Generalized Stokes theorem explained

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,^[1] 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

\R ^{2} or \R ^{3,} and the divergence theorem is the case of a volume in \R ^{3.} ^[2] Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus.^[3]

Stokes' theorem says that the integral of a differential form

\omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega , i.e.,$$\backslash int\_\; \backslash omega\; =\; \backslash int\_\backslash Omega\; \backslash operatorname\backslash omega\backslash ,.$$

Stokes' theorem was formulated in its modern form by Élie Cartan in 1945,^[4] following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.^{[5] [6]}

bf{F} over a surface (that is, the flux of curl bf{F} ) 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

f over the interval [a,b] can be calculated by finding an antiderivative F of f :$$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; F(b)\; -\; F(a)\backslash ,.$$

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 = fdx . The general Stokes theorem applies to higher degree differential forms \omega 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 \partial 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 (fdx = 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 (d\omega ) over n -dimensional manifolds (\Omega ) by considering the antiderivative (\omega ) at the (n -1 ) -dimensional boundaries (\partial\Omega ) of the manifold.

So the fundamental theorem reads:$$\backslash int\_\; f(x)\backslash ,dx\; =\; \backslash int\_\; \backslash ,dF\; =\; \backslash int\_\; \backslash ,F\; =\; \backslash int\_\; F\; =\; F(b)\; -\; F(a)\backslash ,.$$

Formulation for smooth manifolds with boundary

Let

\Omega be an oriented smooth manifold of dimension n with boundary and let \alpha be a smooth n -differential form that is compactly supported on \Omega . First, suppose that \alpha is compactly supported in the domain of a single, oriented coordinate chart \{U,\varphi\} . In this case, we define the integral of \alpha over \Omega as$$\backslash int\_\backslash Omega\; \backslash alpha\; =\; \backslash int\_\; (\backslash varphi^)^*\; \backslash alpha\backslash ,,$$i.e., via the pullback of \alpha to \R ^{n} .

More generally, the integral of

\alpha over \Omega is defined as follows: Let \{\psi _{i\}} be a partition of unity associated with a locally finite cover \{U _{i,\varphi} _{i\}} of (consistently oriented) coordinate charts, then define the integral$$\backslash int\_\backslash Omega\; \backslash alpha\; \backslash equiv\; \backslash sum\_i\; \backslash int\_\; \backslash psi\_i\; \backslash alpha\backslash ,,$$where each term in the sum is evaluated by pulling back to \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

d is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as $\backslash oint\_\; \backslash omega$ to stress the fact that the (n -1 ) -manifold \partial\Omega has no boundary.^[11] (This fact is also an implication of Stokes' theorem, since for a given smooth n -dimensional manifold \Omega , application of the theorem twice gives $\backslash int\_\backslash omega=\backslash int\_\backslash Omega\; d(d\backslash omega)=0$ for any (n -2 ) -form \omega , which implies that \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

\Omega is an embedded oriented submanifold of some bigger manifold, often \R ^{k} , on which the form \omega is defined.

Topological preliminaries; integration over chains

Let be a smooth manifold. A (smooth) singular -simplex in is defined as a smooth map from the standard simplex in to . The group of singular -chains on is defined to be the free abelian group on the set of singular -simplices in . 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 (resp. the singular cohomology group), defined using continuous rather than smooth simplices in .

On the other hand, the differential forms, with exterior derivative,, as the connecting map, form a cochain complex, which defines the de Rham cohomology groups

H ^{k(M,} _{dR} R ) .

Differential -forms can be integrated over a -simplex in a natural way, by pulling back to . Extending by linearity allows one to integrate over chains. This gives a linear map from the space of -forms to the th group of singular cochains,, the linear functionals on . In other words, a -form defines a functional$$I(\backslash omega)(c)\; =\; \backslash oint\_c\; \backslash omega.$$on the -chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative,, 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.,, have zero integral over boundaries, i.e. over manifolds that can be written as, and
2. exact forms, i.e.,, have zero integral over cycles, i.e. if the boundaries sum up to the empty set: .

