Invector calculus anddifferential geometry thegeneralized Stokes theorem (sometimes with apostrophe asStokes' theorem orStokes's theorem), also called theStokes–Cartan theorem,^[1] is a statement about theintegration ofdifferential forms onmanifolds, which both simplifies and generalizes severaltheorems fromvector calculus. In particular, thefundamental theorem of calculus is the special case where the manifold is aline segment,Green’s theorem andStokes' theorem are the cases of asurface in${\displaystyle {\mathbb{R}}^2}$ or⁠${\displaystyle {\mathbb{R}}^3}$⁠, and thedivergence theorem is the case of a volume in⁠${\displaystyle {\mathbb{R}}^3}$⁠. Hence, the theorem is sometimes referred to as thefundamental theorem of multivariate calculus.^[2]

Stokes' theorem says that the integral of a differential form${\displaystyle \omega }$ over theboundary${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ of someorientable manifold${\displaystyle \mathrm{\Omega }}$ is equal to the integral of itsexterior derivative${\displaystyle d\omega }$ over the whole of⁠${\displaystyle \mathrm{\Omega }}$⁠, i.e.,$${\displaystyle {\int }_{\mathrm{\partial }\mathrm{\Omega }}\omega ={\int }_{\mathrm{\Omega }}d\omega .}$$

Stokes' theorem was formulated in its modern form byÉlie Cartan in 1945,^[3] following earlier work on the generalization of the theorems of vector calculus byVito Volterra,Édouard Goursat, andHenri Poincaré.^[4][5]

This modern form of Stokes' theorem is a vast generalization of aclassical result thatLord Kelvin communicated toGeorge Stokes in a letter dated July 2, 1850.^[6][7][8] Stokes set the theorem as a question on the 1854Smith's Prize exam, which led to the result bearing his name. It was first published byHermann Hankel in 1861.^[8][9] This classical case relates thesurface integral of thecurl of avector field${\displaystyle \mathbf{\text{F}}}$ over a surface (that is, theflux of⁠${\displaystyle \text{curl}\mathbf{\text{F}}}$⁠) in Euclidean three-space to theline integral of the vector field over the surface boundary.

Thesecond fundamental theorem of calculus states that theintegral of a function${\displaystyle f}$ over theinterval${\displaystyle [a,b]}$ can be calculated by finding anantiderivative${\displaystyle F}$ of⁠${\displaystyle f}$⁠:$${\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a).}$$

Stokes' theorem is a vast generalization of this theorem in the following sense.

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

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 (⁠${\displaystyle fdx=dF}$⁠) over a 1-dimensional manifold (⁠${\displaystyle [a,b]}$⁠) by considering the anti-derivative (⁠${\displaystyle F}$⁠) at the 0-dimensional boundaries (⁠${\displaystyle \{a,b\}}$⁠), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (⁠${\displaystyle d\omega }$⁠) over⁠${\displaystyle n}$⁠-dimensional manifolds (⁠${\displaystyle \mathrm{\Omega }}$⁠) by considering the antiderivative (⁠${\displaystyle \omega }$⁠) at the⁠${\displaystyle (n-1)}$⁠-dimensional boundaries (⁠${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$⁠) of the manifold.

So the fundamental theorem reads:$${\displaystyle {\int }_{[a,b]}f(x)dx={\int }_{[a,b]}dF={\int }_{\mathrm{\partial }[a,b]}F={\int }_{\{a{\}}^{-}\cup \{b{\}}^{+}}F=F(b)-F(a).}$$

Let${\displaystyle \mathrm{\Omega }}$ be anorientedsmooth manifold ofdimension${\displaystyle n}$ with boundary and let${\displaystyle \alpha }$ be asmooth⁠${\displaystyle n}$⁠-differential form that iscompactly supported on⁠${\displaystyle \mathrm{\Omega }}$⁠. First, suppose that${\displaystyle \alpha }$ is compactly supported in the domain of a single, orientedcoordinate chart⁠${\displaystyle \{U,\phi \}}$⁠. In this case, we define the integral of${\displaystyle \alpha }$ over${\displaystyle \mathrm{\Omega }}$ as$${\displaystyle {\int }_{\mathrm{\Omega }}\alpha ={\int }_{\phi (U)}({\phi }^{-1}{)}^{\ast }\alpha ,}$$i.e., via thepullback of${\displaystyle \alpha }$ to⁠${\displaystyle {\mathbb{R}}^n}$⁠.

