Write

e_{xx} = u_{x, x}

e_{yy} = u_{y, y}

and

2 e_{x y} = u_{x, y} + u_{y, x}

. Differentiating twice and cancelling the mixed partial derivatives gives

e_{xx, yy} - 2 e_{x y, x y} + e_{yy, xx} = 0.

(1)

A convenient set of six three-dimensional Saint-Venant compatibility equations is

e_{11, 22} + e_{22, 11} - 2 e_{12, 12} e_{22, 33} + e_{33, 22} - 2 e_{23, 23} e_{33, 11} + e_{11, 33} - 2 e_{13, 13} e_{11, 23} + e_{23, 11} - e_{12, 13} - e_{13, 12} e_{22, 13} + e_{13, 22} - e_{12, 23} - e_{23, 12} e_{33, 12} + e_{12, 33} - e_{13, 23} - e_{23, 13} = 0, = 0, = 0, = 0, = 0, = 0.

(2)

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} = ϵ_{ik ℓ} ϵ_{jmn} e_{ℓ n, km}

, up to the indicated signs and permutation of components.

They obey the three differential identities

\partial_{i} C_{ij} = 0

, because a symmetric pair of derivatives contracts with an antisymmetric Levi-Civita symbol. Thus 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.