My problem is finding the average of a vector function $\mathbf{f}(\mathbf{x})$ which is a function of a vector field $\mathbf{x}$ and possibly some other variables like time etc.
essentially I want
$\langle \mathbf{f}(\mathbf{x}) \rangle = \int_{0}^{2 \pi} d\phi_1 \int_{0}^{ \pi} d\phi_2 \int_{0}^{2 \pi} d\phi_3 \frac{\sin(\phi_2)}{8 \pi^2} \mathbf{f} ( R(\phi_1,\phi_2,\phi_3) \mathbf{x} ) $
Where $\phi_k$ are some set of Euler angles and $R(\phi_1,\phi_2,\phi_3) $ is a matrix of rotation.
Without any knowledge of the function (except perhaps assuming it is relatively smooth) what is the best way to choose quadrature points for the Euler angles so I can get this average to converge? Or is there a better method for this type of problem?
Many thanks