Unable to understand vector derivative

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Discussion Overview

The discussion revolves around the concept of vector derivatives in different reference frames, particularly when those frames are in motion relative to each other. Participants explore the implications of time-dependent coordinate transformations and how they affect the differentiation of vectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about understanding vector derivatives with respect to different frames, specifically when the origins of the frames do not coincide.
  • Another participant explains that time-dependent vectors are differentiated according to their definitions and provides a mathematical expression for the derivative of a vector in one frame relative to another, noting that the non-coincidence of frame origins does not affect the derivative.
  • A third participant acknowledges a misunderstanding caused by a poorly written paper, suggesting that notation can lead to confusion.
  • A later reply questions the ambiguity in the original question regarding what is meant by the "derivative of the vector R in frame A," indicating a lack of clarity in the terminology used.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity of the original question, and there is uncertainty about the terminology used to describe vector derivatives in different frames.

Contextual Notes

There are limitations in the clarity of the original question, as well as potential misunderstandings arising from notation in external materials. The discussion does not resolve the ambiguity in the terminology used by the original poster.

GiuseppeR7
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Hi guys...i'm having a bad time understanding the concept of vector derivative with respect to different frames.
Suppose i have the vector displayed in the picture, the frame A and B are in motion with respect of each other, i can understand the concept of the derivative of the vector R in the frame A...but what does it mean the derivative of the vector with respect to B since the frame's origins are not coincident?
Thanks!
 

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Vectors that are time dependent are differentiated according to the definition.
If there is a time-dependent coordinate transformation between two frames of reference, that simply comes in:
So if frame A origin as described in frame B is at ##\vec A_0(t) ## and moves with velocity ##\vec v_A##,
and the derivative of ##\vec R## in frame A is ## \vec v_{\rm frame \ a} ## then $$
{d\,\vec R_{\rm frame\ b} \over dt} = { d \left ( \vec R(t) + \vec A_0(t) \right ) \over dt } = \vec v_{\rm frame \ a} + \vec v_A $$

(I do hope this isn't in a relativity course context ?)

That the frame origins are not coincident does not appear in the derivative. ##\vec A_0(0) ## is a constant and drops out. You can check that by writing out the definition in full.
 
Thanks for the reply...a bad paper misleaded me on this topic!
 
How did it manage to do so ?
 
it was using a confusing notation! :p
 
GiuseppeR7 said:
Hi guys...i'm having a bad time understanding the concept of vector derivative with respect to different frames.
Suppose i have the vector displayed in the picture, the frame A and B are in motion with respect of each other, i can understand the concept of the derivative of the vector R in the frame A...but what does it mean the derivative of the vector with respect to B since the frame's origins are not coincident?
You seem to have already solved your problem, however your question is ambiguous: what does "derivative of the vector R in the frame A" means? I know what is a "derivative of a vector with respect to time" or what is a "vector written using different systems of coordinates", but I don't know what it is what you have said.

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