View source for Physics:Kelvin–Stokes theorem

{{Hatnote|Not to be confused with the generalized Stokes theorem.}}[[Image:Stokes' Theorem.svg|thumb|right|An illustration of the Kelvin–Stokes theorem, with surface {{math|Σ}}, its boundary {{math|∂Σ}} and the normal vector {{mvar|n}}.]]
{{Calculus |Vector}}

The '''Kelvin–Stokes theorem''',<ref name="iwahori">
Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12 {{ISBN|978-4-7853-1039-4}}
[http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1039-4.htm](Written in Japanese)</ref><ref name="fujimno">Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
Bai-Fu-Kan(jp)(1979/01) {{ISBN|978-4563004415}} [{{Google books |plainurl=yes |id=nXhDywAACAAJ }}] (Written in Japanese)</ref> named after [[Biography:Lord Kelvin|Lord Kelvin]] and [[Biography:Sir George Stokes, 1st Baronet|George Stokes]], also known as the '''Stokes' theorem''',<ref>{{Cite book|title=Calculus - Early Transcendentals|last=Stewart|first=James|publisher=Brooks/Cole Cengage Learning|year=2012|isbn=978-0-538-49790-9|edition=7th|location=|pages=1122}}</ref> the '''fundamental theorem for curls''' or simply the '''curl theorem''',<ref>{{Cite book|title=Introduction to Electrodynamics|last=Griffiths|first=David|publisher=Pearson|year=2013|isbn=978-0-321-85656-2|location=|pages=34}}</ref> is a theorem in [[Vector calculus|vector calculus]] on <math>\mathbb{R}^3</math>. Given a [[Vector field|vector field]], the theorem relates the [[Surface integral|integral]] of the [[Curl (mathematics)|curl]] of the vector field over some surface, to the [[Line integral|line integral]] of the vector field around the boundary of the surface.

If a vector field  <math>\mathbf{A} = (P(x, y, z), Q(x, y, z), R(x, y, z))</math> is defined in a region with smooth oriented surface <math>\Sigma</math> and has first order continuous partial derivatives then:

:<math>\begin{align}
\iint\limits_\Sigma (\nabla \times \mathbf{A}) \cdot d\mathbf{a} & = \iint\limits_\Sigma \Bigg(\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z} \right)\,dy\,dz +\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\,dz\,dx  +\left (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy\Bigg) \\
& = \oint\limits_{\partial\Sigma} \Big(P\,dx+Q\,dy+R\,dz\Big) = \oint\limits_{\partial\Sigma} \mathbf{A} \cdot d\mathbf{l},
\end{align}</math>

where <math>\partial \Sigma</math> is boundary of region with smooth surface <math>\Sigma</math>.

The above classical Kelvin-Stokes theorem can be stated in one sentence: The [[Line integral|line integral]] of a vector field over a loop is equal to the ''[[Physics:Flux|flux]] of its curl'' through the enclosed surface.

The Kelvin–Stokes theorem is a special case of the "generalized [[Physics:Stokes' theorem|Stokes' theorem]]."<ref name="DTPO">{{cite book |first=Lawrence |last=Conlon |title=Differentiable Manifolds |series=Modern Birkhauser Classics |publisher=Birkhaeuser |location=Boston |year=2008 |url={{Google books |plainurl=yes |id=r2K31Pz5EGcC |page=194 }} }}</ref><ref name="lee">{{cite book |first=John M. |last=Lee |title=Introduction to Smooth Manifolds |series=Graduate Texts in Mathematics |volume=218 |publisher=Springer |year=2002 |url={{Google books |plainurl=yes |id=xygVcKGPsNwC |page=421 }} }}</ref> In particular, a vector field on <math>\mathbb{R}^3</math> can be considered as a [[Differential form|1-form]] in which case its curl is its [[Exterior derivative|exterior derivative]], a 2-form.

==Theorem==
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary.  Surfaces such as the [[Koch snowflake]], for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in [[Lebesgue integration|Lebesgue theory]] cannot be defined for a non-Lipschitz surface.  One (advanced) technique is to pass to a [[Weak formulation|weak formulation]] and then apply the machinery of [[Geometric measure theory|geometric measure theory]]; for that approach see the [[Coarea formula|coarea formula]].  In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of {{math|ℝ{{sup|2}}}}.

