I know that invoking (1) into (2) leads to (3) given $i,j = 1,2$ and $k_{ij}$ is symmetric and they are constant (can be put before the differential operators): $$\tag{1} v_i=k_{ij} \frac{\partial p}{\partial x_j}$$ $$\tag{2} \frac{\partial v_i}{\partial x_i}=0$$ $$\tag{3} k_{11}\frac{\partial^2 p}{\partial x_1^2}+2k_{12}\frac{\partial^2 p}{\partial x_1 \partial x_2}+k_{22}\frac{\partial^2 p}{\partial x_2^2}=0$$
That is, I know I must substitute (1) into (2) prior to expanding (2) [via the Einstein summation convention]. But what error is being made if one were to expand (2) first and then substitute the relation given by (1) for the $v_i$? I'm not sure if there even is a name for this error...If not, I would appreciate your comments as to why this order of operations does not work. Perhaps there are rules that one should follow to not make this type of order of operations mistake?