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Math Help - cylindrical coordinates

  1. #1
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    cylindrical coordinates

    Hi,everyone,I got a problem here that I don't understand.Can anyone please help me?


    The relation between three-dimentional cartesian coordinates (X,Y,Z) and cylindrical coordinates (\rho,\phi,z) is given by
    X=\rho cos\phi
    Y=\rho sin\phi
    Z=z

    Calculate the partial derivatives \frac{\partial \rho}{\partial X}, \frac{\partial \phi}{\partial Y} and \frac{\partial z}{\partial Z} in terms of \rho, \phi, z.


    My current problem here is like a puzzle. I certainly can solve the question and find the answer. For example, I did find \frac{\partial \rho}{\partial X}=cos\phi.Same as solution
    However, my problem occured when I want to calculate \frac{\partial X}{\partial \rho},if I just use X=\rho cos\phi, I would get cos\phi as well!
    I think it is weird, because I thought \frac{\partial X}{\partial \rho} supposed to equal to the reciprocal of \frac{\partial \rho}{\partial X}, which means it supposed to be \frac{1}{cos \phi}.
    So pretty sure \frac{\partial \rho}{\partial X}=cos\phi is the right answer(same as solution),but why the recipocal rule does not apply here? I'm so confused about this puzzle. Same as two other questions.
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  2. #2
    A Plied Mathematician
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    What was your process by which you obtained

    \displaystyle\frac{\partial\rho}{\partial x}?
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  3. #3
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    Oddly enough, both \frac{\partial \rho}{\partial X}= cos(\theta) and \frac{\partial X}{\partial \rho}= cos(\theta) are correct.
    Your "mistake" is in thinking that \frac{\partial \rho}{\partial X} and \frac{\partial X}{\partial \rho} must be reciprocals. For partial derivatives, that simply is not true.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Oddly enough, both \frac{\partial \rho}{\partial X}= cos(\theta) and \frac{\partial X}{\partial \rho}= cos(\theta) are correct.
    Your "mistake" is in thinking that \frac{\partial \rho}{\partial X} and \frac{\partial X}{\partial \rho} must be reciprocals. For partial derivatives, that simply is not true.
    Thanks a lot.
    I felt the reason must be because their are partial derivatives, therefore they are not reciprocals.
    But I wonder the reason, is there any mathematical way can explain that? Thanks a lot.
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