Let {{math|''γ'': [''a'', ''b''] → '''R'''<sup>2</sup>}} be a [[Piecewise|piecewise]] smooth Jordan plane curve. The [[Jordan curve theorem]] implies that {{mvar|γ}} divides {{math|'''R'''<sup>2</sup>}} into two components, a [[Compact space|compact]] one and another that is non-compact. Let {{mvar|D}} denote the compact part; then {{mvar|D}} is bounded by {{mvar|γ}}.  It now suffices to transfer this notion of boundary along a continuous map to our surface in {{math|ℝ{{sup|3}}}}.  But we already have such a map: the [[Parametrization (geometry)|parametrization]] of {{mvar|&Sigma;}}.

Suppose {{math|''ψ'': ''D'' → '''R'''<sup>3</sup>}} is smooth, with {{math|''&Sigma;'' {{=}} ''ψ''(''D'')}}. If {{mvar|Γ}} is the space curve defined by {{math|''Γ''(''t'') {{=}} ''ψ''(''γ''(''t''))}}<ref group="note" name=cgamma>{{mvar|Γ}} may not be a Jordan curve, if the loop {{mvar|γ}} interacts poorly with {{mvar|ψ}}.  Nonetheless, {{mvar|Γ}} is always a [[Loop (topology)|loop]], and topologically a [[Connected sum|connected sum]] of [[Countable set|countably-many]] Jordan curves, so that the integrals are well-defined.</ref>, then we call {{math|Γ}} the boundary of {{mvar|&Sigma;}}, written {{math|&part;''&Sigma;''}}.

With the above notation, if {{math|'''F'''}} is any smooth vector field on {{math|'''R'''<sup>3</sup>}}, then<ref name="Jame">{{cite book|url={{Google books |plainurl=yes |id=btIhvKZCkTsC |page=786 }}|title=Essential Calculus: Early Transcendentals|last=Stewart|first=James|publisher=Cole|year=2010}}</ref><ref name="bath">Robert Scheichl, lecture notes for [[Organization:University of Bath|University of Bath]] mathematics course  [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf]</ref><math display="block">\oint_{\partial\Sigma} \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}}  = \iint_{\Sigma} \nabla\times\mathbf{F}\, \cdot\, d\mathbf{S}. </math>

==Proof==
The proof of the theorem consists of 4 steps. We assume [[Green's theorem]], so what is of concern is how to boil down the three-dimensional complicated problem (Kelvin–Stokes theorem) to a two-dimensional rudimentary problem (Green's theorem).<ref>{{cite book|title=Vector Calculus|last=Colley|first=Susan Jane|edition=4th|publisher=Pearson|year=2002|location=Boston|pp=500-3}}</ref>  When proving this theorem, mathematicians normally deduce it as a special case of a [[Physics:Stokes' theorem|more general result]], which is stated in terms of [[Differential form|differential forms]], and proved using more sophisticated machinery. While powerful, these techniques require substantial background, so the proof below avoids them, and does not presuppose any knowledge beyond a familiarity with basic vector calculus.<ref name="bath"/> At the end of this section, a short alternate proof of the Kelvin-Stokes theorem is given, as a corollary of the generalized Stokes' Theorem.

===Elementary proof===
====First step of the proof (parametrization of integral)====
As in {{slink|Theorem}}, we reduce the dimension by using the natural parametrization of the surface.  Let {{math|''' ''&psi;'' '''}} and {{mvar|&gamma;}} be as in that section, and note that by change of variables

:<math>\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,d\mathbf{l}}=\oint_{\gamma}{\mathbf{F}(\mathbf{\psi}(\mathbf{y}))\cdot\,d\mathbf{\psi}(\mathbf{y})}=\oint_{\gamma}{\mathbf{F}(\mathbf{\psi}(\mathbf{y}))J_{\mathbf{y}}(\mathbf{\psi})\,d\mathbf{y}}</math>

where {{mvar|Jψ}} stands for the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|ψ}}.

