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Follow a conserved mass element from its Lagrangian coordinate $\mathbf{q}$ to its Eulerian coordinate $\mathbf{x}=\mathbf{q}+b\mathbf{p}$. The integrated continuity equation giveswhere $\bar{\rho }={\rho }_0/a^3$ and the Lagrangian labels are chosen with uniform reference mass per $d^{3}q$. These equations hold before shell crossing, when the oriented determinant is positive. If ${\mu }_i$ are the eigenvalues of $\partial p_i/\partial q_j$, thenFor an irrotational displacement the deformation matrix is symmetric, so these eigenvalues are real. Using the conventional positive collapse eigenvalues ${\lambda }_i=-{\mu }_i$, equivalently the eigenvalues of $-\partial p_i/\partial q_j$, givesThere is a sign mismatch in the printed pairing of this last expression with eigenvalues of $+\partial p_i/\partial q_j$: for the displayed plus-sign displacement map, those eigenvalues give plus signs in the determinant. For example, $p_1=c{q}_1$, $p_2=p_3=0$ makes $x_1=(1+bc)q_1$, so mass conservation gives $1+\delta =(1+bc{)}^{-1}$, not $(1-bc{)}^{-1}$. The corrected eigenvalue convention restores the intended formula without changing the map.

Now take $\mathbf{p}=(p(q_1),0,0)$ and $J=1+b{p}^{\prime }$. The physical trajectory is $\mathbf{r}=a\mathbf{x}$. Differentiating at fixed $\mathbf{q}$ givesThe physical derivative in the perturbed direction is $\partial /\partial r_1=[a(1+b{p}^{\prime }){]}^{-1}\partial /\partial q_1$. The two transverse directions still have the background acceleration. ThereforeUse the pressureless, $\Lambda =0$ background acceleration equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ to obtainThis proves the planar exactness of the Zeldovich approximation for the requested full force-divergence equation. The displacement depends on only one coordinate, but the divergence includes all three physical directions; discarding the transverse background terms would give the wrong result. The proof is valid until shell crossing. With the planar gravitational boundary condition fixing the possible spatially uniform force, it is the exact single-stream dust evolution; it does not describe the multistream density by a single invertible map after crossing.