More generally, the integral of${\displaystyle \alpha }$ over${\displaystyle \mathrm{\Omega }}$ is defined as follows: Let${\displaystyle \{{\psi }_i\}}$ be apartition of unity associated with alocally finitecover${\displaystyle \{U_i,{\phi }_i\}}$ of (consistently oriented) coordinate charts, then define the integral$${\displaystyle {\int }_{\mathrm{\Omega }}\alpha \equiv \sum _i{\int }_{U_i}{\psi }_i\alpha ,}$$where each term in the sum is evaluated by pulling back to${\displaystyle {\mathbb{R}}^n}$ 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:

Here${\displaystyle d}$ is theexterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as${\oint }_{\mathrm{\partial }\mathrm{\Omega }}\omega$ to stress the fact that the⁠${\displaystyle (n-1)}$⁠-manifold${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ has no boundary.^{[note 1]} (This fact is also an implication of Stokes' theorem, since for a given smooth⁠${\displaystyle n}$⁠-dimensional manifold⁠${\displaystyle \mathrm{\Omega }}$⁠, application of the theorem twice gives${\int }_{\mathrm{\partial }(\mathrm{\partial }\mathrm{\Omega })}\omega ={\int }_{\mathrm{\Omega }}d(d\omega )=0$ for any⁠${\displaystyle (n-2)}$⁠-form⁠${\displaystyle \omega }$⁠, which implies that⁠${\displaystyle \mathrm{\partial }(\mathrm{\partial }\mathrm{\Omega })=\mathrm{\varnothing }}$⁠.) The right-hand side of the equation is often used to formulateintegral laws; the left-hand side then leads to equivalentdifferential formulations (see below).

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

LetM be asmooth manifold. A (smooth) singulark-simplex inM is defined as asmooth map from the standard simplex inR^k toM. The groupC_k(M,Z) of singulark-chains onM is defined to be thefree abelian group on the set of singulark-simplices inM. These groups, together with the boundary map,∂, define achain complex. The corresponding homology (resp. cohomology) group is isomorphic to the usualsingular homology groupH_k(M,Z) (resp. thesingular cohomology groupH^k(M,Z)), defined using continuous rather than smooth simplices inM.

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

Differentialk-forms can be integrated over ak-simplex in a natural way, by pulling back toR^k. Extending by linearity allows one to integrate over chains. This gives a linear map from the space ofk-forms to thekth group of singular cochains,C^k(M,Z), the linear functionals onC_k(M,Z). In other words, ak-formω defines a functional$${\displaystyle I(\omega )(c)={\oint }_c\omega .}$$on thek-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 thedual 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 overboundaries, i.e. over manifolds that can be written as∂Σ_cM_c; and
2. exact forms, i.e.,ω =dσ, have zero integral overcycles, i.e. if the boundaries sum up to the empty set:∂Σ_cM_c = ∅.

De Rham's theorem shows that this homomorphism is in fact anisomorphism. So the converse to 1 and 2 above hold true. In other words, if{c_i} are cycles generating thekth homology group, then for any corresponding real numbers,{a_i}, 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]

To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example ford = 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.

Let${\displaystyle \gamma :[a,b]\to {\mathbb{R}}^2}$ be apiecewise smoothJordan plane curve. TheJordan curve theorem implies that${\displaystyle \gamma }$ divides${\displaystyle {\mathbb{R}}^2}$ into two components, acompact one and another that is non-compact. Let${\displaystyle D}$ denote the compact part that is bounded by${\displaystyle \gamma }$ and suppose${\displaystyle \psi :D\to {\mathbb{R}}^3}$ is smooth, with⁠${\displaystyle S=\psi (D)}$⁠. If${\displaystyle \mathrm{\Gamma }}$ is thespace curve defined by⁠${\displaystyle \mathrm{\Gamma }(t)=\psi (\gamma (t))}$⁠^{[note 2]} and${\displaystyle \mathbf{\text{F}}}$ is a smooth vector field on⁠${\displaystyle {\mathbb{R}}^3}$⁠, then:^[12][13]$${\displaystyle {\oint }_{\mathrm{\Gamma }}\mathbf{F}\cdot d\mathbf{\Gamma }={\iint }_S\left(\mathrm{\nabla }\times \mathbf{F}\right)\cdot d\mathbf{S}}$$

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$${\displaystyle \left(\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right)\cdot d\mathrm{\Gamma }\to F_{x}dx+F_{y}dy+F_{z}dz}$$

The formulation above, in which${\displaystyle \mathrm{\Omega }}$ 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 twox-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${\displaystyle \mathrm{\Omega }}$ 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${\displaystyle \mathrm{\Omega }}$ and its boundary are well-behaved away from a small set of points (ameasure zero set).

