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Thread: surface of revolution

  1. #1
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    surface of revolution

    Prove that the rotation of a surface of revolution S about its axis are diffeomorphisms of S.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: surface of revolution

    A rotation in space is given by $\displaystyle y=Ax$, where A is orthogonal with determinant 1.
    A rotation of the surface $\displaystyle X(u,v),(u,v)\in D$ is given by $\displaystyle Y(u,v)=AX(u,v),(u,v)\in D$.
    Prove that $\displaystyle S=\{X(u,v),(u,v)\in D\}$ and $\displaystyle R=\{Y(u,v),(u,v)\in D\}$ are diffeomorphic,
    by proving that the map $\displaystyle \phi:S\rightarrow R, p=X(u,v)\rightarrow q=Y(u,v)$ is differentiable
    with full rank.
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