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.
Comments
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 |
Stokes formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stokes_formula&oldid=31288