# Why are stress components on opposite faces identical?

The stress tensor stores information about the stress on each of the faces of an infinitesimal volume of material. But I am confused as to why the stress components on opposite faces should be equal. The textbook I am reading (Fluid Mechanics by Kundu and Cohen) gives the following explanation:

On the opposite face EFGH the stress components have the same value as on ABCD, but their directions are reversed. This is because Figure 2.4 shows the stress at a point. The cube shown is supposed to be of “zero” size, so that the faces ABCD and EFGH are just opposite faces of a plane perpendicular to the x2-axis. That is why the stresses on the opposite faces are equal and opposite. I am not sure why the cube being infinitesimal size implies that the stress components on opposite faces should be the same. Could someone please explain?

The usual argument is the following: If the sides of the cube have length $$L$$ the the net force $$F_x$$ in the $$x$$ direction, say, is equal to $$L^2\Delta \tau_{xx}$$ where $$\Delta \tau_{xx}$$ is the difference in the stress components on the opposite faces. Meanwhile the mass of the cube is $$m=\rho L^3$$ where $$\rho$$ is the density. Thus, from $$F=ma$$ the acceleration is $$a =\frac 1{\rho} \Delta \tau_{xx}L^{-1}.$$ If $$a$$ is to stay finite as $$L\to 0$$, then $$\Delta \tau_{xx}$$ must go to zero $$\propto L$$. Or, to put it another way, if $$\Delta \tau_{xx}$$ were not zero in the limit, the material would move infinitely fast so as make the two stresses equal. A similar argument applied to torque and moments of intertia shows that $$\tau_{ij}=\tau_{ji}$$.