# Complicated triple integral

I have a Euclidean geometric problem whereby the triple integral over SO(3) I am trying to solve seems to confuse many CAS softwares (including Maxima and Mathematica). The problem is

$$a = \int^{\sigma_{max}}_{-\sigma_{max}} \int^\pi_{-\pi} \int^{\theta_{max}}_0 \frac{\sin\theta d\theta d\phi }{|x+R(\theta,\phi,\sigma)y|^3} (x+R(\theta,\phi,\sigma)y) \cdot Q \cdot (x+R(\theta,\phi,\sigma)y) d\sigma,$$

where $a$ is a real number, $x$ and $y$ are 3D, rank-1 vectors, $R$ is a 3D rotation matrix which rotates the vector $y$, {$\theta$, $\phi$, $\sigma$} is a set of $SO(3)$ rotational parameters, and Q is a 3D, rank-2 traceless, symmetric tensor. The rotation matrix $R(\theta, \phi, \sigma)$ elements in this case would be

$$\begin{array}{rcl} R[0, 0] &=& \hphantom{-}\cos\phi\cos\theta\cos(\sigma-\phi) - \sin\phi\sin(\sigma-\phi), \\ R[0, 1] &=& -\cos\phi\cos\theta\sin(\sigma-\phi) - \sin\phi\cos(\sigma-\phi), \\ R[0, 2] &=& \hphantom{-}\cos\phi\sin\theta, \\ R[1, 0] &=& \hphantom{-}\sin\phi\cos\theta\cos(\sigma-\phi) + \cos\phi\sin(\sigma-\phi), \\ R[1, 1] &=& -\sin\phi\cos\theta\sin(\sigma-\phi) + \cos\phi\cos(\sigma-\phi), \\ R[1, 2] &=& \hphantom{-}\sin\phi\sin\theta, \\ R[2, 0] &=& -\sin\theta\cos(\sigma-\phi), \\ R[2, 1] &=& \hphantom{-}\sin\theta \sin(\sigma-\phi), \\ R[2, 2] &=& \hphantom{-}\cos\theta. \end{array}$$

In some cases $\theta_{max}$ is also a function of $\phi$, e.g.

$$\frac{1}{\theta_{max}^2} = \frac{\cos^2\phi}{\theta_x^2} + \frac{\sin^2\phi}{\theta_y^2}$$

Would anyone have any advice as to how I should approach such a problem? Is such a triple integral solvable without reverting to numerical integration?

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(1) Are you asking to evaluate $a$? (2) Can you describe what $R$ and $Q$ are in more detail? – Willie Wong Jul 26 '12 at 10:58
For (1) the question is what is the symbolic value of a? The numerical value of a is not being sought. For (2), R is a standard 3D Euclidean rotation matrix (3D, rank-2, no using Euler angles en.wikipedia.org/wiki/Rotation_matrix) and Q is a 3x3 matrix expression of a tensor (also 3D, rank-2, en.wikipedia.org/wiki/Tensor). Cheers. – bugman Jul 26 '12 at 13:31
Let me ask again to be clear: (1) By symbolic value do you mean in terms of $x,y,Q, \theta_{\text{max}}, \sigma_{\text{max}}$? (2) Please edit your question to give explicitly what the entries of the matrix $R(\theta,\phi,\sigma)$ are. There are many ways to give local coordinates to $SO(3)$. For all I know your $R(\theta,\phi,\sigma)$ could be just be a fixed constant matrix! – Willie Wong Jul 26 '12 at 13:58
Sorry (1), yes, in terms of $x$, $y$, $Q$, $\theta_{max}$, $\sigma_{max}$. (2) The rotation matrix $R(\theta, \phi, \sigma)$ in this case would be $$\left| \begin{array}{ccc} \cos\phi\cos\theta\cos(\sigma-\phi) - \sin\phi\sin(\sigma-\phi) & -\cos\phi\cos\theta\sin(\sigma-\phi) - \sin\phi\cos(\sigma-\phi) & \cos\phi\sin\theta \\ \sin\phi\cos\theta\cos(\sigma-\phi) + \cos\phi\sin(\sigma-\phi) & -\sin\phi\cos\theta\sin(\sigma-\phi) + \cos\phi\cos(\sigma-\phi) & \sin\phi\sin\theta \\ -\sin\theta\cos(\sigma-\phi) & \sin\theta \sin(\sigma-\phi) & \cos\theta \end{array} \right|$$ – bugman Jul 26 '12 at 14:16