I'm looking for detailed examples and practical applications of double surface integrals. I'm particularly interested in parametric surfaces and numerical integration (quadrature/cubature), though different approaches using other surface descriptions and/or analytical integration are also welcomed.
One application I know of are the view factors used in radiative heat transfer (or in radiosity for computer graphics rendering), see e.g. http://en.wikipedia.org/wiki/View_factor#View_factors_of_differential_areas.
[Edit] The concept of double line integrals might be relevant as well, I'd definitely be interested in notes or examples on the topic.
The Total integral curvature $ \int\int K d A $ and total boundary Rotation $ \int k_g d s $ occuring together in Gauss-Bonnet theorem is a good example in 3D geometry. Consider some applications along/inside a closed loop, for instance:
1) a spherical cap
2) the Temple of Viviani
3) quadrangle on a pseudosphere spanned by four asymptotic lines, etc.