View source for Stokes theorem

{{TEX|done}}
{{MSC|58A}}

The term refers, in the modern literature, to the following 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
\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 
[[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.

'''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|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====
{|
|-
|valign="top"|{{Ref|Ap}}|| T.M. Apostol,   "Calculus" , '''I''' , Blaisdell  (1967)  {{MR|0214705}} {{ZBL|0148.28201}} 
|-
|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}}
|-
|valign="top"|{{Ref|dW}}|| C. deWitt-Morette,   "Analysis, manifolds, physics" , North-Holland  (1977)  pp. 205  (Translated from French)
|-
|valign="top"|{{Ref|Sp}}|| M. Spivak,   "Calculus on manifolds" , Benjamin  (1965) {{MR|0209411}} {{ZBL|0141.05403}}
|-
|valign="top"|{{Ref|Tr}}|| H. Triebel,   "Analysis and mathematical physics" , Reidel  (1986)  pp. 375  {{MR|0914975}} {{MR|0880867}} {{ZBL|0607.46047}}
|-
|}