Problem $1$:

The stress tensor $$ at a point $M$ of coordinate $(x_1,x_2,x_2)$ in the Cartesian coordinate system $({}_{}$e1,e$2,e_3)$ is

$=\left[\begin{array}{ccc}0& 0& c{x}_3\\ 0& a{x}_2& b{x}_3\\ c{x}_3& b{x}_3& 0\end{array}\right]$

with $a,b$ and $c$ constants

$\frac{1}{?}$ Determine the relations between a and b $,$ and the value of c such that the equilibrium equations are satisfied. Neglect body forces.

$\frac{2}{?}$ Determine the stress vector at a point $M$ of coordinates $(($:$x_1=0,x_2=1,x_1=1\}$ on a plane of normal $n$ of coordinates $n(\frac{1}{\sqrt[\phantom{2}]{2}},\frac{1}{2},\frac{1}{2}).$

Hint:

Equilibrium equations in Cartesian coordinate system

$\frac{\text{d}\text{e}\text{l}{}_i}{del{x}_j}+F_i=0$

with $Fi=0($Body forces are neglected$),1,j=1..\mathrm{.}3($ all $3$ equations given explicitly in Lecture $13).$