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 are cycles generating the th homology group, then for any corresponding real numbers,, there exist a closed form,, such that$$\backslash oint\_\; \backslash omega\; =\; a\_i\backslash ,,$$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.^[12] Formally stated, the latter reads:^[13]

Underlying principle

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

\gamma:[a,b]\to\R ^{2} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that \gamma divides \R ^{2} into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by \gamma and suppose \psi:D\to\R ^{3} is smooth, with S = \psi(D) . If \Gamma is the space curve defined by \Gamma(t) = \psi(\gamma(t)) ^[14] and bf{F} is a smooth vector field on \R ^{3} , then:^{[15] [16] [17]} $$\backslash oint\_\backslash Gamma\; \backslash mathbf\backslash ,\; \backslash cdot\backslash ,\; d\; =\; \backslash iint\_S\; \backslash left(\backslash nabla\; \backslash times\; \backslash mathbf\; \backslash right)\; \backslash cdot\backslash ,\; d\backslash mathbf$$

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 $$\backslash beginF\_x\; \backslash \backslash \; F\_y\; \backslash \backslash \; F\_z\; \backslash \backslash \backslash end\backslash cdot\; d\backslash Gamma\; \backslash to\; F\_x\; \backslash ,dx\; +\; F\_y\; \backslash ,dy\; +\; F\_z\; \backslash ,dz$$

Generalization to rough sets

The formulation above, in which

\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 \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 \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 Whitney.^[18] Assume that

D is a connected bounded open subset of \R ^{n} . Call D a standard domain if it satisfies the following property: there exists a subset P of \partial D , open in \partial D , whose complement in \partial D has Hausdorff (n -1 ) -measure zero; and such that every point of P has a generalized normal vector. This is a vector bf{v}(x) such that, if a coordinate system is chosen so that bf{v}(x) is the first basis vector, then, in an open neighborhood around x , there exists a smooth function f(x _{2,...,x} _{n)} such that P is the graph \{x _{1=f(x} _{2,...,x} _{n)\}} and D is the region \{x _{1:x} _{1<f(x} _{2,...,x} _{n)\}} . Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff (n -1 ) -measure and a finite or countable union of smooth (n -1 ) -manifolds, each of which has the domain on only one side. He then proves that if D is a standard domain in \R ^{n} , \omega is an (n -1 ) -form which is defined, continuous, and bounded on D\cup P , smooth on D , integrable on P , and such that d\omega is integrable on D , then Stokes' theorem holds, that is,$$\backslash int\_P\; \backslash omega\; =\; \backslash int\_D\; d\backslash omega\backslash ,.$$

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.^[19]

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

See main article: Stokes' theorem.

This is a (dualized) (1 + 1)-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

\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 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, \partial\Sigma , must have positive orientation, meaning that \partial\Sigma points counterclockwise when the surface normal, n , 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,, and 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):

Name	Differential form	Integral form (using three-dimensional Stokes theorem plus relativistic invariance, style\int\tfrac{\partial}{\partial t} ... \to\tfrac{d}{dt} style\int … )
Maxwell–Faraday equation Faraday's law of induction:	\nabla x E =- (\partial B)/(\partial t)	\begin{align} \oint _{C} E ⋅ d l & = \iint _{S} \nabla x E ⋅ d A \\ & = - \iint _{S} (\partial B)/(\partial t) ⋅ d A \end{align} (with and not necessarily stationary)
Ampère's law (with Maxwell's extension):	\nabla x H = J + (\partial D)/(\partial t)	\begin{align} \oint _{C} H ⋅ d l & = \iint _{S} \nabla x H ⋅ d A \ & = \iint _{S} J ⋅ d A + \iint _{S} (\partial D)/(\partial t) ⋅ d A \end{align} (with and 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:^{[20] [21]} respectively, where is the speed of light in vacuum.

