= Airy stress function
{c}
{title2=$\tau_{xx}=A_{yy},\quad\tau_{yy}=A_{xx},\quad\tau_{xy}=-A_{xy}$}
The displayed scalar stress potential automatically satisfies the two in-plane static momentum equations without body force, because the mixed third derivatives cancel. For homogeneous isotropic <linear elasticity>, strain compatibility further makes $A$ biharmonic. Since stresses depend only on second derivatives, $A$ is unique only up to $ax+by+c$ on a connected domain. Conversely a function with all second derivatives zero is affine, proving that this is exactly the gauge freedom. It does not license adding an arbitrary biharmonic function, which generally changes stress.
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