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How to solve a vector integral within a solid angle ?


hi,
I want to plot a picture like Figure 5 in this paper and i need to calculate a function that present at page 8, the H(S) function:
http://users.tricity.wsu.edu/~bobl/personal/mypubs/1993_plausible.pdf
That integration take vector as parameters and integral over a hemisphere range, i want to ask is there a neat statement to do this integral rather than converting it to spherical coordinate than do the integration ?

Thanks. 



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