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Stokes' Theorem


For omega a differential (k-1)-form with compact support on an oriented k-dimensional manifold with boundary M,

 int_Mdomega=int_(partialM)omega,
(1)

where domega is the exterior derivative of the differential form omega. When M is a compact manifold without boundary, then the formula holds with the right hand side zero.

Stokes' theorem connects to the "standard" gradient, curl, and divergence theorems by the following relations. If f is a function on R^3,

 grad(f)=c^(-1)df,
(2)

where c:R^3->R^3^* (the dual space) is the duality isomorphism between a vector space and its dual, given by the Euclidean inner product on R^3. If f is a vector field on a R^3,

 div(f)=^*d^*c(f),
(3)

where * is the Hodge star operator. If f is a vector field on R^3,

 curl(f)=c^(-1)^*dc(f).
(4)

With these three identities in mind, the above Stokes' theorem in the three instances is transformed into the gradient, curl, and divergence theorems respectively as follows. If f is a function on R^3 and gamma is a curve in R^3, then

 int_gammagrad(f)·dl=int_gammadf=f(gamma(1))-f(gamma(0)),
(5)

which is the gradient theorem. If f:R^3->R^3 is a vector field and M an embedded compact 3-manifold with boundary in R^3, then

 int_(partialM)f·dA=int_(partialM)^*cf=int_Md*cf=int_Mdiv(f)dV,
(6)

which is the divergence theorem. If f is a vector field and M is an oriented, embedded, compact 2-manifold with boundary in R^3, then

 int_(partialM)fdl=int_(partialM)cf=int_Mdc(f)=int_Mcurl(f)·dA,
(7)

which is the curl theorem.

de Rham cohomology is defined using differential k-forms. When N is a submanifold (without boundary), it represents a homology class. Two closed forms represent the same cohomology class if they differ by an exact form, omega_1-omega_2=deta. Hence,

 int_Nomega_1-omega_2=int_Ndeta=0.
(8)

Therefore, the evaluation of a cohomology class on a homology class is well-defined.

Physicists generally refer to the curl theorem

 int_S(del xF)·da=int_(partialS)F·ds
(9)

as Stokes' theorem.


See also

Cohomology, Curl Theorem, Differential k-Form, Divergence Theorem, Exterior Algebra, Exterior Derivative, Form Integration, Gradient Theorem, Hodge Star, Jacobian, Manifold, Poincaré's Lemma, Tangent Bundle

Portions of this entry contributed by Todd Rowland

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References

Morse, P. M. and Feshbach, H. "Stokes' Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 43, 1953.

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Stokes' Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StokesTheorem.html

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