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Area of curved space

  • Thread starter Niles
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  • #1
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[SOLVED] Area of curved space

Homework Statement


I have the following metric, which describes a 2D, positive curved space with flat geometry (ie. a sphere):


[tex]

ds^2\,=\,dr^2\,+\,R^2 \sin ^2 (r/R)d\theta ^2

[/tex]

Here ds is the distance between two points (r, theta) and (r + dr, theta + dtheta), R is the radius of the sphere.

I want to find the area of the sphere using this metric.

The Attempt at a Solution


Using the metric, I have found the circumference of the sphere to be 2*Pi*R (big surprise). Now I want sum up "all the circumferences" on the sphere. Is that possible?
 

Answers and Replies

  • #2
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I solved it!

When finding the circumference, I must find all the other circumferences, and r is in the interval [0; Pi*R/2]. Just multiply with 2 in the end (since we only found the first half), and you're set!
 

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