Calculating the Area of a Circle on S^2 in the Spherical Metric

In summary, to find the area of a circle on S^2 of radius p in the spherical metric, divide the circle into ring-shaped slices and integrate their areas. The area of each ring can be found using its thickness and the metric to calculate its perimeter. The sum of all the ring areas will give the total area of the circle, which is 2pi(1-cos p).
  • #1
I need help with this problem:

given a cirlce on S^2 of radius p in the spherical metric, show that its area is 2pi(1-cos p)
 
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  • #2
Welcome to PF!

Hi halvizo1031! Welcome to PF! :smile:

(have a pi: π and a rho: ρ and try using the X2 tag just above the Reply box :wink:)
halvizo1031 said:
given a cirlce on S^2 of radius p in the spherical metric, show that its area is 2pi(1-cos p)

Divide the circular region into ring-shaped slices of thickness ds, and integrate …

what do you get? :smile:
 
  • #3
I'm not sure I understand what you wrote.
 
  • #4
halvizo1031 said:
I'm not sure I understand what you wrote.

Divide the circle into rings …

the area of each ring is its thickness times its length (ie its perimeter) …

use the metric to find the length of the perimeter of each ring …

then add up the areas of all the rings :smile:
 
  • #5
ok I'll give that a try. thanks!
 

What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties and structures of smooth curved spaces. It involves the use of differential calculus and other mathematical tools to analyze and describe curved surfaces and higher dimensional spaces.

What are some applications of differential geometry?

One of the main applications of differential geometry is in the field of physics, particularly in the study of general relativity and the curvature of spacetime. It is also used in other areas such as computer graphics, robotics, and engineering for designing and analyzing curved structures and surfaces.

What is the difference between differential geometry and Euclidean geometry?

Differential geometry deals with curved spaces while Euclidean geometry deals with flat spaces. In differential geometry, the concepts of length, angle, and area are generalized to apply to curved surfaces, whereas in Euclidean geometry they are defined for flat surfaces.

What are some fundamental concepts in differential geometry?

Some fundamental concepts in differential geometry include differentiable manifolds, tangent spaces, vector fields, and Riemannian metrics. These concepts are used to study the properties of curves and surfaces in higher dimensional spaces.

Who are some notable mathematicians who have contributed to differential geometry?

Some notable mathematicians who have made significant contributions to differential geometry include Carl Friedrich Gauss, Georg Friedrich Bernhard Riemann, and Élie Cartan. More recent contributors include Shiing-Shen Chern, Marcel Berger, and Manfredo do Carmo.

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