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Conservative vector field; classification of derivative

  1. Aug 11, 2008 #1


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    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:
    [tex]\left\|\dot{\mathbf{x}}\right\|_1 = 0[/tex]
    (No need to specify this any further).
    If we start off with
    [tex]\left\|\mathbf{x}\right\|_1 = n[/tex]
    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!
  2. jcsd
  3. Aug 12, 2008 #2


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

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