Q$3)(30$ points$)$ The displacement field for a uniform rectangular bar with specific weight $$ is given by:

$u=-\frac{v}{E}xz,v=-\frac{v}{E}yz,w=\frac{}{2E}[(z^2-a^2)+v(x^2+y^2)]$

where $a$ is a known constant. Determine:

$($a$)$ the strain field

$($b$)$ the rigid body rotations

$($c$)$ the stress field

$($d$)$ check the compatibility of strain field