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The Stokes theorem (also Stokes' theorem or Stokes's theorem) asserts that the integral of an exterior differential form on the boundary of an oriented manifold with boundary (or submanifold or chain of such) equals the integral of the de Rham differential of the form on the manifold itself. (The theorem also applies to exterior pseudoforms on a chain of pseudoriented submanifolds.)
Let
be the cosimplicial object of standard -simplices in SmoothMfd: in degree this is the standard -simplex regarded as a smooth manifold with boundary and corners. This may be parameterized as
In this parameterization the coface maps of are
For any smooth manifold a smooth -simplex in is a smooth function
The boundary of this simplex in is the chain (formal linear combination of smooth -simplices)
Let be a degree -differential form on .
(Stokes theorem)
The integral of over the boundary of the simplex equals the integral of its de Rham differential over the simplex itself
It follows that for any -chain in and its boundary -chain, we have
More generally:
(Stokes theorem for fiber integration)
If is any smooth manifold and is a differential form on the Cartesian product, then with respect to fiber-wise integration of differential forms
along we have
(e.g. Gomi & Terashima 2000, remark 3.1)
We discuss here a general abstract formulation of differential forms, their integration and Stokes theorem in the axiomatics of cohesive homotopy type theory (following Bunke-Nikolaus-Völkl 13, theorem 3.2).
Let be a cohesive (∞,1)-topos and write for its tangent cohesive (∞,1)-topos.
Assume that there is an interval object
“exhibiting the cohesion” (see at continuum) in that there is a (chosen) equivalence between the shape modality and the localization at the the projection maps out of Cartesian products with this line
This is the case for instance for the “standard continuum”, the real line in Smooth∞Grpd.
It follows in particular that there is a chosen equivalence of (∞,1)-categories
between the flat modal homotopy-types and the -homotopy invariant homotopy-types.
Given a stable homotopy type cohesion provides two objects
which may be interpreted as de Rham complexes with coefficients in , the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map
which interprets as the de Rham differential . See at differential cohomology diagram for details.
Throughout in the following we leave the “inclusion” of “differential forms regarded as -connections on trivial -bundles” implicit.
Integration of differential forms is the map
which is induced via the homotopy cofiber property of from the counit naturality square of the flat modality on , using that this square exhibits a null homotopy due to the -homotopy invariance of .
Stokes’ theorem holds:
(Bunke-Nikolaus-Völkl 13, theorem 3.2)
In early 20th-century vector analysis? (and even today in undergraduate Calculus courses), the Stokes theorem took various classical forms about vector fields in the Cartesian space :
if and , then this is the second Fundamental Theorem of Calculus: , where are real numbers and is a continuously differentiable function on a neighbourhood of the interval ;
if more generally, then this is a generalized form of the FTC: , where is a continuously differentiable oriented curve in , and are the beginning and ending points (respectively) of , is the unit vector field on tangent to in the direction given by its orientation, and is a continuously differentiable function on a neighbourhood of ;
if and , then this is Green's Theorem (see there for other forms; stated without proof by Cauchy in 1846, proved by Riemann in 1851): , where is a continuously differentiable simple closed curve in (oriented using the standard orientation on ), is the region that it encloses (guaranteed by the Jordan Curve Theorem), and and are continuously differentiable functions of the coordinates and on a neighbourhood of ;
if and , then this is the Kelvin–Stokes Theorem or Curl Theorem (stated without proof by Stokes? in 1854, proved by Hankel? in 1861): , where is a continuously differentiable pseudooriented surface in with a continuously differentiable boundary (oriented to match the pseudoorientation of using the standard orientation on ), is the unit normal vector field on in the direction given by the pseudoorientation of , is the unit tangent vector field on , and is a continuously differentiable vector field on a neighbourhood of ;
if and , then this is the Ostrogradsky–Gauss Theorem or Divergence Theorem (special cases were proved by Gauss in 1813, the general case was proved by Ostrogradsky in 1826): , where is a continuously differentiable closed surface in , is the region that it encloses (guaranteed by the Jordan–Brouwer Separation Theorem), is the outward-pointing unit normal vector field on , and is a continuously differentiable vector field on a neighbourhood of ;
if more generally, then this is the generalized Divergence Theorem (proved by Ostrogradsky in 1836): , where is a continuously differentiable closed hypersurface in , is the region that it encloses, is the outward-pointing unit normal vector field on , and is a continuously differentiable vector field on a neighbourhood of .
Stokes theorem
a special case is Cauchy's integral theorem
The basic statement:
In the generality of manifolds with corners:
John Lee, Theorem 10.32 in: Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)
Brian Conrad, Stokes’ theorem with corners (pdf, pdf)
Statement of the fiberwise Stokes theorem:
Kiyonori Gomi, Yuji Terashima, Remark 3.1 in: A Fiber Integration Formula for the Smooth Deligne Cohomology, International Mathematics Research Notices 2000, No. 13 (pdf, pdf, doi:10.1155/S1073792800000386)
Liviu Nicolaescu, Theorem 3.4.54 of: Lectures on the Geometry of Manifolds, 2018 (pdf, pdf)
and specifically for simplicial differential forms:
Statement of the Stokes theorem in the full generality of fiberwise integration over fibers with corners:
Discussion of chains of smooth singular simplices
Discussion for manifolds with more general singularities on the boundary is in
Discussion in cohesive homotopy type theory is in
Last revised on November 18, 2023 at 14:59:35. See the history of this page for a list of all contributions to it.