Results 1 to 2 of 2

Thread: surface of rev.

  1. #1
    Oct 2010

    surface of rev.


    I try to solve this problem:

    $\displaystyle h(t)=(h_1(t),h_3(t))$ some smooth path in $\displaystyle \mathbb{R}^2$, $\displaystyle t \in (0,1), |h'(t)|=1, h_1(t)>0$ and $\displaystyle g(t,\alpha)=(h_1(t)*cos\alpha, h_1(t)*sin\alpha, h_3(t)), \alpha \in [0,2\pi)$ the corresponding surface of revolution.
    $\displaystyle h_3$ is a diffeomorphism onto its image.

    Find $\displaystyle U \subset \mathbb{R}^3$ and a function f:U->$\displaystyle \mathbb{R}$,s.t. 0 is a regular value of f and Im(g)=$\displaystyle f^{-1}(0)$

    I have try to define some function f on a set U, s.t. the Points on the surface are mapped to 0. But i couldn't find it.
    My results are quite little, we know that the length of the curve h is 1. Therefore we can find a open set U, which contains the surface.

    I try it with some sinus and cosinus relations, like $\displaystyle sin^2+cos^2=1$, but it doesn't help. because in equations like f(x,y,z)=x^2+y^2-... we get $\displaystyle c_1^2$

    How can i define f reasonable?

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Mar 2010
    Beijing, China
    Need only to find a function H of R^2 so that $\displaystyle Im(h)=H^{-1}(0)$. One of the ways to define such a function is as follows:

    Let n(t) be the unit normal vector field on h(t). For t in [0,1] and small values of s, a parametrization of the plane near Im(h) is given by r(t,s)=h(t)+s*n(t). Since Im(h) is compact, the parametrization is well defined on [0,1]*[-d,d], where d is a small positive value. Define H on R^2 as H(t,s)=s, then $\displaystyle Im(h)=H^{-1}(0)$.

    Then rotate the function H to define the function f on R^3 as $\displaystyle f(t,s,\alpha)=H(t,s)$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Surface Area of Surface
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 1st 2010, 09:53 AM
  2. Is this surface possible?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Oct 30th 2010, 01:57 AM
  3. Calculate the surface area of the surface
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jun 26th 2009, 04:03 AM
  4. Help finding surface area of a surface
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 3rd 2008, 04:11 PM
  5. Volume, Surface Area, and Lateral Surface Area
    Posted in the Geometry Forum
    Replies: 1
    Last Post: Apr 14th 2008, 11:40 PM

Search Tags

/mathhelpforum @mathhelpforum