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Spherical Polar Coordinates

  1. Oct 22, 2008 #1
    Express the following vector field in spherical coordinates. (The
    answer should be in a form that uses the unit vectors of the curvilinear coordi-
    nate system and coefficient functions that are written in terms of the curvilinear
    coordinates.)


    [itex] \underline{F} = -y \underline{i} + x \underline{j} + (x^2 + y^2)\underline{k}[/itex]

    OK, so I've obtained the equation:

    [itex]\underline{F} = rsin\theta(-sin\phi\mathbf{i} + cos\phi\underline{j} +rsin\theta\underline{k})[/itex] simply by substituting [itex]x = rsin\theta cos\phi[/itex] etc. into the above equations. Now how do I express this in terms of the unit vectors [itex]\mathbf{e}_r,\mathbf{e}\phi, \mathbf{e}_\theta[/itex] ??
     
  2. jcsd
  3. Oct 22, 2008 #2

    cristo

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    Staff Emeritus
    Science Advisor

    Well, what are the unit vectors in spherical polars in terms of Cartesian unit vectors?
     
  4. Oct 23, 2008 #3
    [itex]\underline{e}_r = sin\theta(cos\phi \underline{i} + sin\phi{j}) + cos\theta\underline{k}[/itex]

    [itex] \underline{e}_{\theta} = cos\theta(cos\phi \underline{i} + sin\phi{j}) - sin\theta\underline{k}[/itex]

    [itex] \underline{e}_{\phi} = -sin\phi \underline{i} + cos\phi{j}[/itex]

    I can't see how to write the above equation in terms of these unit vectors...
     
    Last edited: Oct 23, 2008
  5. Oct 24, 2008 #4

    gabbagabbahey

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    You'll need to solve these 3 equations for i, j, and k. Then substitute the solutions into the equation from your previous post.
     
  6. Oct 24, 2008 #5
    I mean really, I don't mean to sound ungrateful or anything, but how stupid do you think I am? I know what I have to do, I just don't know how to do it. In any event, I've solved it by myself. Note for the future: your method is slightly long winded. Thanks anyway!
     
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