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Mulitvariable Calculus Four Dimensional Sphere

  1. Apr 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Use double polar coordinates x=rcosθ y=rsinθ z=scosφ w=ssinφ in R^4
    to compute the 4-dimensional volume of the ball x^2 +y^2 +z^2+w^2 = R^2

    2. The attempt at a solution
    I first substituted the polar coordinates into the given equation getting: r^2cos^2(θ)+r^2sin^2(θ)+s^2cos^2(φ)+r^2sin^2(φ) = R^2
    this simplified to r^2 + s^2 = R^2 using the identity: cos^2(θ)+sin^2(θ)=1
    After this I'm confused with where to go. Any tips would help, thanks.
    Last edited: Apr 29, 2014
  2. jcsd
  3. Apr 30, 2014 #2


    Staff: Mentor

    To find the volume I think you must setup a 4D integral similar to how one finds the volume of a 3D sphere:

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