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Stokes theorem

From Encyclopedia of Mathematics
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2010 Mathematics Subject Classification: Primary: 58A [MSN][ZBL]

The term refers, in the modern literature, to the following theorem.

Theorem 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 \[ \int_M d \omega = \int_{\partial M} \omega \] (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 Stokes formula can be recovered as particular cases. The latter is also often called Stokes theorem and it is stated as follows.

Theorem 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.

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