Divergence theorem

Likewise, the divergence theorem$$\backslash int\_\backslash mathrm\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; \backslash ,\; d\_\backslash mathrm\; =\; \backslash oint\_\; \backslash mathbf\; \backslash cdot\; d\backslash boldsymbol$$is a special case if we identify a vector field with the

(n -1 ) -form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case bf{F}=f\vec{c} where \vec{c} is an arbitrary constant vector. Working out the divergence of the product gives$$\backslash vec\; \backslash cdot\; \backslash int\_\backslash mathrm\; \backslash nabla\; f\; \backslash ,\; d\_\backslash mathrm\; =\; \backslash vec\; \backslash cdot\; \backslash oint\_\; f\backslash ,\; d\backslash boldsymbol\backslash ,.$$Since this holds for all \vec{c} we find$$\backslash int\_\backslash mathrm\; \backslash nabla\; f\; \backslash ,\; d\_\backslash mathrm\; =\; \backslash oint\_\; f\backslash ,\; d\backslash boldsymbol\backslash ,.$$

Volume integral of gradient of scalar field

Let

f : \Omega \to R be a scalar field. Then$$\backslash int\_\backslash Omega\; \backslash vec\; f\; =\; \backslash int\_\; \backslash vec\; f$$where \vec{n} is the normal vector to the surface \partial \Omega at a given point.

Proof:Let

\vec{c} be a vector. ThenSince this holds for any \vec{c} (in particular, for every basis vector), the result follows.

See also

• Chandrasekhar–Wentzel lemma

Further reading

• Book: Grunsky, Helmut . Helmut Grunsky . The General Stokes' Theorem . Boston . Pitman . 1983 . 0-273-08510-7 . registration .
• Katz . Victor J. . The History of Stokes' Theorem . Mathematics Magazine . May 1979 . 52 . 3 . 146–156 . 2690275 . 10.2307/2690275.
• Book: Advanced Calculus. Sternberg. Shlomo. World Scientific. 2014. 978-981-4583-93-0. Hackensack, New Jersey. Shlomo Sternberg. Loomis. Lynn Harold. Lynn Harold Loomis.
• Book: From Calculus to Cohomology: De Rham cohomology and characteristic classes. Tornehave. Jørgen. Cambridge University Press. 1997 . 0-521-58956-8. Cambridge, UK. Madsen. Ib. Ib Madsen.
• Book: Jerrold E. Marsden. Marsden. Jerrold E.. Anthony. Tromba. Vector Calculus. 5th. W. H. Freeman. 2003.
• Book: Lee, John. Introduction to Smooth Manifolds. 2003. Springer-Verlag. 978-0-387-95448-6.
• Book: Rudin, Walter. Principles of Mathematical Analysis. McGraw–Hill. 1976. 0-07-054235-X. New York, NY. Walter Rudin.
• Book: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus . Spivak. Michael. Calculus on Manifolds (book). Benjamin Cummings . 1965 . 0-8053-9021-9 . San Francisco . Michael Spivak .
• Book: Stewart, James . Calculus: Concepts and Contexts . 2009 . Cengage Learning . 960–967 . 978-0-495-55742-5.
• Book: Stewart, James. Calculus: Early Transcendental Functions. 5th. Brooks/Cole. 2003.
• Book: An Introduction to Manifolds. Tu. Loring W.. Springer. 2011. 978-1-4419-7399-3. 2nd. New York. Loring W. Tu.

