Lemma

If $\nabla g(\mathbf{x}^{*})\not=\mathbf{0}$, then the tangent plane of $S$ at $\mathbf{x}^{*}$ is given by $$T=\{ \mathbf{y}:\nabla g(\mathbf{x}^{*})\mathbf{y}=0 \}$$