Family of curves tangent to a smooth distribution of lines

In summary, the conversation discusses the existence of a unique distribution of curves in R2 that "fills in" the space and is tangent to a smooth distribution of lines at each point. This smooth distribution of lines is referred to as a subbundle of TR2 and is associated with a point p in R2. The conversation also clarifies that the lines do not intersect, but instead represent directions at each point, which is known as a vector field. The existence of an integral curve through any point is supported by the fundamental theorem on solutions of ordinary differential equations.
  • #1
Lodeg
12
0
Hi,
Given a smooth distribution of lines in R2, could we assert that there is a unique distribution of curves such that:
- the family of curves "fill in" R2 completely
- every curve is tangent at every point to one of the smooth distribution of lines
 
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  • #2
Can you be more precise about what you mean by a smooth distribution of lines?
 
  • #3
I mean: to each point p in R2 is associsted a line Lp. The family Lp depends smoothly on p. It is somewhat a smooth subbundle of TR2
 
  • #4
Lodeg said:
I mean: to each point p in R2 is associsted a line Lp. The family Lp depends smoothly on p. It is somewhat a smooth subbundle of TR2
So can these lines intersect? Or do you mean directions at each point not straight lines in space?
 
  • #5
Indeed, directions at each point.
 
  • #6
Lodeg said:
Indeed, directions at each point.
A smooth set of directions is called a vector field and I think a fundamental theorem on solutions of ordinary differential equations guarantees an integral curve through any point.
 

1. What is the definition of a family of curves tangent to a smooth distribution of lines?

A family of curves tangent to a smooth distribution of lines is a group of curves that touch or intersect a set of lines in a continuous and smooth manner. This means that at any point of intersection, the curve and the line have the same slope or direction.

2. What is the significance of a family of curves tangent to a smooth distribution of lines in mathematics?

In mathematics, a family of curves tangent to a smooth distribution of lines is important because it can provide a visual representation of a set of equations or functions. This makes it easier to understand and analyze the relationships between different variables or parameters.

3. How do you determine if a curve belongs to a family of curves tangent to a smooth distribution of lines?

A curve belongs to a family of curves tangent to a smooth distribution of lines if it follows the same pattern as the other curves in the family, and if it intersects or touches the set of lines in a smooth and continuous manner. This can be determined by observing the slope or direction of the curve at different points of intersection with the lines.

4. What is the difference between a family of curves tangent to a smooth distribution of lines and a set of parallel curves?

A family of curves tangent to a smooth distribution of lines and a set of parallel curves are both groups of curves that follow a certain pattern. However, in a family of curves tangent to a smooth distribution of lines, the curves intersect or touch the set of lines at different points, while in a set of parallel curves, the curves never intersect but maintain a constant distance from each other.

5. How is a family of curves tangent to a smooth distribution of lines used in real-life applications?

A family of curves tangent to a smooth distribution of lines has various applications in fields such as physics, engineering, and economics. For example, in physics, it can be used to model the trajectory of a projectile, while in economics, it can represent the relationship between two variables in a demand and supply graph.

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