Do Coordinate Charts Alone Identify a Unit Circle?

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SUMMARY

The discussion centers on the identification of a unit circle using coordinate charts, specifically the equation x² + y² = 1. The unit circle is represented through four coordinate charts, with the first chart defined as U₁: x > 0 and A₁ = y. Participants express uncertainty about whether these charts alone can conclusively indicate a unit circle without additional context or functions. It is established that while the charts can cover the circle, they may also apply to different manifolds, necessitating further specification of functions for clarity.

PREREQUISITES
  • Understanding of coordinate charts in topology
  • Familiarity with the equation of a unit circle (x² + y² = 1)
  • Knowledge of manifold theory
  • Basic concepts of intersections in mathematical functions
NEXT STEPS
  • Research the properties of coordinate charts in topology
  • Study the implications of manifold intersections
  • Learn about the role of functions in defining geometric shapes
  • Explore alternative representations of the unit circle in different coordinate systems
USEFUL FOR

Mathematicians, students of topology, and anyone interested in the geometric representation of functions and manifolds will benefit from this discussion.

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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