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Difference between revisions of "Stokes theorem"

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A theorem which establishes the connection between the flow of a vector field through an oriented surface and the circulation of this field along the boundary of the surface (see [[Stokes formula|Stokes formula]]).
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{{TEX|done}}
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{{MSC|58A}}
  
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The term refers, in the modern literature, to the following theorem.
  
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'''Theorem 1'''
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Let $M$ be a compact orientable [[Differentiable manifold|differentiable manifold]] with boundary (denoted by $\partial M$) and let $k$ be the dimension of $M$. If $\omega$ is a [[Differential form|differential $k-1$-form]], then
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\begin{equation}\label{e:Stokes_1}
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\int_M d \omega = \int_{\partial M} \omega
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\end{equation}
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(see [[Integration on manifolds]] for the definition of integral of a form on a differentiable manifold).
  
====Comments====
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The theorem can be considered as a generalization of the [[Fundamental theorem of calculus]]. The classical
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[[Ostrogradski formula|Gauss-Green theorem]] and the "classical" Stokes formula can be recovered as particular cases.
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The latter is also often called Stokes theorem and it is stated as follows.
  
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'''Theorem 2'''
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Let $\Sigma\subset \mathbb R^3$ be a compact regular $2$-dimensional surface $\Sigma$ that bounds the $C^1$ curve $\gamma$ and let $v$ be a $C^1$ vector field. Then
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\begin{equation}\label{e:Stokes_2}
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\int_\Sigma (\nabla \times v) \cdot \nu = \int_\gamma \tau \cdot v\, ,
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\end{equation}
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where
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* $\nu$ is a continuous unit vector field normal to the surface $\Sigma$
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* $\tau$ is a continuous unit vector field tangent to the curve $\gamma$, compatible with $\nu$
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* $\nabla \times v$ is the [[Curl|curl]] of the vector field $v$.
 +
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The right hand side of \eqref{e:Stokes_2} is called the ''flow of $v$ through $\Sigma$'', whereas the left hand side is called the ''circulation of $v$ along $\gamma$''. The theorem can be easily generalized to surfaces whose boundary consists of finitely many curves: the right hand side of \eqref{e:Stokes_2} is then replaced by the sum of the integrals over the corresponding curves.
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Both \eqref{e:Stokes_1} and \eqref{e:Stokes_2} are often called  ''Stokes formula''. If the vector field of Theorem 2 is given, in the  coordinates $x_1, x_2, x_3$, by $(v_1, v_2, v_3)$ and we introduce the  $1$-form
 +
\[
 +
\omega = v_1 dx_1 + v_2 dx_2 + v_3 dx_3\, ,
 +
\]
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then the right hand side of \eqref{e:Stokes_1} is indeed
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\[
 +
\int_\Sigma d \omega\, ,
 +
\]
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whereas the left hand side is
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\[
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\int_{\partial \Sigma} \omega\, .
 +
\]
 +
 +
The regularity assumptions on $\gamma$ and $\partial M$ in both theorems can be somewhat relaxed. In particular the formulas still hold if such boundaries are piecewise $C^1$, with ''corner''-type singularities.
 +
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'''Remark 3''' The compatibility between the vector fields $\tau$ and $\nu$ in Theorem 2 can be expressed intuitively as follows. The normal $\nu$ identifies a "bottom" and a "top" on the surface $\Sigma$. To an observer which is standing on the top, $\tau$ gives a counterclockwise orientation to the curve $\gamma$. The precise mathematical definition is more cumbersome. Fix $p_0\in \gamma$, let $V\subset \mathbb R^3$ be an open neighborhood of $x_0$ and $U\subset \mathbb R^2$ the intersection of an open neighborhood of $0\in \mathbb R^2$ with the closed upper half plane $\{(x_1, x_2): x_2\geq 0\}$. Assume $\Phi: U\to V$ is a local parametrization of $\Sigma\cap V$ with $\Phi (0) = p_0$, namely that
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* $\Phi$ is $C^1$ and $D\Phi$ has rank 2 at each point of $U$
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* $\Phi$ is an homeomorphism between $U$ and $\Sigma \cap V$
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* $\Phi$ maps $\{x_2=0\}\cap U$ onto $\gamma \cap V$.
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Then the vector field
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\[
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n := \frac{\partial \Phi}{\partial x_1} \times \frac{\partial \Phi}{\partial x_2}
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\]
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is a nonzero vector field normal to the surface $\Sigma$ and therefore the scalar product
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\[
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n (x) \cdot \nu (\Phi (x))
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\]
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is either everywhere positive or everywhere negative. In the first case
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\[
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\tau (x_0) = \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, ,
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\]
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otherwise
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\[
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\tau (x_0) = - \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, .
 +
\]
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'''Remark 4''' Theorem 2 is often used to conclude that a curl-free vector-field $v$ (i.e. such that $\nabla \times v =0$) on a [[Simply-connected domain|simply-connected domain]] $U$ of $\mathbb R^3$ is a gradient. Indeed under such assumptions Theorem 2 guarantees that
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\begin{equation}\label{e:vanishes}
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\int_\gamma \tau \cdot v = 0\qquad \mbox{for every continuous and piecewise } C^1 \mbox{ closed loop } \gamma\subset U\, .
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\end{equation}
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Thus, assuming without loss of generality that $U$ is connected, if we fix a point $p_0$ and an arbitrary $C^1$ arc $\sigma\subset U$ connecting $p_0$ and $q$, we can define the potential
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\[
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f(q) :=\int_\sigma \tau\cdot v\, .
 +
\]
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The condition \eqref{e:vanishes} guarantees that $f$ is well-defined, i.e. that the value $f(q)$ does not depend on the choice of $\sigma$. We then have $v = \nabla f$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Calculus" , '''I''' , Blaisdell  (1967)</TD></TR></table>
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{|
 +
|-
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|valign="top"|{{Ref|Ap}}|| T.M. Apostol,  "Calculus" , '''I''' , Blaisdell  (1967) {{MR|0214705}} {{ZBL|0148.28201}}
 +
|-
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|valign="top"|{{Ref|Ar}}|| V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)  {{MR|}} {{ZBL|0692.70003}}  {{ZBL|0572.70001}} {{ZBL|0647.70001}}
 +
|-
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|valign="top"|{{Ref|dW}}|| C. deWitt-Morette,  "Analysis, manifolds, physics" , North-Holland  (1977)  pp. 205  (Translated from French)
 +
|-
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|valign="top"|{{Ref|Sp}}|| M. Spivak,  "Calculus on manifolds" , Benjamin  (1965) {{MR|0209411}} {{ZBL|0141.05403}}
 +
|-
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|valign="top"|{{Ref|Tr}}|| H. Triebel,  "Analysis and mathematical physics" , Reidel  (1986)  pp. 375  {{MR|0914975}} {{MR|0880867}} {{ZBL|0607.46047}}
 +
|-
 +
|}

