Field lines of a vector field v, around the boundary of an open curved surface with infinitesimal line element dl along boundary, and through its interior with dS the infinitesimal surface element and n the unit normal to the surface. Top: Circulation is the line integral of v around a closed loop C. Project v along dl, then sum. Here v is split into components perpendicular (⊥) parallel ( ‖ ) to dl, the parallel components are tangential to the closed loop and contribute to circulation, the perpendicular components do not. Bottom: Circulation is also the flux of vorticity ω = ∇ × v through the surface, and the curl of v is heuristically depicted as a helical arrow (not a literal representation). Note the projection of v along dl and curl of v may be in the negative sense, reducing the circulation.