Do Coordinate Charts Alone Identify a Unit Circle?

[dyn]

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

 

[Evo]

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?

 

[lavinia]

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.

 

[dyn]

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