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Dis-ambiguate: derivative of a vector field Y on a curve is the covariant of Y

  1. Aug 10, 2010 #1
    1. The problem statement, all variables and given/known data
    This two-part problem is from O'Neill's Elementary Differential Geometry, section 2.5.

    Let W be a vector field defined on a region containing a regular curve a(t). Then W(a(t)) is a vector field on a(t) called the restriction of W to a(t).

    1. Prove that Cov W w.r.t. a'(t) equals (W(a))'(t) where "Cov W w.r.t a'(t)" reads "the covariant of W with respect to a'(t)."

    2. Deduce that the straight line in Definition 5.1 (below) may be replaced by any curve with initial velocity v. Thus the derivative Y' of a vector field Y on a curve a(t) is (almost) Cov Y w.r.t. a'(t).

    Definition 5.1. Let W be a vector field on R^3, and let v be a tangent vector to R^3 at the point p. The the covariant derivative of W with respect to v is the tangent vector (W(p+tv))'(0) at the point p.

    The following definition is useful for part (2) since it distinguishes between a vector field and a vector field on a curve.

    Definition 2.2. A vector field on a curve 'a' from I to R^3 is a function that assigns to each number t in I a tangent vector Y(t) to R^3 at the point a(t).


    2. Relevant equations
    Part (1) was fairly straight-forward, using the definitions of covariant derivative and what it means to differentiate a composition.

    Part (2) has two parts. My approach to the first part is the following, and I believe it to be correct. The idea is to define a function on curves a(t) and show that it agrees with the covariant derivative with respect to a vector v at a point p for all curves a(t) such that a(0)=p and a'(0)=v. Part (1) can be used to show the function is well-defined and that it indeed equals the covariant. The second part, starting at "Thus" is where I'm having trouble. It's with the use of the word "almost." To me, if Y is a vector field on a curve a(t), then using the definition of covariant, straight-line or otherwise, makes no sense because Y(a(t)) is not defined (i.e., Y is not defined on R^3, only R). So I thought this might be the almost part. However, what's 'almost' about it? I was thinking that perhaps given Y on a(t) that a vector field Ybar could be defined such that Ybar equals Y(a(t)), and that then Y' would equal the covariant of Ybar instead of Y. However, I don't think I can 'always' define such a Ybar.


    Thank you in advance for your guided help. This is my first posting.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 11, 2010 #2
    In the "Relevant equations" section I meant to say "Ybar composed with 'a' equals Y," not the other way around.
     
  4. Aug 12, 2010 #3
    Attached is the PDF of an easier-to-read TeX version.
     

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