Consider the following integral over the whole space
I=∫√hij(x)Ai(x)Aj(x)d3x
where
Ai are the components of a highly localized vector field, and
hij is some well behaved Riemannian metric. One can imagine the magnitude of the components of the vector field to scale roughly with a normalized spatial Gaussian with a variance length
σ. In the limit
σ→0, or when the vector field gets localized to almost a delta peak around a certain point
x0, we can write an approximation to the integral as
I≈√hij(x0)∫Ai(x)d3x∫Aj(y)d3y
I believe there are similar higher order corrections to this expression when the field becomes very slightly non-localized. These would involve terms such as ∫(xj−xj0)Aidx,∫(xj−xj0)(xk−xk0)Aidx,... (without having to refer to σ or anything similar) and the metric structure evaluated only at x0. However, I do not know how to methodically search for these corrections.
In other words, I am looking for an approximation scheme which would express the integral I in terms of functionals of Ai(x), its derivatives and the coordinates, while only referring to hij and its derivatives at some point x0 which is chosen so that it is very close to some centroid of the Ais.
How can I do that?
(I have posted a related version of this problem at Math Stackexchange.)