Now let {{math|{'''e'''{{sub|u}},'''e'''{{sub|v}}<nowiki>}</nowiki>}} be an orthonormal basis in the coordinate directions of {{math|ℝ{{sup|2}}}}.  Recognizing that the columns of {{math|''J{{sub|'''y'''}}&psi;''}} are precisely the partial derivatives of {{math|''' ''&psi;'' '''}} at {{math|'' '''y''' ''}}, we can expand the previous equation in coordinates as

:<math>\begin{align}
\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,d\mathbf{l}}&=\oint_{\gamma}{\mathbf{F}(\mathbf{\psi}(\mathbf{y}))J_{\mathbf{y}}(\mathbf{\psi})\mathbf{e}_u(\mathbf{e}_u\cdot\,d\mathbf{y})+\mathbf{F}(\mathbf{\psi}(\mathbf{y}))J_{\mathbf{y}}(\mathbf{\psi})\mathbf{e}_v(\mathbf{e}_v\cdot\,d\mathbf{y})} \\
&=\oint_{\gamma}{\left(\left(\mathbf{F}(\mathbf{\psi}(\mathbf{y}))\cdot\frac{\partial\mathbf{\psi}}{\partial u}(\mathbf{y})\right)\mathbf{e}_u+\left(\mathbf{F}(\mathbf{\psi}(\mathbf{y}))\cdot\frac{\partial\mathbf{\psi}}{\partial v}(\mathbf{y})\right)\mathbf{e}_v\right)\cdot\,d\mathbf{y}}
\end{align}</math>

====Second step in the proof (defining the pullback)====
The previous step suggests we define the function 

:<math>\mathbf{P}(u,v)=\left(\mathbf{F}(\mathbf{\psi}(u,v))\cdot\frac{\partial\mathbf{\psi}}{\partial u}(u,v)\right)\mathbf{e}_u+\left(\mathbf{F}(\mathbf{\psi}(u,v))\cdot\frac{\partial\mathbf{\psi}}{\partial v} \right)\mathbf{e}_v</math>

This is the [[Pullback (differential geometry)|pullback]] of {{math|''' ''F'' '''}} along {{math|''' ''&psi;'' '''}}, and, by the above, it satisfies

:<math>\oint_{\partial\Sigma}{\mathbf{F}(\mathbf{x})\cdot\,d\mathbf{l}}=\oint_{\gamma}{\mathbf{P}(\mathbf{y})\cdot\,d\mathbf{l}}</math>

We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.

====Third step of the proof (second equation)====
First, calculate the partial derivatives appearing in [[Green's theorem]], via the [[General Leibniz rule|product rule]]:

:<math>\begin{align}
\frac{\partial P_1}{\partial v} &= \frac{\partial (\mathbf{F}\circ \psi)}{\partial v}\cdot\frac{\partial \psi}{\partial u} + (\mathbf{F}\circ\psi)\cdot\frac{\partial^2 \psi}{ \partial v \partial u} \\
\frac{\partial P_2}{\partial u} &= \frac{\partial (\mathbf{F}\circ \psi)}{\partial u}\cdot\frac{\partial \psi}{\partial v} + (\mathbf{F}\circ \psi) \cdot\frac{\partial^2 \psi}{\partial u \partial v}
\end{align}</math>

Conveniently, the second term vanishes in the difference, by equality of mixed partials.  So,

:<math>\begin{align}
\frac{\partial P_1}{\partial v} - \frac{\partial P_2}{\partial u} &= \frac{\partial (\mathbf{F}\circ \psi)}{\partial v}\cdot\frac{\partial \psi}{\partial u} - \frac{\partial (\mathbf{F}\circ \psi)}{\partial u}\cdot\frac{\partial \psi}{\partial v} \\
&= \frac{\partial \psi}{\partial u}(J_{\psi(u,v)}\mathbf{F})\frac{\partial \psi}{\partial v} - \frac{\partial \psi}{\partial v}(J_{\psi(u,v)}\mathbf{F})\frac{\partial \psi}{\partial u} && \text{ (chain rule)}\\
&= \frac{\partial \psi}{\partial u}(J_{\psi(u,v)}\mathbf{F}-(J_{\psi(u,v)}\mathbf{F})^{\mathsf{T}})\frac{\partial \psi}{\partial v}
\end{align}
</math>

But now consider the matrix in that quadratic form—that is, <math>J_{\psi(u,v)}\mathbf{F}-(J_{\psi(u,v)}\mathbf{F})^{\mathsf{T}}</math>.  We claim this matrix in fact describes a cross product.

