For two-dimensional potential flow, the continuity equation and the Euler equations (in fact, the compressible Bernoulli's equation due to irrotationality) in Cartesian coordinates  involving the variables fluid velocity , specific enthalpy  and density  are
with the equation of state  acting as third equation. Here  is the stagnation enthalpy,  is the magnitude of the velocity vector and  is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy , which in turn using Bernoulli's equation can be written as .
Since the flow is irrotational, a velocity potential  exists and its differential is simply . Instead of treating  and  as dependent variables, we use a coordinate transform such that  and  become new dependent variables. Similarly the velocity potential is replaced by a new function (Legendre transformation)
such then its differential is , therefore
Introducing another coordinate transformation  for the independent variables from  to  according to the relation  and , where  is the magnitude of the velocity vector and  is the angle that the velocity vector makes with the -axis, the dependent variables become
The continuity equation in the new coordinates become
For isentropic flow, , where  is the speed of sound. Using the Bernoulli's equation we find
where . Hence, we have