Curl theorem

The curl theorem or Stokes-Kelvin theorem states that the integral of the Curl of a three-dimensional Vector field in a Surface is equal to the integral of the field itself projected over the bounding Curve of the surface. For a generic vector field F\mathbf{F} on a surface SS of bounding curve γ\gamma, the theorem states:

∫S(∇×F)⋅da=∮γF⋅ds\int_{S}(\nabla\times\mathbf{F})\cdot d\mathbf{a}=\oint_{\gamma}\mathbf{F}\cdot d\mathbf{s}

where da=n^ dad\mathbf{a}=\hat{\mathbf{n}}\ da is the area element projected with the normal unit vector n^\hat{\mathbf{n}} and ds=τ^ dsd\mathbf{s}=\hat{\boldsymbol{\tau}}\ ds is the line element projected with the tangent unit vector τ^\hat{\boldsymbol{\tau}}.

Mathematical treatment

Let F:U⊂R3→R3F : U ⊂ \mathbb{R}^{3} → \mathbb{R}^{3} be a vector field. Consider a regular surface Σ⊂U\Sigma\subset U oriented with a field of normal unit vectors ν^ν̂. We orient its boundary ∂+Σ∂^+\Sigma positively and assume that it is the support of a closed (piecewise) regular curve. Then

∬Σ(∇×F)⋅ν^  dS=∮∂+ΣF×τ^  ds\iint_{\Sigma}(\nabla\times F)\cdot\hat{\nu}\;dS=\oint_{\partial^+\Sigma}F\times\hat{\tau}\;ds

In the two-dimensional case, it is called Gauss-Green theorem and states: given a vector field F=(F1,F2):U⊆R2→R2F = (F_{1} , F_{2}) : U ⊆ \mathbb{R}^{2} → \mathbb{R}^{2} and a closed domain D⊂UD ⊂ U, then the following holds:

∬D(∂F2∂x−∂F1∂y) dxdy=∮∂+DF×τ^  ds=∮∂+D(F2dx+F1dy)\iint_{D}\left(\frac{\partial F_2}{\partial x}- \frac{\partial F_{1}}{\partial y}\right)\ dxdy=\oint_{\partial^+D}F\times\hat{\tau}\;ds=\oint_{\partial^{+}D}(F_{2}dx+F_{1}dy)