To be precise, let <math>A=(A_{ij})_{ij}</math> be an arbitrary {{math|3 × 3}} matrix and let

:<math>\mathbf{a}=\begin{pmatrix}A_{32}-A_{23} \\ A_{13}-A_{31} \\ A_{21}-A_{12}\end{pmatrix}</math>

Note that {{math|''' ''x'' '''↦''' ''a'' ''' &times; ''' ''x'' '''}} is linear, so it is determined by its action on basis elements.  But by direct calculation
:<math>\begin{align}
(A-A^{\mathsf{T}})\mathbf{e}_1 &= \begin{pmatrix} 0 \\ a_3\\ -a_2 \end{pmatrix} = \mathbf{a}\times\mathbf{e}_1\\
(A-A^{\mathsf{T}})\mathbf{e}_2 &= \begin{pmatrix} -a_3\\ 0\\ a_1 \end{pmatrix} = \mathbf{a}\times \mathbf{e}_{2}\\
(A-A^{\mathsf{T}})\mathbf{e}_3 &= \begin{pmatrix} a_2\\ -a_1\\ 0 \end{pmatrix} = \mathbf{a}\times\mathbf{e}_{3}
\end{align}</math>　

Thus {{math|(''A''-''A''{{sup|T}})''' ''x'' '''{{=}}''' ''a'' ''' &times; ''' ''x'' '''}} for any {{math|''' ''x'' '''}}.  Substituting {{math|''J'' '' '''F''' ''}} for {{mvar|A}}, we obtain 

:<math>({(J_{\psi(u,v)}\mathbf{F})}_{\psi(u,v)} - {(J_{\psi(u,v)}\mathbf{F})}^{\mathsf{T}}) \mathbf{x} =(\nabla\times\mathbf{F})\times \mathbf{x}, \quad \text{for all}\, \mathbf{x}\in\mathbf{R}^{3}</math>

We can now recognize the difference of partials as a [[Triple product|(scalar) triple product]]:

:<math>\begin{align}
\frac{\partial P_1}{\partial v} - \frac{\partial P_2}{\partial u} &= \frac{\partial \psi}{\partial u}\cdot(\nabla\times\mathbf{F})\times\frac{\partial \psi}{\partial v} \\
&=\det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ]
\end{align}</math>

On the other hand, the definition of a [[Surface integral|surface integral]] also includes a triple product—the very same one!

:<math>\begin{align}
\iint_S (\nabla\times\mathbf{F})\cdot \, d^2\mathbf{S} &=\iint_D{(\nabla\times\mathbf{F})(\psi(u,v))\cdot\left(\frac{\partial\psi}{\partial u}(u,v)\times \frac{\partial\psi}{\partial v}(u,v)\,du\,dv\right)}\\
&= \iint_D \det \left [ (\nabla\times\mathbf{F})(\psi(u,v)) \quad \frac{\partial\psi}{\partial u}(u,v) \quad \frac{\partial\psi}{\partial v}(u,v) \right ] \,du \,dv
\end{align}</math>

So, we obtain

:<math> \iint_S (\nabla\times\mathbf{F})\cdot \,d^2\mathbf{S}  =\iint_D \left( \frac{\partial P_2}{\partial u} - \frac{\partial P_1}{\partial v} \right) \,du\,dv </math>

====Fourth step of the proof (reduction to Green's theorem)====
Combining the second and third steps, and then applying [[Green's theorem]] completes the proof.

