Results 1 to 2 of 2

Math Help - how to prove that a parameterization parameterizes a surface....

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    36

    how to prove that a parameterization parameterizes a surface....

    okay this is a docarmo question asking if x(u,v) = (u+v,u-v,4uv) parameterizes the graph z=x^2 - y^2.

    How do we show that it does this.

    Is the question asking does the parameterization give you a regular surface that is the subset of the graph? Considering the next question says 'what parts of the surface does it cover?'

    If my assumption is right.... i guess we just have to show that its 1:1 (dont need to worry about surjectivity since it only covers a part of the surface), that its cts and its inverse is cts... that its twice differentiable and that the differential is injective....

    Using a theorem in docarmo, x is 1:1 implies its inverse is cts...

    My big question is though:

    HOW DO WE GO ABOTU SHOWING THIS PARAMETERIZES THE WHOLE SURFACE?!?

    any help here would be so great!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: how to prove that a parameterization parameterizes a surface....

    Quote Originally Posted by mathswannabe View Post
    okay this is a docarmo question asking if x(u,v) = (u+v,u-v,4uv) parameterizes the graph z=x^2 - y^2.

    How do we show that it does this.

    Is the question asking does the parameterization give you a regular surface that is the subset of the graph? Considering the next question says 'what parts of the surface does it cover?'

    If my assumption is right.... i guess we just have to show that its 1:1 (dont need to worry about surjectivity since it only covers a part of the surface), that its cts and its inverse is cts... that its twice differentiable and that the differential is injective....

    Using a theorem in docarmo, x is 1:1 implies its inverse is cts...

    My big question is though:

    HOW DO WE GO ABOTU SHOWING THIS PARAMETERIZES THE WHOLE SURFACE?!?

    any help here would be so great!
    If you are using DeCarmo I assume you are asking whether \bold{x}:\mathbb{R}^2\to\Gamma_f where f(x,y)=x^2-y^2 is a surface patch completely parametrizing the surface. This entails showing that \bold{x} is a smooth map, and that \text{im }\bold{x}=\Gamma_f. Can you do thsi?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove surface area of cuboid.
    Posted in the Geometry Forum
    Replies: 2
    Last Post: December 27th 2009, 02:24 AM
  2. Parameterization
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 12th 2009, 10:56 AM
  3. parameterization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 11th 2009, 04:01 PM
  4. One-to-one parameterization
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: October 27th 2009, 03:57 AM
  5. Replies: 0
    Last Post: June 18th 2008, 03:51 PM

Search Tags


/mathhelpforum @mathhelpforum