It turns out all of the above discussion about various deformation modes (note that acceleration mode is not a deformation, rather translation mode of motion), can be put under one framework using tensorial representation

To show the power of tensorial representation, first notice that when we talk about flow deformations we are naturally interested in the gradients of the velocity field. Since velocity is a vector its gradient becomes a tensor. Using index-notation that is \[ A_{ij} = (\nabla \vec{V}) = \partial_i v_j, \quad \text{where } i, j \in x, y \]

Now, note that any tensor \( A_{ij} \) can be broken to symmetric and antisymmetric parts. This is easily seen by first writing \( A_{ij} = \frac{1}{2} A_{ij} + \frac{1}{2} A_{ij} \) then adding and subtracting the half of the transpose of the tensor \(\frac{1}{2} A_{ji}\) (notice that finding transpose of any matrix in index-notation is just flipping of the indices). We have therefore \[ A_{ij} = \frac{1}{2} A_{ij} + \frac{1}{2} A_{ji} + \frac{1}{2} A_{ij} - \frac{1}{2} A_{ji} \]

Now taking the first and third terms together, and the second and fourth terms together we have \[ A_{ij} = \frac{1}{2} \underbrace{(A_{ij} + A_{ji})}_{E_{ij}} + \frac{1}{2} \underbrace{(A_{ij} - A_{ji})}_{\Omega_{ij}} \]