Given the displacement vector

${\displaystyle \mathbf {u} =X_{1}^{2}X_{2}{\widehat {\mathbf {e} }}_{1}+X_{3}{\widehat {\mathbf {e} }}_{2}+X_{3}^{2}{\widehat {\mathbf {e} }}_{3}}$

1. Find the strain tensor ${\displaystyle {\boldsymbol {\varepsilon }}}$ and the rotation tensor ${\displaystyle \mathbf {W} \,}$.
2. Write down the compatibility equation in the ${\displaystyle {\widehat {\mathbf {e} }}_{1}}$- ${\displaystyle {\widehat {\mathbf {e} }}_{2}}$ plane. Use index notation with commas representing partial derivatives with respect to Lagrangian material co-orinates.