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For a function $f\colon\mathcal M\to\mathbb R$ the Riemannian gradient $\nabla f(x)$ at $x\in\mathcal M$ is given by the unique tangent vector fulfilling
$\langle \nabla f(x), \xi\rangle_x = D_xf[\xi],\quad \forall \xi \in T_x\mathcal M,$ where $D_xf[\xi]$ denotes the differential of $f$ at $x$ with respect to the tangent direction (vector) $\xi$ or in other words the directional derivative.