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Differential Geometry problem

  1. Apr 20, 2005 #1
    I'm trying to demonstrate the following proposition:

    Let [tex]\vec{\alpha}(s)[/tex] be a natural parametrization of an arc C. Then:

    [tex]\vec{\alpha}(s+h)=\vec{\alpha}(s)+\left(h-\frac{\kappa^2h^3}{6}\right)\hat{t}+\frac{1}{2}\left(\kappa h^2+\frac{\left(\partial_s\kappa\right)h^3}{3}\right)\hat{n}+\frac{1}{6}\kappa\tau h^3 \hat{b}+O(h^4)[/tex]

    where [tex]\kappa[/tex] is the curvature, [tex]\tau[/tex] is the torsion, [tex]\hat{t}[/tex] is the unit tangent vector, [tex]\hat{b}[/tex] is the unit binormal vector and [tex]\hat{n}[/tex] is the unit normal vector.

    I understand this is demonstrated by expanding [tex]\vec{\alpha}(s+h)[/tex] in Taylor series. However, I don't know how to expand a vectorial function in Taylor series. Obviously, after expanding it's only matter of applying the Frenet ecuations to the derivatives of [tex]\vec{\alpha}(s)[/tex]. Then please help me saying:

    HOW TO EXPAND THE VECTORIAL FUNCTION IN ORDEN TO GET THE RESULT?.
     
    Last edited: Apr 21, 2005
  2. jcsd
  3. Apr 21, 2005 #2
    Please try to help me!
     
  4. Apr 21, 2005 #3

    Hurkyl

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    [tex]
    f(\vec{a} + \vec{x}) = f(\vec{a}) + (\vec{x} \cdot \nabla)f(\vec{a}) + \frac{1}{2} (\vec{x} \cdot \nabla)^2f(\vec{a}) + \cdots
    [/tex]


    One way of seeing this is to slice it down into a single-dimensional Taylor series. For example, after selecting [itex]\vec{a}[/itex] and [itex]\vec{x}[/itex], you can define [itex]g(t) = f(\vec{a} + t \vec{x})[/itex] which is a function of t alone, and find its Taylor series.
     
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