If I know the curl, and divergence of a n-component vector in a region, and its normal component around its boundary, is the vector uniquely specified? If yes, how do I prove it? Also, is there a straightforward way to solve these equations, and get the vector?
$\partial_i V_i=s \,\,\,\,\, \partial_iV_j-\partial_jV_i=c_{ij}$
When you know the divergence and curl of a function, it's ripe for using the Green's function to solve. However, I think you're missing one piece of information.
The Green's function $G$ for the derivative $\nabla$ in free space is given by
$$G(r) = \frac{\hat r}{4\pi|r^2|} \implies \nabla \cdot G = \delta(r), \quad \nabla \wedge G = 0$$
Let me stop for a second here and tell you about a lovely thing called geometric calculus. It lets us unify the divergence $\nabla \cdot G$ and the "curl" $\nabla \wedge G$ into a single expression $\nabla G$. This is a useful thing for the following derivation. We still keep track of both the scalar divergence and the pseudovector curl as separate "components" of the expression.
We use the Green's function and the generalized Stokes theorem to reconstruct a vector field from boundary information and information about its divergence and curl. Let $V$ be your vector field such that $\nabla V = s + c$, where $c$ is the pseudovector curl and $s$ is the scalar divergence.
The result is then
$$\oint_{\partial M} |dS'| G(r-r') \hat n' V(r') = - V(r) + \int_M |dV'| G(r-r') \nabla' V(r')$$
where $G \hat n' V = G (\hat n' \cdot V) + G \cdot (\hat n' \wedge V)$ and $G \nabla' V = Gs + G \cdot c$
The bottom line: this means that, unfortunately, you still need $\hat n' \wedge V$, which uses tangential information about the vector field $V$, to fully reconstruct $V$ in a given volume.
Now, just using the volume integral, you will be able to reconstruct $V$ to within the addition of some divergence-free and curl-free function (which come from the surface integral).
If you do ignore the surface integral as I suggest, note that $G \cdot c = G^i c_{ij}$.
There is a decomposition law into curl and divergence. It is closely related to the Greens theorem of the answer already given. But is much more simpler in terms or curland div. It is the Helmholtz decomposition theorem. There are much more physical effects relying on that fact. The concepts of transversal and longitudinal wave that make up each field. The existence of only transversal fields or waves in certain conditions is an often appearing consequence.
You need to know whether von Neumann or Diriclet boundary conditions apply. It is not only important to know about the Greens function there are a lot of special situations to know how to solve where much simpler methodes lead to the solutions.
It is easy to fill plenty of physics books on that topic and in addition a lot of application are of technical importance. A highlight is quantisation of fields and the second Landau speed of sound branch insolids.
For a very rough overwiew take this link www.jpier.org/PIER/pier69/22.06123101.G.Dou.pdf and erwan.deriaz.free.fr/articleACHArw.pdf for higher dimensions.
