Difference between revisions of "Stokes theorem"
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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 | + | [[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
2010 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 |
Stokes theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stokes_theorem&oldid=31279