===Proof via differential forms===
{{math|ℝ&rarr;ℝ{{sup|3}}}} can be identified with the differential 1-forms on {{math|ℝ{{sup|3}}}} via the map 

:<math>F_1\mathbf{e}_1+F_2\mathbf{e}_2+F_3\mathbf{e}_3 \mapsto F_1\,dx+F_2\,dy+F_3dy</math>.

Write the differential 1-form associated to a function {{math|'''F'''}} as {{math|''ω''<sub>'''F'''</sub>}}.  Then one can calculate that 

:<math>\star\omega_{\nabla\times\mathbf{F}}=d\omega_{\mathbf{F}}</math>

where {{math|★}} is the Hodge star and <math>d</math> is the [[Exterior derivative|exterior derivative]].  Thus, by [[Physics:Stokes' theorem|generalized Stokes' theorem]],<ref>{{cite book |last=Edwards |first=Harold M. |title=Advanced Calculus: A Differential Forms Approach |publisher=Birkhäuser |year=1994 |isbn=0-8176-3707-9 }}</ref>

:<math>\oint_{\partial\Sigma}{\mathbf{F}\cdot\,d\mathbf{l}}=\oint_{\partial\Sigma}{\omega_{\mathbf{F}}}=\int_{\Sigma}{d\omega_{\mathbf{F}}}=\int_{\Sigma}{\star\omega_{\nabla\times\mathbf{F}}}=\iint_{\Sigma}{\nabla\times\mathbf{F}\cdot\,d^2\mathbf{S}}
</math>

==Applications==
===In fluid dynamics===
In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.

===Irrotational fields===
<div class="messagebox standard-talk NavFrame">
'''Definition 2-1 (Irrotational field).''' A smooth vector field {{math|'''F'''}} on an [[Open set|open]] {{math|''U'' ⊆ '''R'''<sup>3</sup>}} is '''irrotational''' if {{math|∇ × '''F''' {{=}} 0}}.
</div>

If the domain of '''F''' is simply connected, then {{math|'''F'''}} is a [[Conservative vector field|conservative vector field]].

====Helmholtz's theorems====
In this section, we will introduce a theorem that is derived from the Kelvin–Stokes theorem and characterizes vortex-free vector fields. In fluid dynamics it is called [[Physics:Helmholtz's theorems|Helmholtz's theorems]].

<div class="messagebox standard-talk NavFrame">
'''Theorem 2-1 (Helmholtz's Theorem in Fluid Dynamics).'''<ref name="DTPO"/><ref name=fujimno/>{{rp|142}}  Let {{math|''U'' ⊆ '''R'''<sup>3</sup>}} be an [[Open set|open]] [[Subset|subset]] with a lamellar vector field {{math|'''F'''}} and let {{math|''c''<sub>0</sub>, ''c''<sub>1</sub>: [0, 1] → ''U''}} be piecewise smooth loops. If there is a function {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that
* '''[TLH0]''' {{mvar|H}} is piecewise smooth,
* '''[TLH1]''' {{math|''H''(''t'', 0) {{=}} ''c''<sub>0</sub>(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[TLH2]''' {{math|''H''(''t'', 1) {{=}} ''c''<sub>1</sub>(''t'')}} for all {{math|''t'' ∈ [0, 1]}},
* '''[TLH3]''' {{math|''H''(0, ''s'') {{=}} ''H''(1, ''s'')}}  for all {{math|''s'' ∈ [0, 1]}}.
Then,
:<math>\int_{c_0} \mathbf{F} \, dc_0=\int_{c_1} \mathbf{F} \, dc_1</math>
</div>

Some textbooks such as Lawrence<ref name="DTPO"/> call the relationship between {{math|''c''<sub>0</sub>}} and {{math|''c''<sub>1</sub>}} stated in Theorem 2-1 as "homotopic" and the function {{math|''H'': [0, 1] × [0, 1] → ''U''}} as "homotopy between {{math|''c''<sub>0</sub>}} and {{math|''c''<sub>1</sub>}}". However, "homotopic" or "homotopy" in above-mentioned sense are different (stronger than) [[Homotopy|typical definitions]] of "homotopic" or "homotopy"; the latter omit condition [TLH3].  So from now on we refer to homotopy (homotope) in the sense of Theorem 2-1 as a '''tubular homotopy (resp. tubular-homotopic)'''.{{refn|group="note"|name="TLH"|There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of Theorem 2-1.<ref name="DTPO" />  Indeed, this is very convenient ''for the specific problem'' of conservative forces.  However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that end.}}

