The divergence theorem / Gauss integral theorem states that

$\int dV\; \nabla \cdot \vec F = \int dS\; \hat n \cdot \vec F$

for a vector function $\vec F$, with $dV$ the volume element, $dS$ the surface element, and $\hat n$ a normal vector to the surface.

Is possible to generalize this for the following case,

$\int dV\; \big[ (\nabla \cdot [\nabla F])\; G \pm (\nabla \cdot [\nabla G])\; F \big]$

where $F$ and $G$ are scalar functions that vanish at the boundaries of the volume?

I don't know whether some equivalent of "integration by parts" would be applicable here...?

Any help would be appreciated.


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