Stokes theorem

For the equation governing viscous drag in uids, seeStokes law.

In vector calculus, Stokes theorem (also called thegeneralized Stokes theorem) is a statement about theintegration of dierential forms onmanifolds, which bothsimplies and generalizes several theorems from vectorcalculus. Stokes theorem says that the integral of a dif-ferential form over the boundary of some orientablemanifold is equal to the integral of its exterior deriva-tive d over the whole of , i.e.

Z@

! =

Z

d!:

This modern form of Stokes theorem is a vast general-ization of a classical result. Lord Kelvin communicatedit to George Stokes in a letter dated July 2, 1850.[1][2][3]Stokes set the theorem as a question on the 1854 SmithsPrize exam, which led to the result bearing his name, eventhough it was actually rst published by Hermann Hankelin 1861.[3][4] This classical KelvinStokes theorem relatesthe surface integral of the curl of a vector eld F over asurface in Euclidean three-space to the line integral ofthe vector eld over its boundary :

ZZ

r F d =I@

F dr:

This classical statement, along with the classicaldivergence theorem, fundamental theorem of calculus,and Greens theorem are simply special cases of the gen-eral formulation stated above.

1 IntroductionThe fundamental theorem of calculus states that theintegral of a function f over the interval [a, b] can becalculated by nding an antiderivative F of f :

Z ba

f(x) dx = F (b) F (a):

Stokes theorem is a vast generalization of this theorem inthe following sense.

By the choice of F, dFdx = f(x) . In the parlance ofdierential forms, this is saying that f(x) dx is theexterior derivative of the 0-form, i.e. function, F: inother words, that dF = f dx. The general Stokes the-orem applies to higher dierential forms insteadof just 0-forms such as F.

A closed interval [a, b] is a simple example of a one-dimensional manifold with boundary. Its bound-ary is the set consisting of the two points a and b.Integrating f over the interval may be generalizedto integrating forms on a higher-dimensional mani-fold. Two technical conditions are needed: the man-ifold has to be orientable, and the form has to becompactly supported in order to give a well-denedintegral.

The two points a and b form the boundary of theopen interval. More generally, Stokes theorem ap-plies to oriented manifolds M with boundary. Theboundary M of M is itself a manifold and inher-its a natural orientation from that of the manifold.For example, the natural orientation of the intervalgives an orientation of the two boundary points. In-tuitively, a inherits the opposite orientation as b, asthey are at opposite ends of the interval. So, inte-grating F over two boundary points a, b is takingthe dierence F(b) F(a).

In even simpler terms, one can consider that points canbe thought of as the boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So,just as one can nd the value of an integral (f dx = dF)over a 1-dimensional manifolds ([a, b]) by considering theanti-derivative (F) at the 0-dimensional boundaries ([a,b]), one can generalize the fundamental theorem of calcu-lus, with a few additional caveats, to deal with the value ofintegrals (d) over n-dimensional manifolds () by con-sidering the anti-derivative () at the (n 1)-dimensionalboundaries (d) of the manifold.So the fundamental theorem reads:

Z[a;b]

f(x) dx =Z[a;b]

dF =Zfag[fbg+

F = F (b)F (a):

2 General formulationLet be an oriented smoothmanifold of dimension n andlet be an n-dierential form that is compactly supported

1

2 3 TOPOLOGICAL PRELIMINARIES; INTEGRATION OVER CHAINS

on . First, suppose that is compactly supported in thedomain of a single, oriented coordinate chart {U, }. Inthis case, we dene the integral of over as

Z

=

Z(U)

1

;

i.e., via the pullback of to Rn.More generally, the integral of over is dened as fol-lows: Let {i} be a partition of unity associated with alocally nite cover {Ui, } of (consistently oriented) co-ordinate charts, then dene the integral

Z

Xi

ZUi

i;

where each term in the sum is evaluated by pulling backto Rn as described above. This quantity is well-dened;that is, it does not depend on the choice of the coordinatecharts, nor the partition of unity.Stokes theorem reads: If is an (n 1)-form with com-pact support on and denotes the boundary of withits induced orientation, then

y

xa b

DC

C

C C

1

4

3

2

A normal integration manifold (here called D instead of ) forthe special case n = 2

Z

d! =Z@

!

=

I@

!

:

Here d is the exterior derivative, which is dened usingthe manifold structure only. On the r.h.s., a circle issometimes used within the integral sign to stress the factthat the (n 1)-manifold is closed.[5] The r.h.s. ofthe equation is often used to formulate integral laws; thel.h.s. then leads to equivalent dierential formulations(see below).

