The change of the nucleon density distribution of a heavy nucleus caused by the presence of a second nucleus at a fixed centre-to-centre distance is studied in the framework of the deformed Woods–Saxon mean field. The single-particle problem of one nucleus has been solved by diagonalization in a cylindrical harmonic oscillator basis with the external field of the partner nucleus included in the Hamiltonian before diagonalization. The external field consists of the Coulomb potential of the partner charge distribution and of the nuclear part obtained by double folding of the Migdal density-dependent effective nucleon-nucleon interaction, in the same form as it is used in the dinuclear system approach. The induced density change $\delta \rho (𝐫;R)$ and the corresponding changes of the quadrupole moment, of the mean-square radius and of the interaction energy have been calculated for the ${}^{40}$Ca+${}^{208}$Pb system at separations $R=15.5$–$20$ fm. The results are compared with the first-order (linear-response) treatment of the same external field. Because the mean field is anchored at the origin, the multipole moments are shown to require a translationally invariant definition: a displaced but undeformed distribution acquires a quadrupole moment second order in the displacement, and in this problem that kinematic term is comparable with the induced moment itself. Once it is removed, the intrinsic quadrupole response of ${}^{208}$Pb agrees with the first-order one to better than half a percent, and remains so over the full range of the switching parameter for the Coulomb field at $R=20$ fm, while that of ${}^{40}$Ca retains a genuine higher-order contribution of $12$ to $16$. The polarization of both partners is dominated by the repulsive Coulomb tidal field, which produces an oblate deformation of the density, while the attractive nuclear field acts in the opposite sense and partially cancels the Coulomb contribution at the smallest separations. At the larger separations the induced octupole deformation is close to $40$ of the quadrupole one for ${}^{208}$Pb and to $50$ for ${}^{40}$Ca, and it is the more sensitive indicator of the breakdown of the one-centre picture. The polarization energy is a dipole quantity, the induced quadrupole deformation contributing only a few percent of it; it is verified by coupling-constant integration and is quoted under several correction prescriptions for the spurious translational component and for the absence of the residual interaction. The sensitivity to the geometry of the folding densities and the limits of validity of the one-centre description are established quantitatively.