Vector fields and integral curves

  1. I'm studying about vector fields and integral curves in the space R^n.

    I need some help in proving or getting some comments/feedback on the following propositions:

    1) Find a proof or counterexample: Let K and K' be two vector fields on R^n such that every integral curve of K is also an integral curve of K'. Then K = K'.

    2) State and prove a theorem to the effect that integral curves of a vector field can never cross.

    3) Let K be a vector field, and [tex] \alpha [/tex] a positive function on R^n. Express the integral curves of the vector field [tex] \alpha K [/tex] in terms of those of K. And why did we require that [tex] \alpha [/tex] be positive?

    Thanks in advance.
  2. jcsd
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