Using self adjoint operators to generailze Green's Theorem

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So I have to generalize Green's function, with the definition that $$ \iiint_{V} (vLu - uLv)dτ =\iint_{dV}(p(v\nabla u - u\nabla v)\cdot d\boldsymbol σ $$ Where $$ L = \nabla \cdot [p(\boldsymbol r)\nabla] + q(\boldsymbol r) $$

I managed to simplify the left side down to $$ \iiint_{V} v(\nabla u \cdot p(\boldsymbol r)\nabla u) - u(\nabla v \cdot p(\boldsymbol r)\nabla v)dτ $$ and I'm not sure how this simplifies down to $$ \iint p(v \nabla u - u \nabla v) \cdot d \boldsymbol σ $$

For some clarification, bold letters are supposed to be vectors and σ wouldn't bold in the equation. Also the right hand integral is supposed to be closed. dσ represents the vector area element and dτ represents just the volume element. I'm very new to this website. Any help would be appreciated thanks!