# Stokes's theorem

From AMS Glossary

## Stokes's theorem

The statement that if where

*s*is a surface in three dimensions having a closed curve*c*as its boundary, then the circulation of a vector**V**around*c*is equal to the flux of the vorticity (curl of**V**) through*s*, under certain mathematical conditions on these fields and boundaries:**n**is the unit vector normal to*s*on that side of*s*arbitrarily taken as the positive side. The positive direction along*c*is defined as the direction along which an observer, traveling on the positive side of*s*, would proceed while keeping the enclosed area to his left.*d***r**is a vector line element of*c*.For two-dimensional flow in the where ζ is the vertical component of the vorticity. This states that the circulation around a given curve per unit area enclosed by the curve is equal to the average vorticity within the area.

*x*,*y*plane, Stokes's theorem becomes