=====Proof of the theorem=====
[[File:Domain of singular 2 cube.jpg|thumb|The definitions of {{math|''γ''<sub>1</sub>, ..., ''γ''<sub>4</sub>}}]]

In what follows, we [[Abuse of notation|abuse notation]] and use "{{math|+}}" for concatenation of paths in the [[Fundamental groupoid|fundamental groupoid]] and "{{math|-}}" for reversing the orientation of a path.

Let {{math|''D'' {{=}} [0, 1] × [0, 1]}}, and split {{math|&part;''D''}} into 4 line segments {{math|&gamma;{{sub|j}}}}. 
:<math>\begin{align}
\gamma_1:[0,1] \to D;\quad&\gamma_1(t) = (t, 0) \\
\gamma_2:[0,1] \to D;\quad&\gamma_2(s) = (1, s) \\
\gamma_3:[0,1] \to D;\quad&\gamma_3(t) = (1-t, 1) \\
\gamma_4:[0,1] \to D;\quad&\gamma_4(s) = (0, 1-s)
\partial D = \gamma_1+\gamma_2+\gamma_3+\gamma_4
\end{align}</math>

By our assumption that {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are piecewise smooth homotopic, there is a piecewise smooth homotopy {{math|''H'': ''D'' → ''M''}}
:<math>
\begin{align}
\Gamma_i(t) &= H(\gamma_{i}(t)) && i=1, 2, 3, 4 \\
\Gamma(t) &= H(\gamma(t)) =(\Gamma_1 \oplus \Gamma_2 \oplus \Gamma_3 \oplus \Gamma_4)(t)
\end{align}</math>

Let ''S'' be the image of {{mvar|D}} under ''H''.  That

:<math> \iint_S \nabla\times\mathbf{F}\, dS = \oint_\Gamma \mathbf{F}\, d\Gamma </math>

follows immediately from the Kelvin–Stokes theorem.  {{math|'''F'''}} is lamellar, so the left side vanishes, i.e.

:<math>0=\oint_\Gamma \mathbf{F}\, d\Gamma = \sum_{i=1}^4 \oint_{\Gamma_i} \mathbf{F} \, d\Gamma </math> 

As ''H'' is tubular, {{math|''&Gamma;''{{sub|2}}{{=}}-''&Gamma;''{{sub|4}}}}.  Thus the line integrals along {{math|Γ<sub>2</sub>(''s'')}} and {{math|Γ<sub>4</sub>(''s'')}} cancel, leaving

:<math>0=\oint_{\Gamma_1} \mathbf{F} \, d\Gamma +\oint_{\Gamma_3} \mathbf{F} d\Gamma</math>

On the other hand, {{math|''c''{{sub|1}}{{=}}''&Gamma;''{{sub|1}}}} and {{math|''c''{{sub|3}}{{=}}-''&Gamma;''{{sub|3}}}}, so that the desired equality follows almost immediately.

===Conservative forces===
Helmholtz's theorem gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 2-2, which is a corollary of and a special case of Helmholtz's theorem.

<div class="messagebox standard-talk NavFrame">
'''Lemma 2-2.'''<ref name="DTPO"/><ref name=lee/> Let {{math|''U'' ⊆ '''R'''<sup>3</sup>}} be an [[Open set|open]] [[Subset|subset]], with a Lamellar vector field {{math|'''F'''}} and a piecewise smooth loop {{math|''c''<sub>0</sub>: [0, 1] → ''U''}}. Fix a point {{math|'''p''' ∈ ''U''}}, if there is a homotopy (tube-like-homotopy) {{math|''H'': [0, 1] × [0, 1] → ''U''}} such that
* '''[SC0]''' ''H'' is '''piecewise smooth''',
* '''[SC1]''' ''H''(''t'', 0) = ''c''<sub>0</sub>(''t'') for all ''t'' ∈ [0, 1],
* '''[SC2]''' ''H''(''t'', 1) = '''p''' for all ''t'' ∈ [0, 1],
* '''[SC3]''' ''H''(0, ''s'') = ''H''(1, ''s'') = '''p''' for all ''s'' ∈ [0, 1].
Then,
:<math>\int_{c_0} \mathbf{F} \, dc_0=0</math>
</div>

