Say I have a set of oriented points at locations $\vec{v\_i}$ with each some direction $\vec{n\_i}$, in practice they represent a surface with its normals. How would I calculate the divergence of this vector field? Standard finite-difference methods would obviously not work because there is no grid. 


Can I perhaps assume a grid and use nearest neighbours to infer the vector field at each point? The discretized vector field would then have at each point a weighted average of nearby points. This would then allow me to use standard FD-methods to calculate the divergence at each gridpoint. Is this a common method, or are there better methods?



Numerically calculating the divergence of a set of oriented points
Interpolation of vector fields (a notoriously tricky process) to determine the divergence can be avoided by using a line integral definition of the divergence via Stokes' theorem. You can find a discussion of this method in [On the interpolation of a vector field](http://www.spc.noaa.gov/publications/schaefer/interpol.pdf) (1979). For a more recent discussion see [A Quantitative Comparison between Traditional and Line Integral Methods of Derivative Estimation](http://journals.ametsoc.org/doi/abs/10.1175/1520-0493(2001)129%3C2538%3AAQCBTA%3E2.0.CO%3B2) (2001).



> 
> The “traditional” approach to estimating vector field derivatives
>  extracts a gridded field from the original observations, which are
>  typically not uniformly distributed in space. However, there exist
>  other methods involving derivative estimation via line integral
>  (“triangle”) techniques that do not involve a prior mapping of the
>  field onto a uniform grid. Empirical testing of the differences
>  between vector field derivative estimation using two different schemes
>  is done with prototypical examples of the techniques in the context of meteorological wind data. The test results
>  make it clear that the traditional method is generally inferior to
>  derivative estimates via the line integral methodology.
> 
> 
> 


