classical Stokesโ€™ theorem

Let M be a compactPlanetmathPlanetmath, oriented two-dimensional differentiable manifold (surface) with boundary in โ„3, and ๐… be a C2-smooth vector field defined on an open set in โ„3 containing M. Then


Here, the boundary of M, โˆ‚โกM (which is a curve) is given the induced orientation from M. The symbol โˆ‡ร—๐… denotes the curl of ๐…. The symbol dโข๐ฌ denotes the line element dโขs with a direction parallelMathworldPlanetmathPlanetmathPlanetmath to the unit tangent vector ๐ญ to โˆ‚โกM, while dโข๐€ denotes the area elementMathworldPlanetmath dโขA of the surface M with a direction parallel to the unit outward normal ๐ง to M. In precise terms:


The classical Stokesโ€™ theorem reduces to Greenโ€™s theorem on the plane if the surface M is taken to lie in the xy-plane.

The classical Stokesโ€™ theorem, and the other โ€œStokesโ€™ typeโ€ theorems are special cases of the general Stokesโ€™ theorem involving differential formsMathworldPlanetmath. In fact, in the proof we present below, we appeal to the general Stokesโ€™ theorem.

Physical interpretation

(To be written.)

Proof using differential forms

The proof becomes a triviality once we express (โˆ‡ร—๐…)โ‹…dโข๐€ and ๐…โ‹…dโข๐ฌ in terms of differential forms.


Define the differential forms ฮท and ฯ‰ by

ฮทpโข(๐ฎ,๐ฏ) =โŸจcurlโก๐…โข(p),๐ฎร—๐ฏโŸฉ,
ฯ‰pโข(๐ฏ) =โŸจ๐…โข(p),๐ฏโŸฉ.

for points pโˆˆโ„3, and tangent vectors ๐ฎ,๐ฏโˆˆโ„3. The symbol โŸจ,โŸฉ denotes the dot productMathworldPlanetmath in โ„3. Clearly, the functions ฮทp and ฯ‰p are linear and alternatingPlanetmathPlanetmath in ๐ฎ and ๐ฏ.

We claim

ฮท =โˆ‡ร—๐…โ‹…dโข๐€ on M. (1)
ฯ‰ =๐…โ‹…dโข๐ฌ on โˆ‚โกM. (2)

To prove (1), it suffices to check it holds true when we evaluate the left- and right-hand sides on an orthonormal basis ๐ฎ,๐ฏ for the tangent space of M corresponding to the orientation of M, given by the unit outward normal ๐ง. We calculate

โˆ‡ร—๐…โ‹…dโข๐€โข(๐ฎ,๐ฏ) =โŸจcurlโก๐…,๐งโŸฉโขdโขAโข(๐ฎ,๐ฏ) definition of dโข๐€=๐งโขdโขA
=โŸจcurlโก๐…,๐งโŸฉ definition of volume form dโขA
=โŸจcurlโก๐…,๐ฎร—๐ฏโŸฉ since ๐ฎร—๐ฏ=๐ง

For equation (2), similarly, we only have to check that it holds when both sides are evaluated at ๐ฏ=๐ญ, the unit tangent vector of โˆ‚โกM with the induced orientation of โˆ‚โกM. We calculate again,

๐…โ‹…dโข๐ฌโข(๐ญ) =โŸจ๐…,๐ญโŸฉโขdโขsโข(๐ญ) definition of dโข๐ฌ=๐ญโขdโขs
=โŸจ๐…,๐ญโŸฉ definition of volume form dโขs

Furthermore, dโขฯ‰ = ฮท. (This can be checked by a calculation in Cartesian coordinatesMathworldPlanetmath, but in fact this equation is one of the coordinate-free definitions of the curl.)

The classical Stokesโ€™ Theorem now follows from the general Stokesโ€™ Theorem,



  • 1 Michael Spivak. Calculus on Manifolds. Perseus Books, 1998.
Title classical Stokesโ€™ theorem
Canonical name ClassicalStokesTheorem
Date of creation 2013-03-22 15:27:52
Last modified on 2013-03-22 15:27:52
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 6
Author stevecheng (10074)
Entry type Theorem
Classification msc 26B20
Related topic GeneralStokesTheorem
Related topic GaussGreenTheorem
Related topic GreensTheorem