A version of Stokes' theorem that allows for roughness was proved byHassler Whitney.^[14] Assume that${\displaystyle D}$ is a connected bounded open subset of⁠${\displaystyle {\mathbb{R}}^n}$⁠. Call${\displaystyle D}$ astandard domain if it satisfies the following property: there exists a subset${\displaystyle P}$ of⁠${\displaystyle \mathrm{\partial }D}$⁠, open in⁠${\displaystyle \mathrm{\partial }D}$⁠, whose complement in${\displaystyle \mathrm{\partial }D}$ hasHausdorff⁠${\displaystyle (n-1)}$⁠-measure zero; and such that every point of${\displaystyle P}$ has ageneralized normal vector. This is a vector${\displaystyle \mathbf{\text{v}}(x)}$ such that, if a coordinate system is chosen so that${\displaystyle \mathbf{\text{v}}(x)}$ is the first basis vector, then, in an open neighborhood around⁠${\displaystyle x}$⁠, there exists a smooth function${\displaystyle f(x_2,\dots ,x_n)}$ such that${\displaystyle P}$ is the graph${\displaystyle \{x_1=f(x_2,\dots ,x_n)\}}$ and${\displaystyle D}$ is the region⁠⁠. Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff⁠${\displaystyle (n-1)}$⁠-measure and a finite or countable union of smooth⁠${\displaystyle (n-1)}$⁠-manifolds, each of which has the domain on only one side. He then proves that if${\displaystyle D}$ is a standard domain in⁠${\displaystyle {\mathbb{R}}^n}$⁠,${\displaystyle \omega }$ is an⁠${\displaystyle (n-1)}$⁠-form which is defined, continuous, and bounded on⁠${\displaystyle D\cup P}$⁠, smooth on⁠${\displaystyle D}$⁠, integrable on⁠${\displaystyle P}$⁠, and such that${\displaystyle d\omega }$ is integrable on⁠${\displaystyle D}$⁠, then Stokes' theorem holds, that is,$${\displaystyle {\int }_P\omega ={\int }_{D}d\omega .}$$

The study of measure-theoretic properties of rough sets leads togeometric measure theory. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.^[15]

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 usingCartesian 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.

This is a (dualized)⁠${\displaystyle (1+1)}$⁠-dimensional case, for a 1-form (dualized because it is a statement aboutvector fields). This special case is often just referred to asStokes' theorem in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as thecurl theorem.

The classical Stokes' theorem relates thesurface integral of thecurl of avector field over a surface${\displaystyle \mathrm{\Sigma }}$ in Euclidean three-space to theline integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with⁠${\displaystyle n=2}$⁠) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral,⁠${\displaystyle \mathrm{\partial }\mathrm{\Sigma }}$⁠, must have positiveorientation, meaning that${\displaystyle \mathrm{\partial }\mathrm{\Sigma }}$ points counterclockwise when thesurface normal,⁠${\displaystyle n}$⁠, points toward the viewer.

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

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

Two of the fourMaxwell equations involve curls of 3-D vector fields, and their differential and integral forms are related by the special 3-dimensional (vector calculus) case ofStokes' 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 (seeDifferentiation 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,⁠${\displaystyle \int \frac{\mathrm{\partial }}{\mathrm{\partial }t}\cdots \to \frac{d}{dt}\int \cdots }$⁠)
Maxwell–Faraday equation Faraday's law of induction	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$	(withC andS not necessarily stationary)
Ampère's law (with Maxwell's extension)	${\displaystyle \mathrm{\nabla }\times \mathbf{H}=\mathbf{J}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}}$	(withC andS not necessarily stationary)

The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed inSI units. In other systems of units, such asCGS orGaussian 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:^[16][17]respectively, wherec is thespeed of light in vacuum.

Likewise, thedivergence theorem$${\displaystyle {\int }_{\mathrm{V}\mathrm{o}\mathrm{l}}\mathrm{\nabla }\cdot \mathbf{F}{d}_{\mathrm{V}\mathrm{o}\mathrm{l}}={\oint }_{\mathrm{\partial }\mathrm{Vol}}\mathbf{F}\cdot d\mathbf{\Sigma }}$$is a special case if we identify a vector field with the⁠${\displaystyle (n-1)}$⁠-form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case${\displaystyle \mathbf{\text{F}}=f\overrightarrow{c}}$ where${\displaystyle \overrightarrow{c}}$ is an arbitrary constant vector. Working out the divergence of the product gives$${\displaystyle \overrightarrow{c}\cdot {\int }_{\mathrm{V}\mathrm{o}\mathrm{l}}\mathrm{\nabla }f{d}_{\mathrm{V}\mathrm{o}\mathrm{l}}=\overrightarrow{c}\cdot {\oint }_{\mathrm{\partial }\mathrm{V}\mathrm{o}\mathrm{l}}fd\mathbf{\Sigma }.}$$Since this holds for all${\displaystyle \overrightarrow{c}}$ we find$${\displaystyle {\int }_{\mathrm{V}\mathrm{o}\mathrm{l}}\mathrm{\nabla }f{d}_{\mathrm{V}\mathrm{o}\mathrm{l}}={\oint }_{\mathrm{\partial }\mathrm{V}\mathrm{o}\mathrm{l}}fd\mathbf{\Sigma }.}$$

Let${\displaystyle f:\mathrm{\Omega }\to \mathbb{R}}$ be ascalar field. Then$${\displaystyle {\int }_{\mathrm{\Omega }}\overrightarrow{\mathrm{\nabla }}f={\int }_{\mathrm{\partial }\mathrm{\Omega }}\overrightarrow{n}f}$$where${\displaystyle \overrightarrow{n}}$ is thenormal vector to the surface${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ at a given point.

Proof:Let${\displaystyle \overrightarrow{c}}$ be a vector. ThenSince this holds for any${\displaystyle \overrightarrow{c}}$ (in particular, for everybasis vector), the result follows.

• Chandrasekhar–Wentzel lemma

• 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.
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• "Stokes formula",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Proof of the Divergence Theorem and Stokes' Theorem
• Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation

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