charts, then it would be senseless to take these displacement tuples as vectors. 
The reason I still ask is because I intuitively expected that a single chart should encompass all of space; I would be very surprised if I was working out a physics problem and I couldn’t describe my worldline in the same coordinate system everywhere.