The state of stress throughout a body is given by the stress tensor

${}_{ij}=\left[\begin{array}{ccc}0& C{x}_2& 0\\ C{x}_2& C{x}_3& -C{x}_1\\ 0& -C{x}_1& 0\end{array}\right]$

where $C$ is an arbitrary constant.

$($a$)$ At the point $P(1,1,1),$ calculate the traction vectors on the plane $x_1+x_2-x_3=1,$ and on the sphere

$x_1^2+x_2^2+x_3^2=3$

$($b$)$ Determine the principal stress at point at point $P.$

$($c$)$ According to the results of question $($b$),$ find the corresponding principal axes. Please satisfy the

following relationship.

${}_1>{}_2>{}_3,|vec(n{)}^{(1)}|=|vec(n{)}^{(2)}|=|vec(n{)}^{(3)}|=1,$vec$(n{)}^{(1)}$vec$(n{)}^{(2)}=$vec$(n{)}^{(3)}$

$($d$)$ Use the principal directions to define a new coordinate system. Determine the stress tensor in the new

coordinate system by tensor transformation.

$($e$)$ Suppose the traction vectors calculated in problem $($a$)$ are denoted $t_1$ and $t_2,$ respectively. What are

the components of $t_1$ and $t_2$ in the new coordinate system?