Generalized Stokes theorem

This article is about the generalized theorem. For the classical theorem, see Stokes' theorem. For the equation governing viscous drag in fluids, see Stokes' law.

In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^2} or Template:Tmath, and the divergence theorem is the case of a volume in Template:Tmath. Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.

Stokes' theorem says that the integral of a differential form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} over the boundary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Omega} of some orientable manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is equal to the integral of its exterior derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega} over the whole of Template:Tmath, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\partial \Omega} \omega = \int_\Omega \mathop{}\!d\omega\,.}

Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.

This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861. This classical case relates the surface integral of the curl of a vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{F}} over a surface (that is, the flux of Template:Tmath) in Euclidean three-space to the line integral of the vector field over the surface boundary.

Introduction

The second fundamental theorem of calculus states that the integral of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} over the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} can be calculated by finding an antiderivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} of Template:Tmath: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Template:Tmath, Template:Tmath. In the parlance of differential forms, this is saying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\,dx} is the exterior derivative of the 0-form, i.e. function, Template:Tmath: in other words, that Template:Tmath. The general Stokes theorem applies to higher degree differential forms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} instead of just 0-forms such as Template:Tmath.
• A closed interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Template:Tmath. Integrating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} with boundary. The boundary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial M} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is itself a manifold and inherits a natural orientation from that of Template:Tmath. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} inherits the opposite orientation as Template:Tmath, as they are at opposite ends of the interval. So, "integrating" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} over two boundary points Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is taking the difference Template:Tmath.

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 (Template:Tmath) over a 1-dimensional manifold (Template:Tmath) by considering the anti-derivative (Template:Tmath) at the 0-dimensional boundaries (Template:Tmath), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (Template:Tmath) over Template:Tmath-dimensional manifolds (Template:Tmath) by considering the antiderivative (Template:Tmath) at the Template:Tmath-dimensional boundaries (Template:Tmath) of the manifold.

So the fundamental theorem reads: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} be an oriented smooth manifold of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} with boundary and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} be a smooth Template:Tmath-differential form that is compactly supported on Template:Tmath. First, suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is compactly supported in the domain of a single, oriented coordinate chart Template:Tmath. In this case, we define the integral of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\Omega \alpha = \int_{\varphi(U)} (\varphi^{-1})^* \alpha\,,} i.e., via the pullback of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} to Template:Tmath.

More generally, the integral of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is defined as follows: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\psi_i\}} be a partition of unity associated with a locally finite cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_i,\varphi_i\}} of (consistently oriented) coordinate charts, then define the integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} 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: Template:Math theorem

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \oint_{\partial\Omega} \omega} to stress the fact that the Template:Tmath-manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Omega} has no boundary. (This fact is also an implication of Stokes' theorem, since for a given smooth Template:Tmath-dimensional manifold Template:Tmath, application of the theorem twice gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \int_{\partial(\partial \Omega)}\omega=\int_\Omega d(d\omega)=0} for any Template:Tmath-form Template:Tmath, which implies that Template:Tmath.) 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is an embedded oriented submanifold of some bigger manifold, often Template:Tmath, on which the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 R^k to M. The group C_k(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 H_k(M, Z) (resp. the singular cohomology group H^k(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 Template:Tmath.

Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to R^k. 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, C^k(M, Z), the linear functionals on C_k(M, Z). In other words, a k-form ω defines a functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

1. closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over manifolds that can be written as ∂Σ_c M_c; and
2. exact forms, i.e., ω = dσ, have zero integral over cycles, i.e. if the boundaries sum up to the empty set: ∂Σ_c M_c = ∅.

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 {c_i} are cycles generating the kth homology group, then for any corresponding real numbers, {a_i} , there exist a closed form, ω, such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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. Formally stated, the latter reads:

Underlying principle

File:Stokes patch.svg

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma:[a,b]\to\R^2} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^2} into two components, a compact one and another that is non-compact. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} denote the compact part that is bounded by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} and suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi:D\to\R^3} is smooth, with Template:Tmath. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is the space curve defined by Template:Tmath and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{F}} is a smooth vector field on Template:Tmath, then: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_\Gamma \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}} = \iint_S \left( \nabla \times \mathbf{F} \right) \cdot\, d\mathbf{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} F_x \\ F_y \\ F_z \\ \end{pmatrix}\cdot d\Gamma \to F_x \,dx + F_y \,dy + F_z \,dz} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &\nabla \times \begin{pmatrix} F_x \\ F_y \\ F_z \end{pmatrix} \cdot d\mathbf{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 d\mathbf{S} \to \\[1.4ex] &d(F_x \,dx + F_y \,dy + F_z \,dz) = \left(\partial_y F_z - \partial_z F_y\right) dy \wedge dz + \left(\partial_z F_x -\partial_x F_z\right) dz \wedge dx + \left(\partial_x F_y - \partial_y F_x\right) dx \wedge dy. \end{align}}

Generalization to rough sets

File:Green's-theorem-simple-region.svg

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} and its boundary are well-behaved away from a small set of points (a measure zero set).

