Circulation Integral
A circulation integral measures rotation.
Description
Circulation of a vector field F along a closed curve C is measured with a line integral.
where t̂ is the unit tangent vector. See here for an explanation of dr.
Green's Theorem
When the vector field is given as F = <P(x,y), Q(x,y)>, Green's theorem gives a method for evaluating this.
This can also be easily reformulated into a vector form that uses curl.
where k̂ is the unit basis vector.
Note the closely related normal form of the theorem.
Stoke's Theorem
When the vector field is given as F = <P,Q,R>, Stoke's theorem gives a method for evaluating this.
where n̂ is the unit normal vector. It should be clear then that Green's theorem is a special case of Stoke's theorem, wherein the vector field is constrained to the xy-plane, and therefore k̂ is always the unit normal vector.
Note that, in some texts, the expression n̂ dS is rewritten as dS or dS⃗. This term represents the differential surface area vector.