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Cross product expressed in the cylindrical basis

  1. Jul 5, 2011 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    It's not a homework question but a doubt I have.
    Say I want to write [itex]\vec A \times \vec B[/itex] in the basis of the cylindrical coordinates.
    I already know that the cross product is a determinant involving [itex]\hat i[/itex], [itex]\hat j[/itex] and [itex]\hat k[/itex].
    And that it's worth in my case [itex](A_yB_z-B_yA_z) \hat i +(A_xB_z-B_xA_z)\hat j+(A_xB_y-B_xA_y)\hat k[/itex].
    In order to reach what I'm looking for, can I "translate" [itex]A_x[/itex], [itex]B_y[/itex], etc. into cylindrical coordinates and then replace [itex]\hat i[/itex], [itex]\hat j[/itex] and [itex]\hat z[/itex] by what they are worth when translated in cylindrical coordinates and in function of the cylindrical unit vectors [itex]\hat r[/itex], [itex]\hat \theta[/itex] and [itex]\hat z[/itex]?
     
  2. jcsd
  3. Jul 5, 2011 #2

    lanedance

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    Homework Helper

    i think this is probably ok for the cross product of 2 static vectors as the cylindrical basis is orthonormal 9try it and check)

    however as soon as you look at vectors changing with time or derivatives, you will have a problem as the basis vector change with position, (eg. div, curl, etc.). The reason the cartesian coords are so useful is the basis does not change with position
     
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