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Definition of curve

  1. Jul 1, 2011 #1
    Hi!
    I want to know the precise definition of smooth curve and piecewise smooth curve if the curve indicates a continuous function from an interval [a,b] into a set of real numbers.
     
  2. jcsd
  3. Jul 2, 2011 #2
    A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].
     
  4. Jul 2, 2011 #3

    HallsofIvy

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    It would be better to say "tangent vector" than "all of its derivatives". Also, we require that the tangent vector never be 0.
     
  5. Jul 3, 2011 #4
    In the statement 'the curve is smooth on each of those intervals', can i interpret it as 'each restriction of the curve to a subinterval of the interval is smooth'??
     
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