1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Establishing a smooth differential structure on the ellipsoid

  1. Jan 4, 2013 #1
    1. The problem statement, all variables and given/known data
    Construct a C natural differential structure on the ellipsoid

    [itex]\left\{(x_{1}, x_{2}, x_{3})\in E | \frac{x_{1}^{2}}{a^{2}}+\frac{x_{2}^{2}}{b^{2}}+ \frac{x_{3}^{2}}{c^{2}}=1\right\}[/itex]

    Is this diffeomorphic to S2? Explain.

    2. Relevant equations

    Do I need to prove homeomorphism for my functions mapping E to ℝ2?
    How to/ do I need to prove smoothness for my coordinate transformations, and my diffeomorphism to S2? Are my charts valid?, I use one stereographic projection chart for the ellipsoid minus 1 point, then a "drop the z coordinate" mapping for the top half including the point I missed.

    3. The attempt at a solution
    Here are my charts,

    [itex]U = E-{(0,0,c)}[/itex]
    [itex]φ(x_{1},x_{2},x_{3}) = (\frac{x_{1}}{c-x_{3}},\frac{x_{2}}{c-x_{3}},0)[/itex]

    [itex]V = \left\{(x_{1}, x_{2}, x_{3})\in V | x_{3}>0\right\}[/itex]
    [itex]φ(x_{1},x_{2},x_{3}) = (x_{1},x_{2},0)[/itex]

    To be a differential structure, the coordinate transformation must be smooth
    [itex]φψ^{-1}:ψ(U\cap V)\rightarrowℝ^{2}[/itex]
    It is pretty clear to me these charts are smooth for the values it would need to operate on, need to prove?

    This is where it gets dicey, I need to find a smooth mapping from the ellipsoid to the 2 sphere, will I need multiple charts, here is one for the positive coordinates.
    [itex]F = \left\{(x_{1}, x_{2}, x_{3})\in F | x_{3}>0\right\}[/itex]
    [itex]f(x_{1},x_{2},x_{3}) = (a^{2}x_{1}^{2}, b^{2}x_{2}^{2}, c^{2}x_{3}^{2})[/itex]
    [itex]f(F) = \left\{(x_{1}, x_{2}, x_{3})\in S^{2} | x_{3}>0\right\}[/itex]

    So, am I on the right track to construct this diffeomorphism?
    Last edited: Jan 4, 2013
  2. jcsd
  3. Jan 5, 2013 #2
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook