A solid, isotropic, circular cylinder in the main landing gear of a Boeing $747-81$ is made of a Titanium$-$Aluminum alloy with a Young's modulus E and Poisson's ratio $v.$ It has a radius $b,$ is constrained at the left edge and is subjected to a torque T at the right end. Assume that the deformations are small. The following Lagrangian deformation map is proposed in the cartesian coordinate system $($with $k$ as a constant$)$:

$(x)=(x_1-k{x}_2{x}_3)e_1+(x_2+k{x}_1{x}_3)e_2+x_3{e}_3$

a$.$ Find the deformation gradient F $.$

b$.$ Write the components of infinitesimal strain tensor.