But now if you look carefully (or write down the components of the \( E_{ij} \) and \( \Omega_{ij} \) explicitly) you would notice that they correspond to various deformations we defined earlier \[ E = \frac{1}{2} \begin{bmatrix} 2 \frac{\partial v_x}{\partial x} & \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \\ \frac{\partial v_y}{\partial x} + \frac{\partial v_x}{\partial y} & 2 \frac{\partial v_y}{\partial y} \end{bmatrix} = \begin{bmatrix} \frac{\partial v_x}{\partial x} & E_{xy} \\ E_{yx} & \frac{\partial v_y}{\partial y} \end{bmatrix} \] \[ \Omega = \frac{1}{2} \begin{bmatrix} 0 & \frac{\partial v_x}{\partial y} - \frac{\partial v_y}{\partial x} \\ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} & 0 \end{bmatrix} = \begin{bmatrix} 0 & \omega_z \\ -\omega_z & 0 \end{bmatrix} \] It is always nice to arrive at the same results using different methods!