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2-Plane Distribution in Cylindrical Coords.

  1. Aug 18, 2014 #1

    WWGD

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    Hi all,

    I am trying to describe/understand how to define a 2-plane distribution in R^3 , i.e., an assignment of a 2-plane at each tangent space, when the distribution is given in terms of
    the basis of a plane in (R^3, cylindrical). It has just been a while since I have worked with cylindrical cords.

    The plane at a given point (at each tangent space) is given by its basis in terms of (r,θ,z) (actually, it is given in terms of the basis (∂r,∂θ,∂z)) by: {(0,0,∂z), (r∂r,∂θ,0)} . IIRC ,This basis is pairwise- orthogonal, and the r-axis at a fixed point (r,θ,z) is a line segment that starts at (0,0,z) and ends in (r,θ,z) . The z-axis is the same as that used in Cartesians. That leaves one choice for the θ-axis , using pairwise orthogonality. It seems clear that this distribution is radially-constant, i.e., constant for fixed vaues of r.

    I think we can use, outside of r=0, a sort-of slope given by the ratios of the coefficient vector fields, so that, e.g., in the z-θ plane, the slope is 1, since the ratio of the coefficient fields is itself always 1. And in the r-θ and r-z axes/directions the slopes are both r-to-1.

    Does this work? Is there an easier way?
     
  2. jcsd
  3. Aug 19, 2014 #2
    Why not just make distributions in the standard coordinates and then change to cylindrical coordinates? Or, am I misunderstanding?
     
  4. Aug 19, 2014 #3

    WWGD

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    You're not misunderstanding, you're misunderestimating, I would say ;).

    Part of the problem is that there is no global change of coordinates between the two. But it is a good idea to try that and see what happens with the local changes of coords.
    For the sake of context, FWIW, this distribution is a contact distribution given as the kernel of the (contact, of course) 1-form:

    w:=dr+rdθ

    But it is messy (messi, if you're Argentinian) to work with "moving frames" (don't know if that is the best name/description ).
     
  5. Aug 20, 2014 #4
    Distributions are invariant of the coordinate system, right? Even then, I'm pretty sure you can cover [itex]\mathbb{R}^3[/itex] with only two "cylindrical" charts...

    Plus, isn't [itex]\mathrm{d}\theta[/itex] only locally defined anyway?
     
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