The theorem is often used in situations where is an em-bedded oriented submanifold of some bigger manifold onwhich the form is dened.A proof becomes particularly simple if the submanifold is a so-called normal manifold, as in the gure onthe r.h.s., which can be segmented into vertical stripes(e.g. parallel to the xn direction), such that after a par-tial integration concerning this variable, nontrivial con-tributions come only from the upper and lower bound-ary surfaces (coloured in yellow and red, respectively),where the complementary mutual orientations are visiblethrough the arrows.

3 Topological preliminaries; inte-gration over chains

Let M be a smooth manifold. A smooth singular k-simplex ofM is a smooth map from the standard simplexin Rk to M. The free abelian group, Sk, generated by sin-gular k-simplices is said to consist of singular k-chains ofM. These groups, together with boundary map, , dene achain complex. The corresponding homology (resp. co-homology) is called the smooth singular homology (resp.cohomology) of M.On the other hand, the dierential forms, with exteriorderivative, d, as the connecting map, form a cochain com-plex, which denes de Rham cohomology.Dierential k-forms can be integrated over a k-simplexin a natural way, by pulling back to Rk. Extending bylinearity allows one to integrate over chains. This gives alinear map from the space of k-forms to the k-th group inthe singular cochain, Sk*, the linear functionals on Sk. Inother words, a k-form denes a functional

I(!)(c) =

Ic

!

on the k-chains. Stokes theorem says that this is a chainmap from de Rham cohomology to singular cohomology;the exterior derivative, d, behaves like the dual of onforms. This gives a homomorphism from de Rham coho-mology to singular cohomology. On the level of forms,this means:

1. closed forms, i.e., d = 0, have zero integral overboundaries, i.e. over manifolds that can be writtenas @ PcMc , and

2. exact forms, i.e., = d, have zero integral overcycles, i.e. if the boundaries sum up to the emptyset: Pc @Mc = ? .

De Rhams theorem shows that this homomorphism is infact an isomorphism. So the converse to 1 and 2 above

5.1 KelvinStokes theorem 3

hold true. In other words, if {ci} are cycles generatingthe k-th homology group, then for any corresponding realnumbers, {ai}, there exist a closed form, , such that

Ici

! = ai;

and this form is unique up to exact forms.

4 Underlying principle

To simplify these topological arguments, it is worthwhileto examine the underlying principle by considering an ex-ample for d = 2 dimensions. The essential idea can beunderstood by the diagram on the left, which shows that,in an oriented tiling of a manifold, the interior paths aretraversed in opposite directions; their contributions to thepath integral thus cancel each other pairwise. As a conse-quence, only the contribution from the boundary remains.It thus suces to prove Stokes theorem for sucientlyne tilings (or, equivalently, simplices), which usually isnot dicult.

5 Special casesThe general form of the Stokes theorem using dierentialforms is more powerful and easier to use than the specialcases. The traditional versions can be formulated usingCartesian coordinates without the machinery of dieren-tial geometry, and thus are more accessible. Further, theyare older and their names are more familiar as a result.The traditional forms are often considered more conve-nient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes appar-ent when using other coordinate systems, even familiarones like spherical or cylindrical coordinates. There ispotential for confusion in the way names are applied, andthe use of dual formulations.

5.1 KelvinStokes theoremThis is a (dualized) 1+1 dimensional case, for a 1-form(dualized because it is a statement about vector elds).

z

x y

0

n

An illustration of the KelvinStokes theorem, with surface , itsboundary and the normal vector n.

This special case is often just referred to as the Stokestheorem in many introductory university vector calculuscourses and as used in physics and engineering. It is alsosometimes known as the curl theorem.The classical KelvinStokes theorem:

Z

r F d =I@

F dr;

which relates the surface integral of the curl of a vectoreld over a surface in Euclidean three-space to the lineintegral of the vector eld over its boundary, is a spe-cial case of the general Stokes theorem (with n = 2) oncewe identify a vector eld with a 1 form using the metricon Euclidean three-space. The curve of the line integral,, must have positive orientation, meaning that dr pointscounterclockwise when the surface normal, d, points to-ward the viewer, following the right-hand rule.One consequence of the KelvinStokes theorem is thatthe eld lines of a vector eld with zero curl cannot beclosed contours. The formula can be rewritten as:

ZZ

@R

@y @Q

@z

dydz +

@P

@z @R

@x

dzdx+

@Q

@x @P

@y

dxdy

=

I@

fPdx+Qdy +Rdzg ;

where P, Q and R are the components of F.These variants are rarely used:

Z

[g (r F) + (rg) F] d =I@

gF dr;

4 6 NOTES

Z

[F (r G)G (r F) + (G r)F (F r)G]d =I@

(FG)dr:Z

r(F d) (r F)d =I@

dr F:

5.2 Greens theoremGreens theorem is immediately recognizable as the thirdintegrand of both sides in the integral in terms of P, Q,and R cited above.

