I'm trying to demonstrate the following proposition:(adsbygoogle = window.adsbygoogle || []).push({});

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?.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Differential Geometry problem

**Physics Forums | Science Articles, Homework Help, Discussion**