General Relativity: An Introduction for Physicists · Problem 17.14

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · spatial-derivatives-inverse-radius

Problem

By writing

r \equiv |\vec{x}| = (\delta_{ij}x^i x^j)^{-1/2}

, show that

\partial_i\left(\frac{1}{r}\right) = -\frac{x_i}{r^3}.

Hence show that

\partial_i\partial_j\left(\frac{1}{r}\right) = \frac{3x_i x_j - r^2\delta_{ij}}{r^5}.