A version of Stokes' theorem that allows for roughness was proved by Hassler Whitney. Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is a connected bounded open subset of Template:Tmath. Call Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} a standard domain if it satisfies the following property: there exists a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} of Template:Tmath, open in Template:Tmath, whose complement in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial D} has [[Hausdorff measure|Hausdorff Template:Tmath-measure]] zero; and such that every point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} has a generalized normal vector. This is a vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{v}(x)} such that, if a coordinate system is chosen so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{v}(x)} is the first basis vector, then, in an open neighborhood around Template:Tmath, there exists a smooth function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_2,\dots,x_n)} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is the graph Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_1=f(x_2,\dots,x_n)\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is the region Template:Tmath. Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff Template:Tmath-measure and a finite or countable union of smooth Template:Tmath-manifolds, each of which has the domain on only one side. He then proves that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is a standard domain in Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} is an Template:Tmath-form which is defined, continuous, and bounded on Template:Tmath, smooth on Template:Tmath, integrable on Template:Tmath, and such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega} is integrable on Template:Tmath, then Stokes' theorem holds, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_P \omega = \int_D d\omega\,.}

The study of measure-theoretic properties of rough sets leads to geometric measure theory. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.

Special cases

The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. The traditional versions can be formulated using Cartesian coordinates without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result. The traditional forms are often considered more convenient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. There is potential for confusion in the way names are applied, and the use of dual formulations.

Classical (vector calculus) case

Main article: Stokes' theorem

File:Stokes' Theorem.svg

An illustration of the vector-calculus Stokes theorem, with surface Σ, its boundary ∂Σ and the "normal" vector n.

This is a (dualized) Template:Tmath-dimensional case, for a 1-form (dualized because it is a statement about vector fields). This special case is often just referred to as Stokes' theorem in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the curl theorem.

The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with Template:Tmath) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, Template:Tmath, must have positive orientation, meaning that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Sigma} points counterclockwise when the surface normal, Template:Tmath, points toward the viewer.

One consequence of this theorem is that the field lines of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:

Green's theorem

Green's theorem is immediately recognizable as the third integrand of both sides in the integral in terms of P, Q, and R cited above.

In electromagnetism

Two of the four Maxwell equations involve curls of 3-D vector fields, and their differential and integral forms are related by the special 3-dimensional (vector calculus) case of Stokes' theorem. Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below (see Differentiation under the integral sign):

Differential and integeral forms of Maxwell's electromagnetic equations involving curls of vector fields

Name	Differential form	Integral form (using three-dimensional Stokes theorem plus relativistic invariance, Template:Tmath)
Maxwell–Faraday equation Faraday's law of induction	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \oint_C \mathbf{E} \cdot d\mathbf{l} &= \iint_S \nabla \times \mathbf{E} \cdot d\mathbf{A} \\ &= -\iint_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} \end{align} } (with C and S not necessarily stationary)
Ampère's law (with Maxwell's extension)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \oint_C \mathbf{H} \cdot d\mathbf{l} &= \iint_S \nabla \times \mathbf{H} \cdot d\mathbf{A}\\ &= \iint_S \mathbf{J} \cdot d\mathbf{A} + \iint_S \frac{\partial \mathbf{D}}{\partial t} \cdot d\mathbf{A} \end{align} } (with C and S not necessarily stationary)

The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in SI units. In other systems of units, such as CGS or Gaussian units, the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \nabla \times \mathbf{E} &= -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}\,, \\ \nabla \times \mathbf{H} &= \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{J}\,, \end{align}} respectively, where c is the speed of light in vacuum.

Divergence theorem

Likewise, the divergence theorem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\mathrm{Vol} \nabla \cdot \mathbf{F} \, d_\mathrm{Vol} = \oint_{\partial \operatorname{Vol}} \mathbf{F} \cdot d\boldsymbol{\Sigma}} is a special case if we identify a vector field with the Template:Tmath-form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{F}=f\vec{c}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}} is an arbitrary constant vector. Working out the divergence of the product gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c} \cdot \int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \vec{c} \cdot \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.} Since this holds for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}} we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.}

Volume integral of gradient of scalar field

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \Omega \to \mathbb{R}} be a scalar field. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_\Omega \vec{\nabla} f = \int_{\partial \Omega} \vec{n} f} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n}} is the normal vector to the surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial \Omega} at a given point.

Proof: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}} be a vector. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 0 &= \int_\Omega \vec{\nabla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\ &= \int_\Omega \vec{c} \cdot \vec{\nabla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\ &= \vec{c} \cdot \int_\Omega \vec{\nabla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\ &= \vec{c} \cdot \left( \int_\Omega \vec{\nabla} f - \int_{\partial \Omega} \vec{n} f \right) \end{align} } Since this holds for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}} (in particular, for every basis vector), the result follows.

See also

• Chandrasekhar–Wentzel lemma

Further reading

• Grunsky, Helmut (1983). The General Stokes' Theorem. Boston: Pitman. ISBN 0-273-08510-7.
• Loomis, Lynn Harold; Sternberg, Shlomo (2014). Advanced Calculus. Hackensack, New Jersey: World Scientific. ISBN 978-981-4583-93-0.
• Madsen, Ib; Tornehave, Jørgen (1997). From Calculus to Cohomology: De Rham cohomology and characteristic classes. Cambridge: Cambridge University Press. ISBN 0-521-58956-8.
• Marsden, Jerrold E.; Anthony, Tromba (2003). Vector Calculus (5th ed.). W. H. Freeman.
• Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw–Hill. ISBN 0-07-054235-X.
• Stewart, James (2009). Calculus: Concepts and Contexts. Cengage Learning. pp. 960–967. ISBN 978-0-495-55742-5.
• Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7399-3.

External links

• Template:Springer
• Proof of the Divergence Theorem and Stokes' Theorem
• Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation