This problem arises in fluid dynamics and has me stumped. Consider a scalar function $\delta(\mathbf{x})$ and a vector function $\Phi(\mathbf{x})$ that satisfy $$ \nabla \cdot\left[(1-\delta)\nabla\Phi_i\right] = 0 $$ for each component $i$. Away from the origin, $\delta \to 0$ and $\Phi_i \to x_i$ as $r \to +\infty$. I am trying to prove the following symmetry, for a sufficiently large volume $V$ centered at the origin: $$ \int \left[\left(\Phi_i - x_i\right)\frac{\partial\delta}{\partial x_j} - \left(\Phi_j - x_j\right)\frac{\partial \delta}{\partial x_i}\right]\,\textrm{d}V = 0 $$ It was claimed to me that Gauss's theorem can be used in such a proof.
I have tried a number of approaches. Simpler versions of the problem (piecewise constant $\delta$, 1D simplifications) are trivially true. I have been unable to find any more complex closed analytic forms for $\delta$ and $\Phi$ that satisfy the above PDE, so haven't been able to work with a concrete example. This problem comes from considering wave scattering in an inhomogeneous medium ($\delta$ models the inhomogeneity), so I know from the physical symmetry that the result above must be true. I suspect that the integrand may be expressible as a divergence to which Gauss's theorem can be applied, but I have not been successful.
I appreciate any insight here.
Nevermind, I think I've got it. Apply Gauss's theorem to the PDE, using a cube with one face that cuts the $x_j$ axis at an arbitrary point and where the other faces are far from the origin. This gives an area integral that can be expanded back to a volume integral by integrating over $x_j$. Do similarly by swapping the $i$ and $j$ indices, subtract, and integrate by parts to get the desired result.