Stokes theorem

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