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Average of a quantity in all directions

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data

    I have to compute the average of [itex]A[/itex] in all directions for a sphere of radius [itex]R[/itex].
    For the direction [itex]r[/itex], [itex]A[/itex] is defined as:

    [itex]A=\int_0^R n(r)dr[/itex]
    where [itex]n(r)[/itex] is the density profile of the sphere.
    The answer should be
    [itex]\langle A\rangle=\frac{4}{3}\left\langle n \right\rangle R[/itex]

    2. The attempt at a solution

    I don't know how to compute the average over all directions. I mean for direction r the average value of A is simply
    [itex]\langle A\rangle_r=\left\langle n \right\rangle R[/itex]
    so, where this factor 4/3 comes from?
     
  2. jcsd
  3. Nov 9, 2011 #2
    Well, the way you typically find the average value of some function f(x) on a domain [itex]\Omega[/itex] is the following:

    [tex]<f> = \frac{\int_{\Omega}\ f(x)\ d\Omega}{\int_{\Omega}\ d\Omega[/tex]

    Note that the quantity in the denominator is just the area of the domain
     
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