  In a bounded domain $G$ with smooth border studied boundary value and
spectral problems for operators of the rotor (vortex) and the gradient of the
divergence $+\lambda\,I$ in the Sobolev spaces. For $\lambda\neq 0$ these
operators are reducible ( by B. Veinberg and V. Grushin method) to elliptical
matrices and the boundary value problems satisfy the conditions of V.
Solonnikov's ellipticity. Useful properties of solutions of these spectral
problems follow from the theory and estimates. The $\nabla \text{div}$ and $
\text{rot}$ operators have self-adjoint extensions $\mathcal{N}_d$ and
$\mathcal{S}$ in orthogonal subspaces $\mathcal{A}_{\gamma }$ and
$\mathbf{V}^0$ which formed from potential and vortex fields in
$\mathbf{L}_{2}(G)$. Their eigenvectors forme orthogonal basis in
$\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ elements of which are presented by
Fourier series and operators are transformations of series. We define analogues
of Sobolev spaces $\mathbf{A}^{2k}_{\gamma }$ and $\mathbf{W}^m$ orders of $2k$
and $m$ in classes of potential and vortex fields and classes $ C (2k,m)$ of
their direct sums. It is proved that if $\lambda\neq Sp(\mathrm{rot})$ the
operator $ \text{rot}+\lambda\,I$ displays the class $C(2k,m+1)$ on the class
$C(2k,m)$ one-to-one and continuously. And if $\lambda\neq Sp(\nabla
\mathrm{div})$ operator $\nabla \text{div}+\lambda\,I$ maps class $C(2(k+1),
m)$ on the class $C(2k,m)$, respectivly.
