# Rigid motion in curvilinear coordinates

I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis E={$e_1,e_2,e_3$}. Now also consider new curvilinear coordinates ($y_1,y_2,y_3$) such that $y_i=f_i(x_1,x_2,x_3)$. To each point y there is a local covariant basis $g_i=\frac{\partial x_j}{\partial y_i}e_j$ (summation convention in use. In matrix notation just consider G=J E, with J a $3\times 3$ matrix). The equation relates the Euclidean frame and the local frame.

First question: It is my understanding that vectors live on the tangent space of the base space at point y. Thus a vector at y (or the respective x) lives in a different space than a vector at the origin. In other words, the local basis and the Euclidean basis are defined at different points. How can then, they be associated? i.e. how can G=J E hold since the two frames are defined at different points?

(In a flat space space, one could transport parallely the E frame, in a very trivial way, from the origin to the point x and then associated the two frames at the same point. In general manifolds one should define a connection.)

Second question: Why does this matter? Because in order to describe a rigid motion, you also need the coordinates of the origin i.e. you need an affine space. Thus, should the relation between the E frame (at the origin) and the local frame G (at x) be something like:

$$g_i=\frac{\partial x_j}{\partial y_i}e_j'$$

where,

$$e_j'=e_j+T$$

that is, the transported E at the point x?

EDIT:

Regarding my second question, should the transformation from global to local coordinates be defined as, $$\begin{bmatrix} u_1\\ u_2\\ u_3\\ 0 \end{bmatrix}=\begin{bmatrix} J & \mathbf{x}\\ \mathbf{0}& 1 \end{bmatrix}\begin{bmatrix} v_1\\ v_2\\ v_3\\ 0 \end{bmatrix}$$

where $v_i$ are the components of a vector located at x in the global frame , and $u$ the components in the local frame at x, and $J=[\frac{\partial x_j}{\partial y_i}]$ (i.e. G=J E), using of course homogenous coordinates.

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