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If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for the altitude one just needs to integrate from $0$ to $\theta$ where $\theta$ is the angle that refers to the position. Spherical cap

Now I need to find out how to do this, if the cap is not symmetric around the z-axis but somewhere on the sphere. So I am looking for the most general way to find the integration interval for integration over the surface of a spherical cap. (Notice, I just need the interval). Since the integrand is highly ugly and unsymmetric I cannot reduce it to this general case with azimuthal symmetry.

Therefore, does anybody know how to integrate over the surface of an arbitrarily positioned spherical cap?

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  • $\begingroup$ What do you mean by "not symmetric around the z-axis, but somewhere on the sphere"? Do you mean somewhere else on the sphere? $\endgroup$
    – rurouniwallace
    Aug 14, 2013 at 12:51
  • $\begingroup$ yes, that was what i meant $\endgroup$
    – user66906
    Aug 14, 2013 at 12:55

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You can rotate the vector field to match the re-positioning of the cap. Just multiply the field by a rotation matrix:

$$\vec{f}'(x,y,z)=R\vec{f}(x,y,z)$$

Then integrate $\vec{f}'(x,y,z)$ over the cap symmetric about the z-axis.

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  • $\begingroup$ thank you, actually this is a helpful trick $\endgroup$
    – user66906
    Aug 14, 2013 at 14:51

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