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Math Help - Proof of uniqueness part of the Fundamental Theorem of the local theory of curves

  1. #1
    Senior Member x3bnm's Avatar
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    Proof of uniqueness part of the Fundamental Theorem of the local theory of curves

    In DoCarmo's Differential Geometry book there's an excerpt (on page 19-20) like this:


    ------------------------------Beginning of Excerpt------------------------------------------------------------------------
    Fundamental theorem of the local theory of curves:

    Given differentiable functions k(s) > 0 and \tau (s), where s \in I, there exists a regular parametrized curve \alpha \colon I \rightarrow \mathrm{R^3}
    such that s is the arc length, k(s) is the curvature, and \tau(s) is the torsion of \alpha. Moreover, any other curve \bar{\alpha}, satisfying the
    same conditions, differs from \alpha by a rigid motion; that is, there exists an orthogonal linear map \rho of \mathrm{R^3}, with positive determinant,
    and a vector c such that \bar{\alpha} = \rho \circ \alpha + c.


    .........

    Proof of the Uniqueness Part of the Fundamental Theorem:

    We first remark that arc length, curvature, and torsion are invariant under rigid motions; that means, for instance, that if M \colon \mathrm{R^3} \rightarrow \mathrm{R^3}
    is a rigid motion and \alpha = \alpha(t) is a parametrized curve, then

     \int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt

    That is plausible, since these concepts are defined by using inner or vector products of certain derivatives (the derivatives are invariant under translations, and the
    inner and vector products are expressed by means of lengths and angles of vectors, and thus also invariant under rigid motions).
    -------------------------------End of Excerpt------------------------------------------------------------------------

    My question is:
    why's that
     \int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt ?


    How did the author prove this? Is it possible to help me prove this? I'm completely lost at the last line of the excerpt given above.
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  2. #2
    Senior Member x3bnm's Avatar
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    Re: Proof of uniqueness part of the Fundamental Theorem of the local theory of curves

    I found the solution at this website on page 8:

    http://www.math.uregina.ca/~mareal/cs2.pdf


    The proof is something like this:

    To prove the uniqueness part, we choose the linear orthogonal transformation M which maps the Frenet fame of \alpha at c to the Frenet frame of \beta at c.

    Note that both frames are orthonormal systems of vectors with the last vector equal to the vector product of the previous two: thus the transformation M mentioned above exists and moreover, has
    \text{det}(M) = 1 \neq 0.

    We have that \beta'(s) = M\alpha'(s), has unique solution for all s, by using a uniqueness property of the solutions of the system.[Q.E.D]
    Last edited by x3bnm; May 20th 2012 at 01:57 PM.
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