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Math Help - calculus in euclidean space, working with curves in R3

  1. #1
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    calculus in euclidean space, working with curves in R3

    Deduce from \alpha ' (t) [f] = \frac {d(f(\alpha))}{dt} (t) that in the definition of directional derivative, {\bf v_p}[f] = \frac {d}{dt} (f({\bf p} + t{\bf v}) \mid _{t = 0}, the straight line t \to {\bf p} + t{\bf v} can be replaced by any curve \alpha with initial velocity {\bf v_p}, that is, such that \alpha (0) = {\bf p} and \alpha ' (0) = {\bf v_p}

    Not quite sure how to proceed. Any help would be appreciated.
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  2. #2
    Senior Member Tinyboss's Avatar
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    Try replacing f(p+tv) with f(\alpha(t)) and see what you get. In the standard directional derivative definition, you're traveling from p along a straight line in the direction of v, and taking the limit as t goes to zero. They're saying that you could travel t units along the path alpha, and take the same t->0 limit, and get the same answer, provided alpha has the right properties.
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