I Finding position vector in general local basis

farfromdaijoubu
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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?
 
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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.
 
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