My question is, that given a vector field only numerically discrete in space, is there a way to estimate its vector potential?


Theoretically, I see [this](http://en.wikipedia.org/wiki/Vector_potential#Theorem) which requires the vector field over all of $\mathbb{R}^3$. Further, I also see [Will Jagy's answer using Poincare's lemma](https://math.stackexchange.com/questions/81405/anti-curl-operator) which I like much more. But in this case I still need to know the entire vector field from $(0,0,0)$ to the point $(x,y,z)$ where I want to estimate the vector potential because of $t\in(0,1)$. Better than $\mathbb{R}^3$ but still requiring too much information.


Is there any way to use only some local information from a vector field and estimate the local vector potential? I don't need to know the entire vector potential and I don't have the entire vector field anyway. It seems to me it should be possible because from what I understand, [the curl is defined](http://en.wikipedia.org/wiki/Curl_%28mathematics%29#Definition) as a limit and seems to be a local property. So can I reverse it locally? Any insight or references would be appreciated.


Just to clarify, I only have discrete (in space) numerical measurements of the vector field in a very small finite corner of $\mathbb{R}^3$ away from the origin. And I am hoping to use them to estimate the local vector potential numerically in the same corner.



Estimating the vector potential
Let's put the origin in your corner of ${\mathbb R}^3$, and say you have measurements of the field $\vec{F}$ at lattice points $\epsilon [i,j,k]$ for integers $0 \le i \le m$, $0 \le j \le n$, $0 \le k \le p$. One version of the 
vector potential at $(x,y,z)$ is 
$$ {\vec V}(x,y,z) = \left[ \int\_0^z F\_2(x,y,t)\ dt, \int\_0^x F\_3(s,y,0)\ ds - \int\_0^z F\_1(x,y,t)\ dt, 0\right]$$
The integrals (when $x,y,z$ are lattice points) can then be approximated by sums
over lattice points by the usual numerical methods.


