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Partial differentiation of cos (in vector calculus)

  1. Dec 5, 2011 #1
    1. The problem statement, all variables and given/known data

    So using standard spherical polar co-ordinates, my notes define a sphere as

    r(s,t) = aCos(s)Sin(t) i + aSin(s)Sin(t) j + aCos(t) k

    and the normal to the surface is given by the cross product of the two partial differentials:

    [itex]\partial[/itex]r[itex]/\partial[/itex]s X [itex]\partial[/itex]r[itex]/dt[/itex]

    So my issue is what is in my notes where:

    [itex]\partial[/itex]r[itex]/\partial s[/itex]= -aSin(s)Sin(t) i + aCos(s)Sin(t) j + aCot(t) k

    It is the final part that I don't understand. Why is the partial differential of Cos(t) with respect to s Cot(t)? I would have calculated it as zero.

    2. Relevant equations

    [itex]\partial[/itex][itex]/[/itex][itex]\partial[/itex]s (aCos(t)) = aCot(t) ???

    3. The attempt at a solution

    All I can think is that it's using a trig identity? As I understand it the k component should only depend upon t, not just from the above equations but in spherical polar co-ordinates in general, at least how I'm picturing it.



    Thanks in advance for any help.
     
  2. jcsd
  3. Dec 6, 2011 #2

    Fredrik

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    I agree with you. That looks completely wrong. It should be 0. And it's such a weird mistake that I kept staring at it for some time, wondering if I had misunderstood something. I guess that's what you're doing too. :smile:
     
  4. Dec 6, 2011 #3
    I keep thinking it must be a mistake too, but if it is a mistake then it is deliberate, as it has been repeated in a number of places. And during lectures where we go through the notes it was never picked up on.
     
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