  We present some new discoveries on the mathematical foundation of linear
hydrodynamic stability theory. The new discoveries are: 1. Linearized Euler
equations fail to provide a linear approximation on inviscid hydrodynamic
stability. 2. Eigenvalue instability predicted by high Reynolds number
linearized Navier-Stokes equations cannot capture the dominant instability of
super fast growth. 3. As equations for directional differentials, Rayleigh
equation and Orr-Sommerfeld equation cannot capture the nature of the full
differentials.
