Q $1$: On a $3-$D block experiences deformation characterized by

$x=A{x}_2$hat$(e{)}_1+B{x}_1$hat$(e{)}_2+C{x}_3$hat$(e{)}_3$

where $A,B$ and $C$ are constants. The Cauchy stress tensor components at certain point of the block are given by

$[]=|[10,5,0],[5,20,0],[0,0,20]||$

Determine the Cauchy stress vector $f$ on a plane whose normal in the current configuration is hat$(n)={}_1,$ the first Piola $-$ Kirchhoff stress tensor $P.$ Kirchhoff stress tensor $r$ and second Piola$-$Kirchhoff stress tensor $S.$ Also, write the balance of linear momentum equation in terms First Piola$-$Karchhoff stress and secund Piola$-$Kirchhoff stress. solve it carefully