Question

how define a system is irrotational

Answer 1

Flow is said to be irrotational when the vorticity  has the magnitude zero everywhere. It immediately follows, from Equation (4.77), that the circulation around any arbitrary loop in an irrotational flow pattern is zero (provided that the loop can be spanned by a surface that lies entirely within the fluid). Hence, from Kelvin's circulation theorem, if an inviscid fluid is initially irrotational then it remains irrotational at all subsequent times.