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Relation between parameters of a vector field and it's projection

  1. Nov 16, 2013 #1
    Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?
     
  2. jcsd
  3. Nov 19, 2013 #2

    fzero

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    The projection of Y orthogonal to X can be expressed as (a scalar multiple of)

    $$Y' = Y - \frac{\langle X, Y\rangle X }{ \langle X, X \rangle} ,$$

    wherever ## \langle X, X \rangle\neq 0##. We can define the flow curves via ##\dot{\gamma}_Y(t) = Y \gamma(t)## and ##\dot{\gamma}_{Y'}(t) = Y' \gamma(t)##. In principle we can use the same parameter ##t## to describe both of these curves, but the curve for ##Y'## will also depend on the vector field ##X##, so I don't think there's a general answer to your question. If ##X## and ##Y## are given and nice enough, then we can just solve for the curves to work out the relationship.
     
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