# 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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The text passage discusses the transformation between Euclidean and curvilinear coordinates in a 3D space, focusing on the relationship between the Euclidean basis and the local covariant basis. It poses two specific questions about the association of these bases and the transformation required for describing rigid motions.
multiple_questions

1. the first question asks how the euclidean basis and the local covariant basis can be associated when they are defined at different points.
2. the second question asks why this association matters and suggests a transformation involving the transport of the euclidean basis to the point of interest.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.

-
user
The text passage discusses the transformation between Euclidean and curvilinear coordinates in a 3D space, focusing on the relationship between the Euclidean basis and the local covariant basis. It poses two specific questions about the association of these bases and the transformation required for describing rigid motions.

TYPE: MULTIPLE_QUESTIONS

1. The first question asks how the Euclidean basis and the local covariant basis can be associated when they are defined at different points.
2. The second question asks why this association matters and suggests a transformation involving the transport of the Euclidean basis to the point of interest.