How to Expand a Vectorial Function into a Taylor Series?

[elessar_telkontar]

I'm trying to demonstrate the following proposition:

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

\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)

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

I understand this is demonstrated by expanding \vec{\alpha}(s+h) 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 \vec{\alpha}(s). Then please help me saying:

HOW TO EXPAND THE VECTORIAL FUNCTION IN ORDEN TO GET THE RESULT?.

 

[elessar_telkontar]

Please try to help me!

 

[Hurkyl]

<br /> 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<br />


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