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Volume of a 2-sphere on a 3-sphere

  1. Apr 16, 2012 #1
    1. The problem statement, all variables and given/known data

    On a 3-sphere of radius 1 defined using the coordinate system (r=1, χ,θ,[itex]\phi[/itex]) where chi and theta run from 0 to π and [itex]\phi[/itex] runs from 0 to 2π. This coordinate system therefore has the metric

    ds^2=dχ^2=[sin(χ)]^2*[dθ^2+[sin(θ)]^2*d[itex]\phi[/itex]^2]

    2. Relevant equations

    How do I find the surface area of a 2-sphere defined by χ=χ0?

    3. The attempt at a solution

    3. I think wht I need to do is take the integral of ds over theta and phi. dχ=0, which leaves me with the following integral:

    A=[sin(χ0)]^2 [itex]\int \sqrt{dθ^2+[sin(θ)]^2*d\phi^2}[/itex]

    I am not sure how to do this integral, as it does not even seem to be in the normal integral format. Am I going about this wrong? If not, how do I do this integral?
     
  2. jcsd
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