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Triple Integral in Spherical Coordinates

  1. Sep 15, 2004 #1


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    I have a hemispherical surface of radius R with it's base centred on the origin. We are using the convention:

    r is the radius i.e. the magnitude of the position vector of a point: its distance from the origin.

    theta is the polar angle

    phi is the azimuthal angle

    I am asked to calculate the integral of the divergence of a given vector field v over this volume (enclosed by the hemisphere).

    So far what I have done is to say that a full sphere would be given by the equation:

    r = R

    What about a hemisphere? It seems to me that the angle theta must be restricted so that points below the x-y plane are not part of the domain. So what are the allowed values of theta? I can't seem to figure out whether it should be -pi/2 < theta < pi/2, or something else? I need to know this to set my bounds of integration for one of the three integrals. Thanks.
    Last edited: Sep 15, 2004
  2. jcsd
  3. Sep 15, 2004 #2
    For a hemisphere : r goes from 0 to R
    theta goes from : 0 to 90°
    phi goes from : 0 to 360°

    Ofcourse, you know that you need to express the angles in radials

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