Stokes formula

A formula that expresses the connection between the flow of a vector field through a two-dimensional oriented manifold and the circulation of this field along the correspondingly oriented boundary of this manifold. Let be an oriented piecewise-smooth surface, let be the unit normal to (at those points, of course, where it exists), which defines the orientation of , and let the boundary of consist of a finite number of piecewise-smooth contours. The boundary of is denoted by , and is oriented by means of the unit tangent vector , such that the orientation of obtained is compatible with the orientation of .

If is a continuously-differentiable vector field in a neighbourhood of , then

(*)

( is the area element of , is the differential of the arc length of the boundary of ) or, in coordinate form,

Stated by G. Stokes (1854).

Stokes' formula is also the name given to a generalization of formula , which represents the equality between the integral of the exterior differential of a differential form over an oriented compact manifold and the integral of the form itself along the boundary of (the orientation of is taken to be compatible with that of ):

Other particular cases of this formula are the Newton–Leibniz formula, the Green formulas and the Ostrogradski formula.

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[a2]	M. Spivak, "Calculus on manifolds" , Benjamin (1965) MR0209411 Zbl 0141.05403
[a3]	C. deWitt-Morette, "Analysis, manifolds, physics" , North-Holland (1977) pp. 205 (Translated from French)
[a4]	H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. 375 MR0914975 MR0880867 Zbl 0607.46047