Derive the following relations:

$($a$)$ in cylindrical coordinates, between the physical components of Green strain $e_{11},e_{22}$

and $e_{13}$ and the phsical componens of displacement

$($b$)$ in spherical coordinates, between the physical components of Green strain $e_{22},e_{12}$ and

$e_{13}$ and the physical components of displacement.

$($c$)$ Write the part $($a$)$ and $($b$)$ in the case of small displacement gradients $.$

$($d$)$ Compare the results of $($c$)$ with the results available in the elasticity books.