In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states:
Stated mathematically:
where  is the circulation around a material contour  Stated more simply, this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal.
This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example the Coriolis force) or non-barotropic pressure-density relations.