Latest revision as of 09:41, 29 January 2014

2020 Mathematics Subject Classification: Primary: 58A [MSN][ZBL]

The term refers, in the modern literature, to the following theorem.

Theorem 1 Let $M$ be a compact orientable differentiable manifold with boundary (denoted by $\partial M$) and let $k$ be the dimension of $M$. If $\omega$ is a differential $k-1$-form, then \begin{equation}\label{e:Stokes_1} \int_M d \omega = \int_{\partial M} \omega \end{equation} (see Integration on manifolds for the definition of integral of a form on a differentiable manifold).

The theorem can be considered as a generalization of the Fundamental theorem of calculus. The classical Gauss-Green theorem and the "classical" Stokes formula can be recovered as particular cases. The latter is also often called Stokes theorem and it is stated as follows.

Theorem 2 Let $\Sigma\subset \mathbb R^3$ be a compact regular $2$-dimensional surface $\Sigma$ that bounds the $C^1$ curve $\gamma$ and let $v$ be a $C^1$ vector field. Then \begin{equation}\label{e:Stokes_2} \int_\Sigma (\nabla \times v) \cdot \nu = \int_\gamma \tau \cdot v\, , \end{equation} where

  • $\nu$ is a continuous unit vector field normal to the surface $\Sigma$
  • $\tau$ is a continuous unit vector field tangent to the curve $\gamma$, compatible with $\nu$
  • $\nabla \times v$ is the curl of the vector field $v$.

