Conservative vector field; classification of derivative

[Prz]

Dear forum-members,

Pestered by many (in my opinion, fundamental) questions and no literature at hand to answer them, I resort to posing my questions here. Let me start with the following. (Hopefully I have the correct subsection.)

I am inspecting a dynamical, autonomous and conservative system driven by a 'conservative' vector-field:
$$\left\|\dot{\mathbf{x}}\right\|_1 = 0$$
(No need to specify this any further).
If we start off with
$$\left\|\mathbf{x}\right\|_1 = n$$
Then the system we inspect is a vector-field on an n-simplex.

To be honest I am in doubt about this being a proper manifold, since the tangent has to abide the conservation constraints, while the neighborhood of any point in the state space is not perfectly Euclidean.
In line with this: how accurate is a Taylor-expansion of a subspace cut off at the m-th term?

Any response would be very much appreciated!

 

[Prz]

Let me address one topic at a time, and let me put the first issue differently.

Is a space
$$\Delta^r = \{ \mathbf{y} \; | \mathbf{y} \in R^r, \sum_i y_i = n \; \}$$
a proper manifold?