Results 1 to 2 of 2

Math Help - Sphere orientation form

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    8

    Sphere orientation form

    Hello,

    i have read in wikipedia, that a orientation form (volume form) of the sphere S^m is given by: w=\sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \wedge...\wedge dx_{i-1} \wedge dx_{i+1} \wedge...\wedge dx_{n+1}

    I don't understand the notation. Ok w is a map from the sphere into the set of alternating tensors:
    w:S^m->\Lambda^m (S^m).

    but what is w(x_1,...,x_m+1)=?
    and what does dx_i mean?

    Can you please explain it to me?

    Regards
    Last edited by Bongo; February 7th 2011 at 11:50 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23
    Giving examples may help you to understand. Let n=2 we have the standard sphere S^2 embedded in R^3:
    \{(x,y,z)|x^2+y^2+z^2=1\}. Thus
    \omega=x dy\wedge dz+y dz \wedge dx + z dx \wedge dy.
    This is a top form defined on S^2 and you can easily check that it is nowhere zero. So it is a volume form thus defines an orientation of S^2.
    Actually it is the volume element so that integrating it will get the surface area: \oint_{S^2} \omega = \int_{D^3} d\omega = \int_{D^3} 3 dx\wedge dy \wedge dz= 3 Volume(D^3) = 4\pi
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. integration of a 2-form over en sphere
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 13th 2011, 11:29 PM
  2. Orientation of submanifold
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 9th 2011, 04:21 AM
  3. orientation of manifolds
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 5th 2011, 06:31 AM
  4. Calculating sphere orientation?
    Posted in the Geometry Forum
    Replies: 7
    Last Post: October 24th 2010, 01:45 AM
  5. Orientation of 3D Polygon
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: February 11th 2008, 02:34 AM

Search Tags


/mathhelpforum @mathhelpforum