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Comparing differentials

  1. Feb 19, 2013 #1
    I recently did a problem with some electron constraint to move on a hoop. It kind of surprised me that you just could take the old Schrödinger-equation with and let your
    dx ->dβ, where β is the distance along the hoop.
    Saying it in a less mathematical way, isn't a differential distance along something curved larger than a differential distance in a fixed direction? I do realize that a rigorous mathematician would shoot me for saying something like this, so how would he say it?
     
  2. jcsd
  3. Feb 19, 2013 #2
    In a problem like this the best suited thing to do is to pass to polar coordinates, so you can describe a loop more simply. Then in these coordinates with origin at the center of the loop which is at a fixed radius r, you have [itex]dx^2+dy^2=r^2d\theta^2=d\beta^2[/itex]. So it's simply the old good polar coordinates.
     
  4. Feb 19, 2013 #3
    Remember this? ## \displaystyle \lim_{x \rightarrow 0 } \frac {\sin x} {x} = 1 ##.

    It ensures that the length of a chord and the corresponding arc are about the same when they are small, let alone differential. It might be useful for you to follow the proof of the statement.
     
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