$(4).$ The strain$-$displacement equation provides a direct relationship between the strain field and

the displacement field.

Why are the compatibility equations necessary?

If the strain field is given, is it necessary to ensure compatibility to obtain the displace$-$

ment field? Why or why not?

If the displacement field is given, is it necessary to ensure compatibility to obtain the

strain field? Why or why not?

Suppose the strain field $lon=lo{n}_{ij}{e}_{i}ox{e}_j$ is given by

$[lo{n}_{ij}(x)]=([3x_2^2,3x_1{x}_2+2x_1{x}_2^2,0],[3x_1{x}_2+2x_1{x}_2^2,4x_1^2{x}_2,0],[0,0,0])$

Check whether the compatibility conditions are satisfied?