Understanding proof of stress tensor symmetry

Question

Hi I'm trying to understand basic physics but with a more formal scheme. I'm reading P.K.Kundu book of mechanical fluids. In page 90 he proves that stress tensor is symmetric. But first applies torques to a infinitesimal volume element. I cannot understand why the torque in z direction gives that relation, it does not explain term by term, I'm very lost. I understand that $\tau_{ij}$ is force per unit area, the force is applied at center of cube's faces, and that we are interested in the ones that gives angular momenta to the volume element. The first term okay, then why appears again in third term no idea. Second and fourth term maybe are added and subtracted on purpose?. Same last terms. Here an image of the page.

Answer 1

(Why show text that refers to Fig. 4.6 and then show only Fig. 4.5?)

I don't like this presentation. The stresses are shown to act on the faces in Fig. 4.5, but in Fig. 4.6, they're clearly considered to act on the cube center, since the stresses are essentially Taylor expanded to $$\tau(\text{edge})=\tau(\text{center})+\frac{\partial\tau}{\partial (\text{distance})}(\text{center–edge distance, or half the side length}),$$

or

$$\tau=\tau_{12}+\frac{\partial\tau_{12}}{\partial x_i}\frac{dx_1}{2}$$

in one example. OK, so call that the stress on the surface with normal unit vector $+x_1$. Hopefully it's then clear that the corresponding force is $\tau dx_2dx_3=\tau dx_2$ (because $dx_3$ is taken as 1), and then the moment arm is $\frac{x_1}{2}$, so that particular torque component is

$$\left(\tau_{12}+\frac{\partial\tau_{12}}{\partial x_i}\frac{dx_1}{2}\right)dx_2\frac{dx_1}{2}.$$

There's the first term.

But then the authors write $\tau_{12}$ on the opposite face, which should strictly be $\tau_{(-1)2}$ according to their indicial convention (with the arrow pointing toward $+x_2$), but then seem to assume implicitly that $\tau_{(-1)2}=\tau_{1(-2)}$ so that they can flip the arrow to obtain the label $\tau_{12}$. Maybe I'm missing something, but that seems to assume symmetry already.

All this is underexplained in the main text, as you note. But if you accept the indicial flip-flops, then the rest of the terms are straightforward, as the Taylor expansion when moving in the $-x_1$ direction to the plane pointing with normal unit vector $-x_2$ should be

$$\tau=\tau_{12}-\frac{\partial\tau_{12}}{\partial x_i}\frac{dx_1}{2},$$

for example.