Results 1 to 6 of 6

Math Help - Smooth Function R^3 -> R^4

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    16

    Smooth Function R^3 -> R^4

    Is the function F: R^3 -> R^4 defined by F(x,y,z) = (sqrt(x), x,y,z) if x >= 0 and (sqrt(-x), x,y,z) if x < 0 smooth on all of R^3?

    I'm thinking it is not, because taking the square root messes up smoothness at x=0, is this correct? Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2010
    Posts
    16
    OK. The problem is I am trying to find a smooth bijection F: S^2 (unit sphere in R^3) --> U where U is the subset of R^4 defined as:

    U = {(x,y,z,w) in R^4 such that x^2 + y = 0 and y^2 + z^2 + w^2 = 1}

    All bijections I have tried involve taking a square root in the first component so I have not managed to find a smooth one.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Magus01 View Post
    OK. The problem is I am trying to find a smooth bijection F: S^2 (unit sphere in R^3) --> U where U is the subset of R^4 defined as:

    U = {(x,y,z,w) in R^4 such that x^2 + y = 0 and y^2 + z^2 + w^2 = 1}

    All bijections I have tried involve taking a square root in the first component so I have not managed to find a smooth one.
    You may find it easier to visualise what to do if you consider the same problem in one dimension lower: Find a smooth bijection F: S^1 (unit sphere in R^2) --> U where U is the subset of R^3 defined as:

    U = {(x,y,z) in R^3 such that x^2 + y = 0 and y^2 + z^2 = 1}.


    Here, x^2 + y = 0 is a cylindrical parabola and y^2 + z^2 = 1 is a circular cylinder. Their intersection looks like a distorted circle. In two dimensions, the circle x^2+y^2=1 is smoothly mappable to the curve x^4+y^2=1 (which also looks like a distorted circle). If you can produce a smooth map (x,y)\mapsto (z,w) implementing this equivalence, then the map (x,y)\mapsto(z,-z^2,w) will be a smooth bijection from the circle to the set U.

    If you succeed in doing that, you should have no trouble jacking up the dimension by 1 to solve the given problem.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2010
    Posts
    16
    Thanks a lot. For the smooth map from the circle to the curve, I drew them both on the plane and for (x,y) in S1 defined r(x,y) to be the point on the curve that meets the straight line starting from the origin and going through (x,y). Is this the smooth map you had in mind?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Magus01 View Post
    Thanks a lot. For the smooth map from the circle to the curve, I drew them both on the plane and for (x,y) in S1 defined r(x,y) to be the point on the curve that meets the straight line starting from the origin and going through (x,y). Is this the smooth map you had in mind?
    Yes it is. It's easy to visualise that map geometrically, and it's "obvious" that it must be smooth. But I haven't found a clean way to describe it analytically or to prove that it's smooth.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. smooth function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 20th 2011, 06:15 AM
  2. Replies: 1
    Last Post: March 26th 2011, 09:10 AM
  3. Torus and smooth function
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 28th 2010, 05:39 PM
  4. Replies: 1
    Last Post: October 7th 2010, 08:54 PM
  5. Smooth function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 6th 2008, 02:40 PM

Search Tags


/mathhelpforum @mathhelpforum