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A simple (?) vector problem

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine the vector [tex]\bm{B}=\left(\frac{\partial A_{\theta}}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta}\right)\hat{\phi}[/tex]

    [tex]A_r[/tex] and [tex]A_{\theta}[/tex] are the components of the basis vectors [tex]\hat{r}[/tex] and [tex]\hat{\theta}[/tex].

    3. The attempt at a solution

    I just calculated the differentials in the expression for B above, but that gave me a factor [tex]1/r[/tex] too much in the answer. My textbook rewrites B as

    [tex]\bm{B}=\frac{1}{r}\left(\frac{\partial (A_{\theta}r)}{\partial r}-\frac{\partial A_r}{\partial \theta}\right)\hat{\phi}[/tex].

    They've broken out a factor [tex]1/r[/tex] before differentiating, but I don't understand the

    [tex]\frac{\partial (A_{\theta}r)}{\partial r}[/tex]

    part. Why isn't it

    [tex]r\frac{\partial (A_{\theta})}{\partial r}[/tex]?
     
  2. jcsd
  3. Oct 14, 2009 #2

    Dick

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    I agree with you and I don't agree with the textbook.
     
  4. Oct 15, 2009 #3
    Unfortunately for us, I have another textbook that agrees with the first. I don't think it's likely that two professional physicists makes exactly the same mistake on such a (I guess) basic thing.
     
  5. Oct 15, 2009 #4

    Dick

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    Well, if I put A_theta=1 and A_r=1, I certainly don't get the same result for B for the two expressions. That's about all I can say.
     
  6. Oct 15, 2009 #5
    I know, it's a mystery. Perhaps I should provide some more information. B is the cross product of the operator del and another vector, A, expressed in polar coordinates (B is only a function of [tex]r[/tex] and [tex]\theta[/tex]).
     
  7. Oct 15, 2009 #6

    Dick

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    It sounds like you are computing the curl of A. If you are working in cylindrical coordinates, and A is independent of z, and the z component of A is independent of r and theta, then your second expression is the curl if you replace the theta hat with a z hat. That's a near as I can get to figuring out what you are up to. http://mathworld.wolfram.com/CylindricalCoordinates.html
     
  8. Oct 15, 2009 #7
    Yes, it's the curl I'm trying to calculate. I see now that I've been working with an incorrect http://hyperphysics.phy-astr.gsu.edu/Hbase/curl.html#c2". I still don't know where the factor [tex]r[/tex] in front of [tex]\bm{E}_\theta[/tex] in the determinant (see the link I provided) comes from, though.
     
    Last edited by a moderator: Apr 24, 2017
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