Stokes' theorem

Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of a surface to the behavior of its curl on the surface itself. In its usual three-dimensional form, it says that the total circulation of a vector field around a closed curve is equal to the total curl of the field through a surface bounded by that curve.

If Σ is an oriented surface with boundary ∂Σ, Stokes' theorem is commonly written as

∮

∂

Σ

F

⋅

d

r

=

∬

Σ

(

∇

×

F

)

⋅

n

d

S

.

{\displaystyle \oint _{\partial \Sigma }\mathbf {F} \cdot d\mathbf {r} =\iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot \mathbf {n} \,dS.}

Here the left side is the line integral of the vector field around the boundary curve, while the right side is the surface integral of its curl over the surface. Informally, the theorem says that adding up the local rotation of a vector field across a surface gives the net circulation around its edge.

The theorem is also called the fundamental theorem for curls, the curl theorem, or the rotor theorem. It is a special case of the generalized Stokes theorem. In the language of differential forms, the vector field corresponds to a 1-form and its curl corresponds to the exterior derivative of that form.

Theorem

Let

Σ

{\displaystyle \Sigma }

be a smooth oriented surface in

R

3

{\displaystyle \mathbb {R} ^{3}}

, parametrized by

Σ

(

u

,

v

)

{\displaystyle \mathbf {\Sigma } (u,v)}

, with boundary

∂

Σ

≡

Γ

{\displaystyle \partial \Sigma \equiv \Gamma }

, parametrized by

Γ

(

t

)

{\displaystyle \mathbf {\Gamma } (t)}

. If a vector field

F

(

x

,

y

,

z

)

=

(

F

x

(

x

,

y

,

z

)

,

F

y

(

x

,

y

,

z

)

,

F

z

(

x

,

y

,

z

)

)

{\displaystyle \mathbf {F} (x,y,z)=(F_{x}(x,y,z),F_{y}(x,y,z),F_{z}(x,y,z))}

has continuous first-order partial derivatives in

Σ

{\displaystyle \Sigma }

, then

∬

Σ

(

∇

×

F

)

⋅

d

Σ

=

∮

∂

Σ

F

⋅

d

Γ

{\displaystyle \iint _{\Sigma }(\nabla \times \mathbf {F} )\cdot d\mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot d\mathbf {\Gamma } }

with the shorthands for the line element

d

Γ

=

d

Γ

d

t

d

t

{\displaystyle d\mathbf {\Gamma } ={\frac {d\mathbf {\Gamma } }{dt}}dt}

and the surface element

d

Σ

=

n

d

Σ

=

(

∂

Σ

∂

u

×

∂

Σ

∂

v

)

d

u

d

v

{\displaystyle d\mathbf {\Sigma } =\mathbf {n} d\Sigma =\left({\frac {\partial \mathbf {\Sigma } }{\partial u}}\times {\frac {\partial \mathbf {\Sigma } }{\partial v}}\right)dudv}

where

n

(

u

,

v

)

{\displaystyle \mathbf {n} (u,v)}

is the vector orthogonal to the surface at the point

Σ

(

u

,

v

)

{\displaystyle \mathbf {\Sigma } (u,v)}

.