Saint-Venant's compatibility condition

In the mathematical theory of elasticity the strain ${\displaystyle \varepsilon }$ is related to a displacement field ${\displaystyle \ u}$ by

where ${\displaystyle 1\leq i,j\leq 3}$. Barré de Saint-Venant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form, this has now been generalized to higher rank symmetric tensor fields on spaces of dimension ${\displaystyle n\geq 2}$

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