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Math Help - Velocities of curves.

  1. #1
    Junior Member
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    Velocities of curves.

    Calculate the velocities v1,v2 of alpha1, alpha2 respectively at p = (1,0,0).

    alpha1 = (-Pi, Pi) -> Reals(3), defined by alpha1(t) = (cost,sint,t)

    alpha2 = (-Pi, Pi) -> R(Reals)(3), defined bby alpha2(t) = (cost, sint, -t).
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  2. #2
    Super Member Showcase_22's Avatar
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    \alpha_1(t)=(cos(t), \; sin(t), \; t)
    \alpha_2(t)=(cos(t), \; sin(t), \; -t)
    p=(1, \; 0, \; 0)
    These are expressions for the position of a particle at a time t.

    Since p=(1, \; 0, \; 0) we need a t value. We can find this by:

    (1, \; 0, \; 0)=(cos(t), \; sin(t), \; t) \Rightarrow \ t=0.

    We only have expressions for position. Therefore we need to differentiate them to have an expression for the velocity.

    \alpha_1(t)=(cos(t), \; sin(t), \; t)
    \bigtriangledown \alpha_1(t)=(-sin(t), \; cos(t), \; 1)

    \alpha_2(t)=(cos(t), \; sin(t), \; -t)
    \bigtriangledown \alpha_2(t)=(-sin(t), \; cos(t), \; -1)

    (Can someone post back and tell me if this is notationally correct. I'm not sure if I should be using \bigtriangledown here.)

    When t=0:

    \bigtriangledown \alpha_1(t)=(-sin(0), \; cos(0), \; 1)=(0, \; 1, \; 1)
    \bigtriangledown \alpha_2(t)=(-sin(0), \; cos(0), \; -1)=(0, \; 1, \; -1)
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