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Math Help - quick question vector fields

  1. #1
    Junior Member
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    quick question vector fields

    Define the vector field on the complement of the z axis by
    F(x ,y, z) = (-y i + x j)/(x^2 + y^2)

    Im trying to find the curl and i know how to do it but for my k direction im confused because of this whole complement of z axis thing, do i just take x^2 + y^2 to be k direction??
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  2. #2
    MHF Contributor Calculus26's Avatar
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    The vector field is defined at all points except where x = y = 0.

    This is precisely all points on the z axis.

    So when you say :
    Define the vector field on the complement of the z axis by
    F(x ,y, z) = (-y i + x j)/(x^2 + y^2)
    You are simply saying that F is defined everywhere but along the z axis.

    So simply compute the curl as you normally would with 0 k component.

    In General if F =f i + gj

    Then curl F is[dg/dx - df/dy] k

    Remember Green's Theorem in the plane ?
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