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]

$$f(\vec{a} + \vec{x}) = f(\vec{a}) + (\vec{x} \cdot \nabla)f(\vec{a}) + \frac{1}{2} (\vec{x} \cdot \nabla)^2f(\vec{a}) + \cdots$$


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.