= Solution
Write $e_{xx}=u_{x,x}$, $e_{yy}=u_{y,y}$ and $2e_{xy}=u_{x,y}+u_{y,x}$. Differentiating twice and cancelling the <mixed partial derivatives> gives
$$
\boxed{e_{xx,yy}-2e_{xy,xy}+e_{yy,xx}=0.}
$$
A convenient set of six three-dimensional <Saint-Venant compatibility equations> is
$$
\boxed{\begin{aligned}
e_{11,22}+e_{22,11}-2e_{12,12}&=0,\\
e_{22,33}+e_{33,22}-2e_{23,23}&=0,\\
e_{33,11}+e_{11,33}-2e_{13,13}&=0,\\
e_{11,23}+e_{23,11}-e_{12,13}-e_{13,12}&=0,\\
e_{22,13}+e_{13,22}-e_{12,23}-e_{23,12}&=0,\\
e_{33,12}+e_{12,33}-e_{13,23}-e_{23,13}&=0.
\end{aligned}}
$$
They follow by inserting $e_{ij}=(u_{i,j}+u_{j,i})/2$ and commuting <derivatives>. They form the six symmetric components of the incompatibility <tensor> $C_{ij}=\epsilon_{ik\ell}\epsilon_{jmn}e_{\ell n,km}$, up to the indicated signs and permutation of components.
They obey the three differential identities $\partial_iC_{ij}=0$, because a symmetric pair of <derivatives> contracts with an antisymmetric <Levi-Civita symbol>. Thus \b[there are three independent differential constraints], represented symmetrically by six component equations; they are not six unrelated constraints on the six <strain> functions. For example, at a nonzero <wavevector> directed along the third axis, compatibility imposes $e_{11}=e_{22}=e_{12}=0$ while $e_{13},e_{23},e_{33}$ can arise from the three <displacement> components. This confirms <rank> three. One cannot simply discard the same arbitrary three component equations in every direction; their independence is differential, with the stated identities. On a <simply connected> domain these are the local <Saint-Venant compatibility equations> for <strain> to be a symmetric <displacement> <gradient>, while domains that are <multiply connected regions> require global closure conditions as in the screw example. This is <Saint-Venant strain compatibility and differential identities>.
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