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The properties of the curl and the gradient of divergence operators ( $\text{rot}$ and $\nabla\text{div}$ ) are studied in the space $ \mathbf {L}_{2}(G) $ in a bounded domain $ G \subset \textrm {R}^3 $ with a smooth boundary $\Gamma$ and in the classes $ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G)\oplus \mathbf{W}^m(G)$. The space $ \mathbf {L}_{2} (G) $ is decomposed intoorthogonal subspaces $ \mathcal{A} $ and $ \mathcal {B} $:$\mathbf{L}_{2}(G)=\mathcal{A}\oplus \mathcal{B}$. In turn, $ \mathcal{A}=\mathcal{A}_H\oplus \mathbf{A}^0$ and $\mathcal{B}=\mathcal{B}_H \oplus\mathbf{V}^0$, where $\mathcal{A}_H $ and $\mathcal{B}_H $ are null spaces ofoperators $\nabla \text{div}$ and $ \text{rot}$ in $\mathcal{A}$ and$\mathcal{B}$; the dimensions of $\mathcal{A}_H $ and $\mathcal{B}_H $ arefinite and determined by the topology of the boundary; $\mathcal{A}_H=\emptyset$ and $\mathcal{B}_H= \emptyset $ if the domain $\Omega $ is a ball. Theorthonormal basis are constructed in the class $ \mathbf{A}^0$ (resp., In$\mathbf{V}^0$ ) by eigenfields $\mathbf{q}_{j}(\mathbf{x})$ of $\nabla\text{div}$ operator (resp., $\mathbf{q}^{\pm }_{j}(\mathbf{x})$ of $\text{rot}$ operator) with nonzero eigenvalues $\mu_{j}$ (resp., $\pm\lambda_{j}$ ). The operators $\nabla\mathrm{div}$ and $\mathrm{rot}$ canceleach other out and project $\mathbf{L}_{2}(G) $ onto $ \mathcal {A} $ and $\mathcal { B} $, and $ \mathrm {rot} \, \mathbf {u} = 0 $ for $ \mathbf {u} \in\mathcal {A} $, and $ \nabla \mathrm div \mathbf {v} = 0 $ for $ \mathbf {v}\in \mathcal {B} $ \cite{hw}. Laplace matrix operator expressed through them:$\mathrm{\Delta} \mathbf {v} \equiv \nabla \mathrm{div}\,\mathbf {v}-(\mathrm{rot})^2\, \mathbf {v}$.

Curl and gradient of divergence operators in Spaces $ \mathbf{W}^{m}$  and $\mathbf{A}^{2k}$ vortex and potential fields and in the classes  $\mathbf{C}(2k, m)$