Mathematical foundation of Liutex theory

Liutex is a mathematical definition of vortex, which is called the third generation of vortex definition and identification. This paper introduces the mathematical foundation of the Liutex theoretical system including differences in definition and operations between tensor/vector and matrix. The right version of velocity gradient tensor matrix is given to correct the old version which has been widely distributed by many mathematics and fluid dynamics textbooks. A unique velocity gradient principal matrix is provided. The mathematical foundation for Liutex definition is given. The coordinate rotation ( Q - and P - rotation) for principal coordinate system and principal matrix is derived, which is the key issue of the new fluid kinematics. The divergence of velocity gradient tensor is given in different forms which may be beneficial in developing new governing equations for fluid dynamics.