Finding position vector in general local basis

[farfromdaijoubu]

How do you derive the position vector in a general local basis?

For example, in spherical coordinates, it's $\vec r =r \hat {\mathbf e_r}$, not an expression that involves that involves the vectors ${\hat {\mathbf e_{\theta}}}$ and $\hat {{\mathbf e_{\phi}}}$. But how would you show this?

 

[pasmith]

This follows from the definition of the coordinate system. For example, in spherical polar coordinates, $r$ is by definition the distance of a point from the origin and $\mathbf{e}_r(\theta,\phi)$ is the unit vector in the direction of that point. Hence $\mathbf{r} = r\mathbf{e}_r(\theta,\phi)$.

Otherwise, if you have a global Cartesian basis then you can express the cartesian coordinates and basis vectors in terms of the curvilinear coordinates and basis vectors.