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Forum index » Science and Technology » Math
Curves and Vector Fields
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Maury Barbato
science forum Guru Wannabe


Joined: 13 Jun 2005
Posts: 169

PostPosted: Tue Jun 27, 2006 8:17 pm    Post subject: Curves and Vector Fields Reply with quote
Hello,
let F be a family of regular C^1 curves in R^n, A an open subset of R^n. Let us suppose that:
(I) every two curves of F don't intersect in A;
(II) for every point P of A there's a curve in F that
passes through P.
Is the field of tangent versors defined by F in A a continuous vector field?
Thank you very very much for your ideas.
My Best Regards,
Maury
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Lionel Brits
science forum beginner


Joined: 26 Apr 2006
Posts: 4

PostPosted: Wed Jun 28, 2006 1:06 am    Post subject: Re: Curves and Vector Fields Reply with quote
They are non-intersecting parameterized curves. I believe that makes
them a congruence. See:
http://en.wikipedia.org/wiki/Congruence_%28general_relativity%29

I am not sure how to do the proof. Maybe use Frobenius' theorem?

- DSA

Maury Barbato wrote:
Quote:
Hello,
let F be a family of regular C^1 curves in R^n, A an open subset of R^n. Let us suppose that:
(I) every two curves of F don't intersect in A;
(II) for every point P of A there's a curve in F that
passes through P.
Is the field of tangent versors defined by F in A a continuous vector field?
Thank you very very much for your ideas.
My Best Regards,
Maury
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Wed Jun 28, 2006 1:19 am    Post subject: Re: Curves and Vector Fields Reply with quote
Maury Barbato <mauriziobarbato@aruba.it> writes:

Quote:
Hello,
let F be a family of regular C^1 curves in R^n, A an open subset of R^n. Let us suppose that:
(I) every two curves of F don't intersect in A;
(II) for every point P of A there's a curve in F that
passes through P.
Is the field of tangent versors defined by F in A a continuous vector field?

Not necessarily.

Let n = 3 and take the standard coordinates x, y, z on A=R^3. For each
real c and strictly positive real r, throw into the family F the circle
on which z is identically c and x^2+y^2=r^2. Complete the family F
with the z-axis.

Lee Rudolph
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Google
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