Lemma 2-2 follows from Theorem 2-1. In Lemma 2-2, the existence of {{mvar|H}} satisfying [SC0] to [SC3] is crucial. If ''U'' is simply connected, such ''H'' exists. The definition of [[Simply connected space]] follows:

<div class="messagebox standard-talk NavFrame">
'''Definition 2-2 (Simply Connected Space).'''<ref name="DTPO"/><ref name=lee/> Let {{math|''M'' ⊆ '''R'''<sup>''n''</sup>}} be non-empty and [[Connected space#Path connectedness|path-connected]].  {{mvar|M}} is called simply connected if and only if for any continuous loop, {{math|''c'': [0, 1] → ''M''}} there exists a continuous tubular homotopy {{math|''H'': [0, 1] × [0, 1] → ''M''}} from {{mvar|c}} to a fixed point {{math|''p'' ∈ ''c''}}; that is,
* '''[SC0']''' ''H'' is '''continuous''',
* '''[SC1]''' ''H''(''t'', 0) = ''c''(''t'') for all ''t'' ∈ [0, 1],
* '''[SC2]''' ''H''(''t'', 1) = '''p''' for all ''t'' ∈ [0, 1],
* '''[SC3]''' ''H''(0, ''s'') = ''H''(1, ''s'') = '''p'''  for all ''s'' ∈ [0, 1].
</div>

The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately.  But recall that simple-connection only guarantees the existence of a ''continuous'' homotopy satisfiying [SC1-3]; we seek a piecewise smooth hoomotopy satisfying those conditions instead.

However, the gap in regularity is resolved by the Whitney approximation theorem.<ref name=lee/>{{rp|136,421}}<ref name="ptr">L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American Mathematical Society Translations, Ser. 2, Vol. 11, American Mathematical Society, Providence, R.I., 1959, pp. 1–114. {{MR|0115178}} (22 #5980 [http://www.ams.org/mathscinet-getitem?mr=0115178]).  See theorems 7 & 8.</ref>  We thus obtain the following theorem.

<div class="messagebox standard-talk NavFrame">
'''Theorem 2-2.'''<ref name="DTPO"/><ref name=lee/> Let {{math|''U'' ⊆ '''R'''<sup>3</sup>}} be [[Open set|open]] and simply connected with an irrotational vector field {{math|'''F'''}}. For all piecewise smooth loops {{math|''c'': [0, 1] → ''U''}}
:<math>\int_{c_0} \mathbf{F} \, dc_0=0</math>
</div>

===Maxwell's equations===
In the physics of [[Physics:Electromagnetism|electromagnetism]], the Kelvin-Stokes theorem provides the justification for the equivalence of the differential form of the Maxwell–Faraday equation and the [[Physics:Ampère's circuital law|Maxwell–Ampère equation]] and the integral form of these equations. For Faraday's law, the Kelvin-Stokes theorem is applied to the electric field, <math>\mathbf{E}</math>.

<math>\oint_{\partial\Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{\nabla}\times \mathbf{E} \cdot \mathrm{d} \mathbf{S}</math>

For Ampère's law, the Kelvin-Stokes theorem is applied to the magnetic field, <math>\mathbf{B}</math>.

<math>\oint_{\partial\Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{\nabla}\times \mathbf{B} \cdot \mathrm{d} \mathbf{S}</math>

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

{{DEFAULTSORT:Kelvin-Stokes theorem}}
[[Category:Electromagnetism]]
[[Category:Mechanics]]
[[Category:Vectors (mathematics and physics)]]
[[Category:Vector calculus]]

{{Sourceattribution|Kelvin–Stokes theorem|1}}