Cauchy Stress Tensor: Equilibrium Equations

[ARTjoMS]

''..Cauchy stress tensor in every material point in the body satisfy the equilibrium equations.''

$\sigma_{ij,j}+F_i=0$

I would appreciate if you could write out what it means.

Also there is this notation which I don't understand:
$\frac{\partial}{\partial y_j}\sigma_{ij}(x,y)$

 

[nejibanana]

The first two indices refer to the tensor, (row and column of the matrix) the ",j" then refers to the jth coordinate derivative of the tensor, as is shown in the second notation. From the second notation, I would guess j = 1 -> x and j= 2 -> y. Looks like this is in Einstein summation notation, so since there is a repeated index, j, you need to sum over all values of j. So you'd have (d/dx (sigma_i1) + d/dy (sigma_i2) = -F_i).

One way to check an equation and make sure you are interpreting it right is to look at the free indices. If you move the F to the right side, you have a free index of i on the right, and i and j on the right. So a summation over j will reduce it to a free index of only i on the left, which makes it a proper equation as the free indices have to match up on each side.