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Math Help - Normal to a sphere using jacobian

  1. #1
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    Normal to a sphere using jacobian

    Hi. I need a bit of help with the following question:

    Calculate the normal to the sphere of radius
    R and centred at the origin, using the appropriate Jacobians and the spherical coordinates. Show that the normal is directed everywhere along the radial vector from the sphere centre, r = (x, y, z).


    So initially I desrcibed the sphere using polar coordinates:

    x = Rcos(phi)sin(phi) y = Rsin(phi)sin(phi)and z = Rcos(phi) ;

    Then I got three jacobians:

    Jx=d(y,z)/d(phi,z) Jy=d(z,x)/d(phi,z) Jz=d(x,y)/d(phi,z)

    For this I got:
    Jx=2Rsin(phi)cos(phi)
    Jy=Rsin(phi)sin(phi)-Rcos(phi)cos(phi)
    Jz=0

    So N=2Rsin(phi)cos(phi) i + Rsin(phi)sin(phi)-Rcos(phi)cos(phi) j

    Just wanted to know if I have done this coorectly, and how to do the final 'show that' part of the question.

    Cheers.
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  2. #2
    MHF Contributor

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    No, what you have is NOT correct. For one thing, the normal will have 0 k component only in the xy-plane and you are supposed to have the normal at any point on the sphere.

    In fact, since any line through the origin is a normal to this sphere, the normal at point (x,y,z) should be just a multiple of xi+ yj+ zk. That is what you are asked to verify in the last part.

    I think your fundamental problem is your expression of the spherica coordinates:
    "x = Rcos(phi)sin(phi) y = Rsin(phi)sin(phi)and z = Rcos(phi)".

    You must have two parameters, \phi and \theta:
    x = Rcos(\theta)sin(\phi)
    y = Rsin(\theta)sin(\phi)
    and z = Rcos(\phi)
    (Note: engineering notation reverses " \theta" and " \phi".)

    Now, you want your Jacobians to be
    Jx=d(y,z)/d(\theta, \phi), Jy=d(z,x)/d(\theta,\phi), and Jz=d(x,y)/d(\theta,\phi).
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