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Math Help - Transformation of a Vector

  1. #1
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    Question Transformation of a Vector

    Greetings,

    My question is from the book "Tensor Analysis" by Barry Spain. I am asked to show that what the components of a vector become upon transforming from rectangular Cartesian coordinates to polar coordinates. I have attached the question in jpeg format. I have came up with a solution but the angular component in my solution is r^2 times the angular component given in the book. I have checked some other books on this subject and found out that both the solution given in the attachment and the one I found exist. I am pretty confused about this, and I assume that this book is wrong or I am doing a terrible mistake. I will be grateful if someone can provide some insight.

    Note: I found it appropriate to post this in differential geometry section since it is related with tensors somehow.
    Attached Thumbnails Attached Thumbnails Transformation of a Vector-ss.jpg  
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  2. #2
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    Re: Transformation of a Vector

    Any ideas? I can post my solution method if it is necessary.
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  3. #3
    Super Member ILikeSerena's Avatar
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    Re: Transformation of a Vector

    Hi Vesnog!

    See here for a derivation.
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  4. #4
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    Re: Transformation of a Vector

    Okay my derivation was similar and it is included in many textbooks but my problem is that there is an additional factor of r^2 in the angular part of the vector in my solution and in Wikipedia's solution. My question is this, if it is not clear I can explain it again.
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  5. #5
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    Re: Transformation of a Vector

    The solution for this transformation is also given as in my book on a book available through Google. I am attaching a screenshot to pinpoint my point.
    Attached Thumbnails Attached Thumbnails Transformation of a Vector-ss.jpg  
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Transformation of a Vector

    Ah, now I understand what you mean.
    But it is not a factor of r^2, but a factor of r.

    When we use polar coordinates (r,\theta), a small change is expressed as \mathbf{dr} = dr \mathbf{\hat r} + r d\theta \boldsymbol{\hat \theta}.
    This change is identified by (dr, d\theta).
    Note that the vector formula has an extra r in it where the change in angle is identified, but when we identify them as a coordinate pair, the r is left out.
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  7. #7
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    Re: Transformation of a Vector

    Thanks for your replies. First of all that should be a factor of r sorry for that, I am glad you got my point; however, I still could not understand your last statement(sentence).
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  8. #8
    Super Member ILikeSerena's Avatar
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    Re: Transformation of a Vector

    It's an arbitrary decision to take a factor r out.
    It does not have any particular meaning or significance.
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  9. #9
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    Re: Transformation of a Vector

    It may be significant because of dimensional analysis by the way the book should explicitly give a reason why it ignores r in my opinion. There may be a more subtle reason behind this.
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