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Averaging a function in spherical polar coordinates? Watch

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    Hello, I've come across this formula in some notes but I can't seem to find where it has come from.

    The notes read:

    We can find the average of any quantity  f from

    \langle f\rangle=\dfrac{\int f(\theta,\phi)\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi}{\int\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi}.

    Can someone explain where this comes from? Or direct me to somewhere online I can find it? Google searches of "averaging a function in spherical polar coordinates" don't seem to get anywhere

    Thanks.
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    (Original post by KeyFingot)
    Hello, I've come across this formula in some notes but I can't seem to find where it has come from.

    The notes read:

    We can find the average of any quantity  f from

    \langle f\rangle=\dfrac{\int f(\theta,\phi)\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi}{\int\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi}.

    Can someone explain where this comes from? Or direct me to somewhere online I can find it? Google searches of "averaging a function in spherical polar coordinates" don't seem to get anywhere

    Thanks.
    Your formula assumes that the radius is a constant.

    An area element on a sphere in spherical polar co-ordinates is r^2\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi

    Since r is a constant you can pull it out of the integral.

    Your formula is the integral of f over the given area, divided by the given area, which is just the integral of the area element. The r^2 will cancel, leaving your formula.
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    Well, normally, the average of a function f on some manifold M with volume form d_{\mathrm{vol}} is:

    \frac{\int_M f d_{\mathrm{vol}}}{\int_M  d_{\mathrm{vol}}} = \frac{1}{\mathrm{volume(M)}} \int_M f d_{\mathrm{vol}}

    In this instance, I am assuming that your function is on the 2-sphere: \sin(\theta)d\theta d\phi is the volume form for the 2-sphere and thus the average is just the normal one as mentioned above.

    This just comes from the determinant of the Jacobian with respect to the change of coordinates from cartesian to spherical.
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    (Original post by ghostwalker)
    Your formula assumes that the radius is a constant.

    An area element on a sphere in spherical polar co-ordinates is r^2\sin\theta\ \mathrm{d}\theta\ \mathrm{d}\phi

    Since r is a constant you can pull it out of the integral.

    Your formula is the integral of f over the given area, divided by the given area, which is just the integral of the area element. The r^2 will cancel, leaving your formula.
    I thought it was that, but I was confused because, like you say, there were no r's in the formula. The case this is being applied to, r is constant

    Thanks!
 
 
 
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