- #1
- 7,679
- 12,406
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?
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?
