Do Coordinate Charts Alone Identify a Unit Circle?

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Discussion Overview

The discussion revolves around the identification of a unit circle defined by the equation x² + y² = 1 through the use of coordinate charts. Participants explore whether the four coordinate charts alone are sufficient to indicate that they refer specifically to a unit circle, or if additional information is necessary.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the four coordinate charts alone can demonstrate that they refer to a unit circle, suggesting that extra information may be required.
  • Another participant expresses confusion about the notation used, specifically the meaning of A = y, and seeks clarification.
  • A later reply notes that while four charts can cover the circle, they could also intersect in ways that pertain to different manifolds, indicating that the charts alone do not uniquely identify the unit circle.
  • Another participant agrees that the function or inverse function needs to be specified to clarify the reference to the unit circle.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the four coordinate charts alone can identify the unit circle, with multiple viewpoints and uncertainties expressed regarding the necessity of additional information.

Contextual Notes

Participants highlight the potential for different intersections of charts on various manifolds, suggesting that the identification of the unit circle may depend on specific definitions or additional context.

dyn
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Hi. I have been looking at the coordinate charts for the unit circle x^2 + y^2 = 1. In the notes I have the circle is split into 4 coordinate charts the first being -

##U_1## : x>0 , ##A_1## = y (PS without the symbols tab I have used A for the letter phi )

There are 3 more similar charts. My question is - do the 4 charts on their own show that they are referring to a unit circle ? I can't see that they do without providing extra information
 
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dyn said:
Hi. I have been looking at the coordinate charts for the unit circle x^2 + y^2 = 1. In the notes I have the circle is split into 4 coordinate charts the first being -

##U_1## : x>0 , ##A_1## = y (PS without the symbols tab I have used A for the letter phi )

There are 3 more similar charts. My question is - do the 4 charts on their own show that they are referring to a unit circle ? I can't see that they do without providing extra information
Is there anything else that you can add that could help get you an answer?
 
dyn said:
Hi. I have been looking at the coordinate charts for the unit circle x^2 + y^2 = 1. In the notes I have the circle is split into 4 coordinate charts the first being -

##U_1## : x>0 , ##A_1## = y (PS without the symbols tab I have used A for the letter phi )

There are 3 more similar charts. My question is - do the 4 charts on their own show that they are referring to a unit circle ? I can't see that they do without providing extra information

Dont understand what A = y means. Can you explain that?The circle can be covered by four charts which intersect in connected segments. If one keeps track of the intersections then one can show that one has a circle. But four charts could intersect in a different way on a different manifold. Also you can cover the circle with four charts in more than one way.

Hope that helps.
 
Last edited:
φ1 = y is how it should read. What I don't understand is how or if the 4 coordinate charts alone specifically refer only to the unit circle. It seems to me the function or inverse function needs to be specified as well
 

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