Euler equations in an intrinsic coordinate system

The Euler equations are the equations of motion for an ideal fluid, that is, an inviscid fluid. They are easily deduced from the Navier-Stokes equations, simply by omitting the viscous terms. 

General properties of the presstrre field come to the fore when the Euler equations ate written in an irttrirrsic coonlirtate system. Such a formrrlation is especially helpful when the flow is steady. In that case, the strearrrlines and particle paths are the same curved lines. 

The intrinsic coordinate system is associated with the path of a particle. Considering a point M in the flrrid that rrroves in time, its path follows the crrrve OM (t). The movement of the particle along its path can be characterized by the 

Some notions of differential geometry ate necessary to prove what follows. So these can be recollected withont proving them. If the position of point M is 

At every point M, the intrinsic coordinate system comprises the unit vectors t and . It is important because velocity is tangential to the path  

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