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Chain rule violated for arc length?

  1. Jan 15, 2012 #1
    ->ds/dt where s is the arc length in cartesian coordinates is ((dx/dt)^2+(dy/dt)^2)^(1/2).

    -> Therefore by the chain rule ds/dt = ds/dp * dp/dt, but if I substitute dx/dt=dx/dp* dp/dt and dy/dt= dy/dp* dp/dt in the formula above, I get ds/dt=ds/dp * |dp/dt|??
    What is happening?

    ->Even by elementary thinking, ds/dt and ds/dp are always positive whereas dp/dt need not always be. So, how is the chain rule being followed here?
    Please explain. I have spent a full evening thinking over this.
     
  2. jcsd
  3. Jan 15, 2012 #2
    It only works for a monotonic parameter changes, i.e. [itex]p = p(t)[/itex] is a monotonic increasing function. Otherwise, you run into the trouble that different values of t give back one and the same value of p, so p is not a good parameter of the curve.

    Under these circumstances, it works like this:
    [tex]
    \frac{ds}{dt} = \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2 }
    [/tex]
    [tex]
    = \sqrt{ \left( \frac{dx}{d t} \, \frac{d p}{d t} \right)^2 + \left( \frac{d y}{d t} \, \frac{d p} {d t} \right)^2 + \left( \frac{d z}{d t} \, \frac{d p} {d t} \right)^2}
    [/tex]
    [tex]
    = \sqrt{ \left( \frac{d p}{d t} \right)^2 \, \left[ \left( \frac{dx}{dp} \right)^2 + \left( \frac{dy}{dp} \right)^2 + \left( \frac{dz}{dp} \right)^2 \right]}
    [/tex]
    [tex]
    = \left\vert \frac{d p}{d t} \right\vert \, \frac{d s}{d p} = \frac{d p}{d t} \, \frac{d s}{d p}, \ p'(t) > 0
    [/tex]
     
  4. Jan 15, 2012 #3
    I know what you wrote! that, was my question itself.
    The chain rule, in ordinary circumstances does not require monotonic parametrization, but for the arc length ,it does, so what is the reason for this apparent difference?
     
  5. Jan 16, 2012 #4

    chiro

    User Avatar
    Science Advisor

    It would not make sense if this quantity were negative.

    Arc-length is always a positive quantity when we are dealing with proper metrics. The arc-length is strictly an increasing function as you would expect from measuring the change in length of a string with respect to time even if it is transformed.

    Dickfore did a good job of showing this mathematically and did a good job of answering your question.
     
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