2-Plane Distribution in Cylindrical Coords.

[WWGD]

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?

 

[Pond Dragon]

Is there an easier way?

Why not just make distributions in the standard coordinates and then change to cylindrical coordinates? Or, am I misunderstanding?

 

[WWGD]

Why not just make distributions in the standard coordinates and then change to cylindrical coordinates? Or, am I misunderstanding?

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 ).

 

[Pond Dragon]

Part of the problem is that there is no global change of coordinates between the two.

Distributions are invariant of the coordinate system, right? Even then, I'm pretty sure you can cover $\mathbb{R}^3$ with only two "cylindrical" charts...

Plus, isn't $\mathrm{d}\theta$ only locally defined anyway?

 

[blue_raver22]

I can confirm that your understanding and approach to defining a 2-plane distribution in cylindrical coordinates is correct. The basis you have described is indeed pairwise-orthogonal and the distribution is radially-constant. Your method of using ratios of the coefficient vector fields to determine the slope in each direction is a valid approach. However, there may be other ways to define and understand this distribution, and it ultimately depends on the specific context in which it is being studied. I would recommend consulting with other experts in the field or conducting further research to explore alternative methods.