Stokes theorem

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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}\tag{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}\tag{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 (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 (2) is then replaced by the sum of the integrals over the corresponding curves.

Both (1) and (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 (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}\tag{3} \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 (3) guarantees that $f$ is well-defined, i.e. that the value $f(q)$ does not depend on the choice of $\sigma$. We then have $v = \nabla f$.


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How to Cite This Entry:
Stokes formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article