Integration over 3D vector variables

[arnyt...@gmail.com]

Is there a way in SymPy to either analytically integrate, or numerically integrate (with some symbolic variables retained), involving vectors?

Integrate over vectors p1, p3
( e-i p1 . x (p12 + p32) )  / ( (p3 - p1)2 (p32a2 +1)2 )

(with vector x , and scalar "a" remaining as free variables after integration)

If absolutely required for numerical integration, i.e. no other way to get SymPy to perform a semi-numerical integration, then vector x, and the scalar variable "a" can also be specified a value, but x should remain a vector. 

Of course, if the integration above can be attempted analytically if possible in SymPy, or even partly analytically (e.g. if the vectors are expressed in spherical polar co-ordinates, and some of the variables like Sin \theta etc. can be integrated over), any suggestions towards such an approach would be very useful as well. 

[Max Linke]

Is the value of the function you are trying to integrate a scalar?

If so I have recently done a integration using polar coordinates. Here is a example to calculate the average of v.T * U * v.

```python
phi = Symbol('phi', real=True)
theta = Symbol('theta', real=True)

a = Symbol('a')
b = Symbol('b')
c = Symbol('c')

v = Matrix([cos(phi) * sin(theta), sin(phi)*sin(theta), cos(theta)])
U = Matrix([[a, 0, 0], [b, 0, b], [c, 0, c]])

integrate(integrate(v.T * U * v, (phi, 0, 2*pi)) * sin(theta), (theta, 0, pi)) / (4 * pi)

```