The right hand side of \eqref{e:Stokes_2} is called the flow of $v$ through $\Sigma$, whereas the left hand side is called the circulation of $v$ along $\gamma$. The theorem can be easily generalized to surfaces whose boundary consists of finitely many curves: the right hand side of \eqref{e:Stokes_2} is then replaced by the sum of the integrals over the corresponding curves.

Both \eqref{e:Stokes_1} and \eqref{e:Stokes_2} are often called Stokes formula. If the vector field of Theorem 2 is given, in the coordinates $x_1, x_2, x_3$, by $(v_1, v_2, v_3)$ and we introduce the $1$-form \[ \omega = v_1 dx_1 + v_2 dx_2 + v_3 dx_3\, , \] then the right hand side of \eqref{e:Stokes_1} is indeed \[ \int_\Sigma d \omega\, , \] whereas the left hand side is \[ \int_{\partial \Sigma} \omega\, . \]

The regularity assumptions on $\gamma$ and $\partial M$ in both theorems can be somewhat relaxed. In particular the formulas still hold if such boundaries are piecewise $C^1$, with corner-type singularities.

Remark 3 The compatibility between the vector fields $\tau$ and $\nu$ in Theorem 2 can be expressed intuitively as follows. The normal $\nu$ identifies a "bottom" and a "top" on the surface $\Sigma$. To an observer which is standing on the top, $\tau$ gives a counterclockwise orientation to the curve $\gamma$. The precise mathematical definition is more cumbersome. Fix $p_0\in \gamma$, let $V\subset \mathbb R^3$ be an open neighborhood of $x_0$ and $U\subset \mathbb R^2$ the intersection of an open neighborhood of $0\in \mathbb R^2$ with the closed upper half plane $\{(x_1, x_2): x_2\geq 0\}$. Assume $\Phi: U\to V$ is a local parametrization of $\Sigma\cap V$ with $\Phi (0) = p_0$, namely that

  • $\Phi$ is $C^1$ and $D\Phi$ has rank 2 at each point of $U$
  • $\Phi$ is an homeomorphism between $U$ and $\Sigma \cap V$
  • $\Phi$ maps $\{x_2=0\}\cap U$ onto $\gamma \cap V$.

Then the vector field \[ n := \frac{\partial \Phi}{\partial x_1} \times \frac{\partial \Phi}{\partial x_2} \] is a nonzero vector field normal to the surface $\Sigma$ and therefore the scalar product \[ n (x) \cdot \nu (\Phi (x)) \] is either everywhere positive or everywhere negative. In the first case \[ \tau (x_0) = \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, , \] otherwise \[ \tau (x_0) = - \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, . \]

Remark 4 Theorem 2 is often used to conclude that a curl-free vector-field $v$ (i.e. such that $\nabla \times v =0$) on a simply-connected domain $U$ of $\mathbb R^3$ is a gradient. Indeed under such assumptions Theorem 2 guarantees that \begin{equation}\label{e:vanishes} \int_\gamma \tau \cdot v = 0\qquad \mbox{for every continuous and piecewise } C^1 \mbox{ closed loop } \gamma\subset U\, . \end{equation} Thus, assuming without loss of generality that $U$ is connected, if we fix a point $p_0$ and an arbitrary $C^1$ arc $\sigma\subset U$ connecting $p_0$ and $q$, we can define the potential \[ f(q) :=\int_\sigma \tau\cdot v\, . \] The condition \eqref{e:vanishes} guarantees that $f$ is well-defined, i.e. that the value $f(q)$ does not depend on the choice of $\sigma$. We then have $v = \nabla f$.

References

[Ap] T.M. Apostol, "Calculus" , I , Blaisdell (1967) MR0214705 Zbl 0148.28201
[Ar] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[dW] C. deWitt-Morette, "Analysis, manifolds, physics" , North-Holland (1977) pp. 205 (Translated from French)
[Sp] M. Spivak, "Calculus on manifolds" , Benjamin (1965) MR0209411 Zbl 0141.05403
[Tr] H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. 375 MR0914975 MR0880867 Zbl 0607.46047
How to Cite This Entry:
Stokes theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stokes_theorem&oldid=17962
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article