External links

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

Notes and References

1. Book: Physics of Collisional Plasmas – Introduction to . Michel Moisan . Jacques Pelletier . Springer. en.
2. "The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD
3. Book: Spivak, Michael . Calculus on manifolds : a modern approach to classical theorems of advanced calculus . 1965 . 0-8053-9021-9 . New York . 187146 . Avalon Publishing.
4. Book: Cartan, Élie. Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques. Hermann. 1945. Paris.
5. Katz. Victor J.. 1979-01-01. The History of Stokes' Theorem. 2690275. Mathematics Magazine. 52. 3. 146–156. 10.2307/2690275.
6. Book: Katz, Victor J. . History of Topology . Elsevier . 1999 . 9780444823755. James . I. M. . Amsterdam . 111–122 . 5. Differential Forms.
7. See:
   • Victor J.. Katz . May 1979. The history of Stokes' theorem. Mathematics Magazine . 52 . 3. 146–156. 10.1080/0025570x.1979.11976770.
   • The letter from Thomson to Stokes appears in: Book: The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869. Stokes. George Gabriel. 1990. Cambridge University Press. Wilson. David B.. Cambridge, England. 96–97. William. Thomson. 9780521328319. William Thomson, 1st Baron Kelvin. George Gabriel Stokes.
   • Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in 1861 in: Book: Hankel, Hermann. Zur allgemeinen Theorie der Bewegung der Flüssigkeiten . 1861. Dieterische University Buchdruckerei. Göttingen, Germany . 34–37. On the general theory of the movement of fluids. Hermann Hankel. Hankel doesn't mention the author of the theorem.
   • In a footnote, Larmor mentions earlier researchers who had integrated, over a surface, the curl of a vector field. See: Book: Stokes, George Gabriel . Mathematical and Physical Papers by the late Sir George Gabriel Stokes. 1905. University of Cambridge Press . 5 . Cambridge, England . 320–321 . George Gabriel Stokes. Joseph. Larmor. John William . Strutt.
8. Book: Darrigol, Olivier. Electrodynamics from Ampère to Einstein. 146. 0198505930 . Oxford, England . OUP Oxford . 2000.
9. Spivak (1965), p. vii, Preface.
10. See:
    • The 1854 Smith's Prize Examination is available online at: Clerk Maxwell Foundation. Maxwell took this examination and tied for first place with Edward John Routh. See: Book: James. Clerk Maxwell. James Clerk Maxwell. P. M.. Harman. The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862. Cambridge, England. Cambridge University Press. 1990. 237, footnote 2. 9780521256254. See also Smith's prize or the Clerk Maxwell Foundation.
    • Book: Clerk Maxwell, James. James Clerk Maxwell. A Treatise on Electricity and Magnetism. Oxford, England. Clarendon Press. 1873. 1. 25–27. In a footnote on page 27, Maxwell mentions that Stokes used the theorem as question 8 in the Smith's Prize Examination of 1854. This footnote appears to have been the cause of the theorem's being known as "Stokes' theorem".
11. For mathematicians this fact is known, therefore the circle is redundant and often omitted. However, one should keep in mind here that in thermodynamics, where frequently expressions as \oint _{W\{d} _{totalU\}} appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path W is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where U is a function of the temperature \alpha _{1=T} , the volume \alpha _{2=V} , and the electrical polarization \alpha _{3=P} of the sample, one has$$\backslash \; =\; \backslash sum\_^3\backslash frac\backslash ,d\backslash alpha\_i\backslash ,,$$and the circle is really necessary, e.g. if one considers the differential consequences of the integral postulate$$\backslash oint\_W\backslash ,\backslash \backslash ,\; \backslash stackrel\backslash ,0\backslash ,.$$
12. Book: Renteln, Paul. Manifolds, Tensors, and Forms. Cambridge University Press. 2014. 9781107324893. Cambridge, UK. 158–175.
13. Book: Lee, John M.. Introduction to Smooth Manifolds. Springer. 2013. 9781441999818. New York . 481.
14. \gamma and \Gamma are both loops, however, \Gamma is not necessarily a Jordan curve
15. Book: Stewart, James . [{{Google books |plainurl=yes |id=btIhvKZCkTsC |page=786 }} Essential Calculus: Early Transcendentals ]. Cole . 2010.
16. This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) http://www.maths.bath.ac.uk/~masrs/ma20010/, please refer the http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf
17. Web site: This proof is also same to the proof shown in .
18. Whitney, Geometric Integration Theory, III.14.
19. Harrison. J.. Stokes' theorem for nonsmooth chains. Bulletin of the American Mathematical Society . New Series. 29. 2. October 1993. 235–243. 10.1090/S0273-0979-1993-00429-4. math/9310231. 1993math.....10231H. 17436511.
20. Book: Jackson, J.&nbsp;D.. Classical Electrodynamics. registration . 2nd. Wiley. New York, NY. 1975. 9780471431329 .
21. Book: M.. Born. E.. Wolf. Principles of Optics. 6th. Cambridge University Press. Cambridge, England. 1980.