  Traditional Cauchy-Stokes decomposition of velocity gradient tensor gives a
symmetric and an anti-symmetric subtensors which are called the strain-rate and
vorticity tensors. There are two problems with Cauchy-Stokes decomposition. The
first one is that the anti-symmetric or vorticity tensor cannot represent the
fluid rotation or vortex. The second one is that the symmetric (strain-rate)
tensor cannot distinguish the stretching (compression) and shear. The
stretching and shear are dependent on the coordinate or are not Galilean
invariant. Since vorticity cannot distinguish between the non-rotational shear
and the rigid rotation, vorticity has been decomposed to a rigid rotation
called Liutex and anti-symmetric shear in our previous work. A Liutex-based
principal coordinate was developed and the velocity gradient tensor was
decomposed in the principal coordinate as a rigid rotation (Liutex tensor), a
pure shear tensor and a stretching (compression) tensor, which is called the
principal decomposition. However, the principal decomposition is made at each
point which has own principal coordinate different from other points. This
paper derives the principal decomposition in the original xyz coordinate
system, and, therefore, provides a new tool for fluid kinematics to conduct the
velocity gradient tensor decomposition to rigid rotation, pure shear, and
stretching (compression) which is Galilean invariant and has clear physical
meanings. The new velocity gradient tensor decomposition could become a
foundation for new fluid kinematics.
