How to inverse surface integral of a vector field

[Sify]

Assume that I know the value of $$\iint_{S} \overrightarrow{F} \cdot \hat{n} dS$$ over any surface in $$\mathbb{R}^3$$, where $$\overrightarrow{F}(x,y,z)$$ is a vector field in $$\mathbb{R}^3$$ and $$\hat{n}$$ is the normal to the surface at any point considered.
Using that I would like to compute $$\overrightarrow{F}$$. How can this be done?

P.S. I think I suffices to know $$\iint_{S} \overrightarrow{F} \cdot \hat{n} dS$$ over the planes perpendicular to the axes.

 

[Petr Mugver]

Consider a small circumference centered at a point x_0, with radius R and normal directed along a generic direction n (|n|=1):

$$S(\mathbf^{x}_0,R,\mathbf{n})=\{\mathbf{x}\in\mathbb{R}^3:(\mathbf{x}-\mathbf{x}_0)\cdot\mathbf{n}=0,|\mathbf{x}-\mathbf{x}_0|<R\}$$

then, if F is sufficiently smooth, you can calculate it's components

$$\mathbf{F}(\mathbf{x}_0)\cdot\mathbf{n}=\lim_{R\rightarrow 0}\,\,\frac{1}{\pi R^2}\,\,\int_{S(\mathbf^{x}_0,R,\mathbf{n})} \mathbf{F} \cdot d\mathbf{S}$$

 

[Sify]

Great!
This works perfectly. Thank You.

 

[Petr Mugver]

Great!
This works perfectly. Thank You.

Glad to help. Note that you are not forced to take small circumferences, but every small surface you like, depending on what's more useful: squares, triangles...