Consider the simple shear flow described as \(v_{x}=\dot{\gamma} y\) and \(v_{y}=0\), where \(\dot{\gamma}\) is the rate of strain.

Verify that the rate of strain has only shear components as shown in the text.

Now consider a coordinate system that is rotated by an angle \(\theta\) to the \(x\)-axis. Find the components of the vector \(v\) in this system. Then write the expression for the velocity-gradient, rate-of-strain, and vorticity tensors.

Find the angle \(\theta\) such that the rate of strain has only diagonal components. This represents a case where the strain is purely elongational. Show that the vorticity tensor remains unchanged by the rotation of the coordinates.