Let \(Y\) be a stationary real-valued Gaussian process on \(\mathbb{R}^{d}\) with isotropic covariance function \(\exp \left(-\left\|x-x^{\prime}ight\|^{2} / 2ight)\). Show that the gradient field \(\partial Y\) is an \(\mathbb{R}^{d}\) valued Gaussian process with covariance function

\(K_{r s}\left(x, x^{\prime}ight)=\operatorname{cov}\left(\partial_{r} Y(x), \partial_{s} Y\left(x^{\prime}ight)ight)=\exp \left(-\left\|x-x^{\prime}ight\|^{2} / 2ight)\left(\delta_{r s}-\left(x_{r}-x_{r}^{\prime}ight)\left(x_{s}-x_{s}^{\prime}ight)ight)\).

Show also that \(K\left(R x, R x^{\prime}ight)=R K\left(x, x^{\prime}ight) R^{\prime}\) for \(R \in \mathcal{O}_{d}\), and hence that the gradient process is hydrodynamically symmetric.