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The term refers, in the modern literature, to the following theorem.
 
The term refers, in the modern literature, to the following theorem.
  
'''Theorem'''
+
'''Theorem 1'''
 
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| differential $k-1$-form]], then
 
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| differential $k-1$-form]], then
\[
+
\begin{equation}\label{e:Stokes_1}
 
\int_M d \omega = \int_{\partial M} \omega
 
\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).
 
(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  
 
The theorem can be considered as a generalization of the [[Fundamental theorem of calculus]]. The classical  
[[Ostrogradski formula|Gauss-Green theorem]] and the [[Stokes formula]] can be recovered as particular cases.
+
[[Ostrogradski formula|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.
 
The latter is also often called Stokes theorem and it is stated as follows.
  
'''Theorem'''
+
'''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
 
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}
 
\begin{equation}\label{e:Stokes_2}
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* $\nabla \times v$ is the [[Curl|curl]] of the vector field $v$.
 
* $\nabla \times v$ is the [[Curl|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
+
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.  
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
 +
\[
 +
\int_{\partial \Sigma} \omega\, .
 +
\]
 +
 
 +
'''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
 +
\[
 +
\tau (x_0) = - \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, .
 +
\]
  
 
====References====
 
====References====

Revision as of 12:38, 28 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 \[ \int_{\partial \Sigma} \omega\, . \]

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 \[ \tau (x_0) = - \left|\frac{\partial \Phi}{\partial x_1} (0)\right|^{-1} \frac{\partial \Phi}{\partial x_1} (0)\, . \]

References

[Ap] T.M. Apostol, "Calculus" , I , Blaisdell (1967) MR0214705 Zbl 0148.28201
[Sp] M. Spivak, "Calculus on manifolds" , Benjamin (1965) MR0209411 Zbl 0141.05403
How to Cite This Entry:
Stokes theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stokes_theorem&oldid=31279
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article