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: Prove that subset of regular surface is also a regular surface

  1. Apr 14, 2009 #1
    Prove that subset of regular surface is also a regular surface(updated)

    Dear Sirs and Madam's

    I have following problem which I hope you go assist me in. I have been recommended this forum because I heard its the best place with the best science experts in the world.

    Anyway the problem is as follows

    1. The problem statement, all variables and given/known data

    Let [tex]A \subset S [/tex] be a subset of regular surface S. Prove that A itself is a regular surface iff A is open in S. Where [tex]A = U \cap S[/tex] and where U is open in [tex]\mathbb{R}^3[/tex]

    I am using a pretty old book by a guy named Do Carmo so just as now. on page 52 there is a definition of a regular surface:

    A subset of [tex]S \subset \mathbb{R}^3[/tex] is a regular surface if for eac [tex]p \in S[/tex] there exist a neighbourhood V in [tex]\mathbb{R}^3[/tex] and a map [tex]x: U \rightarrow V \cap S[/tex] of a open set [tex]U \subset \mathbb{R}^2[/tex] onto [tex]V \cap S \subset \mathbb{R}^3[/tex]

    Such that

    1) x is differentiabel

    2) x is an homomorphism.

    3) For each q in U the differential [tex]dx_q : \mathbb{R}^2 \rightarrow \mathbb{R}^3[/tex] is onto-one.
    2. Relevant equations

    3. The attempt at a solution

    condition 2) By the definition above let [tex]p \in A[/tex]. Next assume that [tex]x: U \rightarrow S [/tex]. Where U is open subset of [tex]\mathbb{R}^3[/tex]. Then

    [tex]x^{-1}(A \cap x(U)) \subset U [/tex] is a regular surface and by the definition its open in S.

    condition 3)

    Again we assume that [tex]p \in A[/tex] Next [tex] x: u \rightarrow A[/tex] where U is a subset of [tex]\mathbbb{R}^3[/tex]. next we assume that x(q) = p and that

    [tex]dxq: \mathbb{R}^2 \rightarrow \mathbb{R}^3[/tex] and is thusly one-to-one.

    How does this look???
    Last edited: Apr 14, 2009
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted