I'd like to integrate a function over a hemisphere (obtained from a sphere of radius $\delta$) whose flat side is normal to an arbitrary vector $\mathbf{n}$. I thought a change to spherical coordinates would be an appropriate way to solve this; however, the limits of integration are giving me a bit of trouble. I considered a modification of spherical coordinates so that the north pole is aligned with $\mathbf{n}$; however, this ended up being rather messy. If someone has a reference or a more elegant solution I would really appreciate it.
2026-03-27 06:51:13.1774594273
Integral over hemisphere not aligned with principle axis.
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