Stress Force - Understanding Cauchy Stress Tensor

how on earth.....? A possible way to look at it goes like this, let's consider a little cube length $l$, then the stress force in the $i$th direction acting on $j$th surface element is $-\sigma_{ji}(x_i,x_j,x_k)*dA$ where $dA=l^2$, the force in the $i$th direction acting on the other surface element parallel to the first is $\sigma_{ji}(x_i+dx_i,x_j,x_k)*dA$, so the total stress force acting in the $i$th direction is $$(\sigma_{ji}(x_i+dx_i,x_j,x_k)-\sigma_{ji}(x_i,x_i,x_j,x_k))*dA$$ Now divide by the volume element $dV=dx_id_xjdx_k$ and consider $dA=dx_idx_k$ to obtain $$f_i^{(j)}=\frac{\sigma_{ji}(x_i+dx_i,x_j,x_k)-\sigma_{ji}(x_i,x_j,x_k)}{dx_j}=\frac{\partial\sigma_{ji}}{\partial x_j}$$ Then total force per unit volume in the $i$th direction is given by your desired equation