Stokes theorem

In vector calculus and differential geometry, the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,





∫

∂
Ω


ω
=

∫

Ω


d
ω

.


{\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.}
Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861. This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface (that is, the flux of curl F) in Euclidean three-space to the line integral of the vector field over its boundary (also known as the loop integral).
Simple classical vector analysis example
Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t)) and F is a smooth vector field on R3, then:





∮

Γ



F


⋅

d


Γ


=

∬

S


∇
×

F


⋅

d

S



{\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }
This classical statement, is a special case of the general formulation stated above after making an identification of vector field with a 1-form and its curl with a two form through






(




F

x







F

y







F

z





)


⋅
d
Γ
→

F

x



d
x
+

F

y



d
y
+

F

z



d
z


{\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\\\end{pmatrix}}\cdot d\Gamma \to F_{x}\,dx+F_{y}\,dy+F_{z}\,dz}





∇
×


(




F

x







F

y







F

z





)


⋅
d

S

=


(




∂

y



F

z


−

∂

z



F

y







∂

z



F

x


−

∂

x



F

z







∂

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F

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−

∂

y



F

x





)


⋅
d

S

→


{\displaystyle \nabla \times {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}\cdot d\mathbf {S} ={\begin{pmatrix}\partial _{y}F_{z}-\partial _{z}F_{y}\\\partial _{z}F_{x}-\partial _{x}F_{z}\\\partial _{x}F_{y}-\partial _{y}F_{x}\\\end{pmatrix}}\cdot d\mathbf {S} \to }





d
(

F

x



d
x
+

F

y



d
y
+

F

z



d
z
)
=
(

∂

y



F

z


−

∂

z



F

y


)

d
y
∧
d
z
+
(

∂

z



F

x


−

∂

x



F

z


)

d
z
∧
d
x
+
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∂

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F

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−

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F

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d
x
∧
d
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{\displaystyle d(F_{x}\,dx+F_{y}\,dy+F_{z}\,dz)=(\partial _{y}F_{z}-\partial _{z}F_{y})\,dy\wedge dz+(\partial _{z}F_{x}-\partial _{x}F_{z})\,dz\wedge dx+(\partial _{x}F_{y}-\partial _{y}F_{x})\,dx\wedge dy}
.Other classical generalisations of the fundamental theorem of calculus like the divergence theorem, and Green's theorem are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).

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