Let \(\mathbb{S}^{d-1}_{R}\) denote the \((d-1)\)-sphere with radius \(R>0\). By a similar computation as in (0x692a804f) , we find
\[ \Delta f = \frac{1}{R^2} r^{-d} \partial_r (r^d \partial_r f) + r^{-2} \Delta_{\mathbb{S}^{d-1}_R} (r^d f). \]Let \(\mathbb{S}^{d-1}_{R}\) denote the \((d-1)\)-sphere with radius \(R>0\). By a similar computation as in (0x692a804f) , we find
\[ \Delta f = \frac{1}{R^2} r^{-d} \partial_r (r^d \partial_r f) + r^{-2} \Delta_{\mathbb{S}^{d-1}_R} (r^d f). \]