A vector field u is said to be irrotational if curl u = 0.
This condition is equivalent to saying that the field is conservative. It is also equivalent to saying that there is a scalar potential U (unique up to an additive constant) such that u = grad U.
Many fields that we are interested in are irrotational, eg gravitional fields, and time-independent electric and magnetic fields, and corresponding potentials may be used to facilitate the theory. In fluid dynamics it is easy to show that an irrotational flow in an incompressible fluid will remain irrotational in the absence of viscous effects, allowing the introduction of a velocity potential.