Consider the velocity fields

$v_1={?}^{\text{n}\text{o}\text{t}}[\frac{x}{1+t},\frac{2y}{1+t},\frac{3z}{1+t}{]}^T,$ and $,v_2={?}^{\text{n}\text{o}\text{t}}[\frac{x}{1+t},\frac{2Y}{1+t},\frac{3Z}{1+t}{]}^T$

Determine:

$($a$)$ The material description of $v_1$ and the spatial description of $v_2($consider $t=0$ is the reference configuration$).$

$($b$)$ The density distribution in both cases $($consider ${}_0$ is the initial density$)$

$($c$)$ The material and spatial descriptions of the displacement field as well as the material, $($Green$-$Lagrange$)$ and spatial $($Almansi$)$ strain tensors for the velocity field $v_1.$

$($d$)$ Repeat c $)$ for configurations close to the reference configuration $(t0).$

$($e$)$ Prove that the two strain tensors coincide for the conditions stated in d$).$