# Generalized Stokes theorem

Classical generalizations of the fundamental theorem of calculus like the divergence theorem, and Green's theorem from vector calculus are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).

Contents 1 Introduction 2 Formulation for smooth manifolds with boundary 3 Topological preliminaries; integration over chains 4 Underlying principle 5 Classical vector analysis example 6 Generalization to rough sets 7 Special cases 7.1 Classical (vector calculus) case 7.2 Green's theorem 7.2.1 In electromagnetism 7.3 Divergence theorem 8 See also 9 Footnotes 10 References 11 Further reading 12 External links Introduction The second fundamental theorem of calculus states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: {displaystyle int _{a}^{b}f(x),dx=F(b)-F(a),.} Stokes' theorem is a vast generalization of this theorem in the following sense.

By the choice of F, dF / dx = f(x). In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies to higher differential forms ω instead of just 0-forms such as F. A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and the form has to be compactly supported in order to give a well-defined integral. The two points a and b form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientation from that of M. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, a inherits the opposite orientation as b, as they are at opposite ends of the interval. So, "integrating" F over two boundary points a, b is taking the difference F(b) − F(a).

In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral (f dx = dF) over a 1-dimensional manifold ([a, b]) by considering the anti-derivative (F) at the 0-dimensional boundaries ({a, b}), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (dω) over n-dimensional manifolds (Ω) by considering the antiderivative (ω) at the (n − 1)-dimensional boundaries (∂Ω) of the manifold.

So the fundamental theorem reads: {displaystyle int _{[a,b]}f(x),dx=int _{[a,b]},dF=int _{partial [a,b]},F=int _{{a}^{-}cup {b}^{+}}F=F(b)-F(a),.} Formulation for smooth manifolds with boundary Let Ω be an oriented smooth manifold with boundary of dimension n and let α be a smooth n-differential form that is compactly supported on Ω. First, suppose that α is compactly supported in the domain of a single, oriented coordinate chart {U, φ}. In this case, we define the integral of α over Ω as {displaystyle int _{Omega }alpha =int _{varphi (U)}(varphi ^{-1})^{*}alpha ,,} i.e., via the pullback of α to Rn.

More generally, the integral of α over Ω is defined as follows: Let {ψi} be a partition of unity associated with a locally finite cover {Ui, φi} of (consistently oriented) coordinate charts, then define the integral {displaystyle int _{Omega }alpha equiv sum _{i}int _{U_{i}}psi _{i}alpha ,,} where each term in the sum is evaluated by pulling back to Rn as described above. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity.

The generalized Stokes theorem reads: Theorem (Stokes–Cartan) — Let {displaystyle omega } be a smooth {displaystyle (n-1)} -form with compact support on an oriented, {displaystyle n} -dimensional manifold-with-boundary {displaystyle M} , where {displaystyle partial M} is given the induced orientation.Then {displaystyle int _{M}domega =int _{partial M}omega .} Here {displaystyle d} is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as {textstyle oint _{partial Omega }omega } to stress the fact that the {displaystyle (n-1)} -manifold {displaystyle partial Omega } has no boundary.[note 1] (This fact is also an implication of Stokes' theorem, since for a given smooth {displaystyle n} -dimensional manifold {displaystyle Omega } , application of the theorem twice gives {textstyle int _{partial (partial Omega )}omega =int _{Omega }d(domega )=0} for any {displaystyle (n-2)} -form {displaystyle omega } , which implies that {displaystyle partial (partial Omega )=emptyset } .) The right-hand side of the equation is often used to formulate integral laws; the left-hand side then leads to equivalent differential formulations (see below).

The theorem is often used in situations where {displaystyle Omega } is an embedded oriented submanifold of some bigger manifold, often {displaystyle mathbf {R} ^{k}} , on which the form {displaystyle omega } is defined.

Topological preliminaries; integration over chains Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in Rk to M. The group Ck(M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex. The corresponding homology (resp. cohomology) group is isomorphic to the usual singular homology group Hk(M, Z) (resp. the singular cohomology group Hk(M, Z)), defined using continuous rather than smooth simplices in M.

On the other hand, the differential forms, with exterior derivative, d, as the connecting map, form a cochain complex, which defines the de Rham cohomology groups {displaystyle H_{dR}^{k}(M,mathbf {R} )} .

Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to Rk. Extending by linearity allows one to integrate over chains. This gives a linear map from the space of k-forms to the kth group of singular cochains, Ck(M, Z), the linear functionals on Ck(M, Z). In other words, a k-form ω defines a functional {displaystyle I(omega )(c)=oint _{c}omega .} on the k-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means: closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over manifolds that can be written as ∂Σc Mc, and exact forms, i.e., ω = dσ, have zero integral over cycles, i.e. if the boundaries sum up to the empty set: Σc Mc = ∅.

De Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to 1 and 2 above hold true. In other words, if {ci} are cycles generating the kth homology group, then for any corresponding real numbers, {ai} , there exist a closed form, ω, such that {displaystyle oint _{c_{i}}omega =a_{i},,} and this form is unique up to exact forms.

Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa.[10] Formally stated, the latter reads:[11] Theorem (Stokes' theorem for chains) — If c is a smooth k-chain in a smooth manifold M, and ω is a smooth (k − 1)-form on M, then {displaystyle int _{partial c}omega =int _{c}domega .} Underlying principle To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example for d = 2 dimensions. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains. It thus suffices to prove Stokes' theorem for sufficiently fine tilings (or, equivalently, simplices), which usually is not difficult.

Classical vector analysis example Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t))[note 2] and F is a smooth vector field on R3, then:[12][13][14] {displaystyle oint _{Gamma }mathbf {F} ,cdot ,d{mathbf {Gamma } }=iint _{S}left(nabla times mathbf {F} right)cdot ,dmathbf {S} } This classical statement, is a special case of the general formulation after making an identification of vector field with a 1-form and its curl with a two form through {displaystyle {begin{pmatrix}F_{x}\F_{y}\F_{z}\end{pmatrix}}cdot dGamma to F_{x},dx+F_{y},dy+F_{z},dz} {displaystyle {begin{aligned}&nabla times {begin{pmatrix}F_{x}\F_{y}\F_{z}end{pmatrix}}cdot dmathbf {S} ={begin{pmatrix}partial _{y}F_{z}-partial _{z}F_{y}\partial _{z}F_{x}-partial _{x}F_{z}\partial _{x}F_{y}-partial _{y}F_{x}\end{pmatrix}}cdot dmathbf {S} to \[1.4ex]&d(F_{x},dx+F_{y},dy+F_{z},dz)=left(partial _{y}F_{z}-partial _{z}F_{y}right)dywedge dz+left(partial _{z}F_{x}-partial _{x}F_{z}right)dzwedge dx+left(partial _{x}F_{y}-partial _{y}F_{x}right)dxwedge dy.end{aligned}}} Generalization to rough sets A region (here called D instead of Ω) with piecewise smooth boundary. This is a manifold with corners, so its boundary is not a smooth manifold.

The formulation above, in which Ω is a smooth manifold with boundary, does not suffice in many applications. For example, if the domain of integration is defined as the plane region between two x-coordinates and the graphs of two functions, it will often happen that the domain has corners. In such a case, the corner points mean that Ω is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above does not apply. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. This is because Ω and its boundary are well-behaved away from a small set of points (a measure zero set).