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Thread: Normal to a sphere using jacobian

  1. #1
    Apr 2010

    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:

    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.

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  2. #2
    MHF Contributor

    Apr 2005
    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, $\displaystyle \phi$ and $\displaystyle \theta$:
    $\displaystyle x = Rcos(\theta)sin(\phi)$
    $\displaystyle y = Rsin(\theta)sin(\phi)$
    and $\displaystyle z = Rcos(\phi)$
    (Note: engineering notation reverses "$\displaystyle \theta$" and "$\displaystyle \phi$".)

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