1. Not finding help here? Sign up for a free 30min 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!

Differential Geometry Theorem on Surfaces

  1. Jun 18, 2009 #1
    1. The problem statement, all variables and given/known data

    I am having difficulty understanding the proof of the following theorem from Differential Geometry

    Theorem

    [tex]S\subset \mathbb{R}^3[/tex] and assume [tex]\forall p\in S \exists p\in V\subset\mathbb{R}^3[/tex] [tex]V[/tex] open such that

    [tex] f:V\rightarrow\mathbb{R}^3[/tex] is [tex]C^1[/tex]

    [tex] V\cap S=f^{-1}(0)[/tex]

    [tex]\forall\in V\cap S,\quad \nabla_x f\neq 0[/tex]

    [tex]\Rightarrow f[/tex] is a surface


    2. Relevant equations



    3. The attempt at a solution

    The proof that I’m trying to understand goes as follows

    Write out [tex] \nabla_p f[/tex] in coordinates, since [tex]\nabla_p f[/tex] is non-zero at least one of the components of the gradient is non-zero. W.L.O.G take it to be the z coord

    [tex] \frac{\partial f}{\partial x_3}(p)\neq 0[/tex] so that

    [tex]p\in \left\{ q | \frac{\partial f}{\partial x_3}(q)\neq 0\}\subset V[/tex]

    [tex]V[/tex] open

    We then Construct a function

    [tex]\psi :V_1\rightarrow \mathbb{R}^3[/tex]

    [tex]\psi(x_1,x_2,x_3)=(x_1,x_2,f(x_1,x_2,x_3))[/tex]

    Calculating the Jacobian [tex]d_p\psi[/tex] we find that its determinant is non-zero, so that we can apply the Inverse function theorem to [tex]\psi[/tex] but I don’t understand the rest of the proof below

    [tex]\exists p\in V_2\subset V_1,\quad \psi |_{V_2}\rightarrow W_2[/tex]

    Choose [tex]\epsilon[/tex] so that [tex]\psi(p)+(-\epsilon,\epsilon)^3=W_3\subset W_2[/tex]

    What might this mean here?

    [tex]\left(\psi_{V_2}\right)^{-1}\left|_{W_3}\rightarrow V_2[/tex]

    Then we construct another map

    [tex] W_3\cap\left(\mathbb{R}^2\times 0\right)\rightarrow S[/tex]

    [tex] W_3\cap(\mathbb{R}^2\times 0)=\psi(p) +(-\epsilon,\epsilon)^2\times 0[/tex]

    Construct another map
    [tex] \phi : (-\epsilon, \epsilon)^2\rightarrow S[/tex]

    [tex] (u,v)\rightarrow \phi (\psi(p)+(u,v,0))[/tex]
    is a parameterization around p

    I can’t seem to understand what is going on here, maybe there is another way to prove the theorem. I have an intuitive proof using Taylor’s theorem and showing that f is like a plane locally but its not a rigorous proof
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Differential Geometry Theorem on Surfaces
Loading...