Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 3 / b / Solution

Fix ${\mathbf{x}}^{\prime }$ inside $V$ and put $\psi (\mathbf{x})=G(\mathbf{x},{\mathbf{x}}^{\prime })$ in Green's second identity. The volume integral becomesHere Laplace's equation eliminates the second term, and the Dirac delta evaluates the first. Reversing the signs in the boundary integral givesAll boundary derivatives act on the integration variable $\mathbf{x}$, not on the fixed evaluation point ${\mathbf{x}}^{\prime }$.