5.2.1 In electromagnetism

Two of the four Maxwell equations involve curls of 3-Dvector elds and their dierential and integral forms arerelated by the KelvinStokes theorem. Caution must betaken to avoid cases with moving boundaries: the par-tial time derivatives are intended to exclude such cases.If moving boundaries are included, interchange of in-tegration and dierentiation introduces terms related toboundary motion not included in the results below (seeDierentiation under the integral sign):The above listed subset of Maxwells equations are validfor electromagnetic elds expressed in SI units. In othersystems of units, such as CGS or Gaussian units, the scal-ing factors for the terms dier. For example, in Gaussianunits, Faradays law of induction and Ampres law takethe forms[6][7]

r E = 1c

@B@t

;

rH = 1c

@D@t

+4

cJ;

respectively, where c is the speed of light in vacuum.

5.3 Divergence theoremLikewise, the OstrogradskyGauss theorem (also knownas the divergence theorem or Gausss theorem)

ZVolr F dVol =

I@Vol

F d

is a special case if we identify a vector eld with the n1form obtained by contracting the vector eld with the Eu-clidean volume form. An application of this is the caseF = fc where c is an arbitrary constant vector. Workingout the divergence of the product gives

c ZVolrf dVol = c

I@Vol

fd

Since this holds for all ~c; we nd

ZVolrf dVol =

I@Vol

fd

6 Notes[1] See:

Victor J. Katz (May 1979) The history of Stokestheorem,Mathematics Magazine, 52 (3): 146156.

The letter from Thomson to Stokes appears in:William Thomson and George Gabriel Stokes withDavid B. Wilson, ed., The Correspondence betweenSir George Gabriel Stokes and Sir William Thom-son, Baron Kelvin of Largs, Volume 1: 18461869 (Cambridge, England: Cambridge UniversityPress, 1990), pages 9697.

Neither Thomson nor Stokes published a proof ofthe theorem. The rst published proof appeared in1861 in: Hermann Hankel, Zur allgemeinen The-orie der Bewegung der Flssigkeiten [On the gen-eral theory of the movement of uids] (Gttingen,(Germany): Dieterische University Buchdruckerei,1861); see pages 3437. Hankel doesn't mentionthe author of the theorem.

In a footnote, Larmor mentions earlier researcherswho had integrated, over a surface, the curl of a vec-tor eld. See: George G. Stokes with Sir JosephLarmor and John Wm. Strutt (Baron Rayleigh),ed.s, Mathematical and Physical Papers by the lateSir George Gabriel Stokes, ... (Cambridge, England:University of Cambridge Press, 1905), vol. 5, pages320321.

[2] Olivier Darrigol,Electrodynamics fromAmpere to Einstein,p. 146,ISBN 0198505930 Oxford (2000)

[3] Spivak (1965), p. vii, Preface.

[4] See:

The 1854 Smiths Prize Examination is availableon-line at: Clerk Maxwell Foundation. Maxwelltook this examination and tied for rst place withEdward John Routh in the Smiths Prize examina-tion of 1854. See footnote 2 on page 237 of: JamesClerk Maxwell with P. M. Harman, ed., The Sci-entic Letters and Papers of James Clerk Maxwell,Volume I: 18461862 (Cambridge, England: Cam-bridge University Press, 1990), page 237; see alsoWikipedias article "Smiths prize" or the ClerkMaxwell Foundation.

James Clerk Maxwell, A Treatise on Electric-ity and Magnetism (Oxford, England: ClarendonPress,1873), volume 1, pages 2527. In a footnoteon page 27, Maxwell mentions that Stokes used thetheorem as question 8 in the Smiths Prize Exami-nation of 1854. This footnote appears to have beenthe cause of the theorems being known as Stokestheorem.

5[5] For mathematicians this fact is known, therefore the circleis redundant and often omitted. However, one should keepin mind here that in thermodynamics, where frequentlyexpressions as

HW

fd totalUg appear (wherein the totalderivative, see below, should not be confused with the ex-terior one), the integration path W is a one-dimensionalclosed line on a much higher-dimensional manifold. Thatis, in a thermodynamic application, where U is a functionof the temperature 1 := T , the volume 2 := V; andthe electrical polarization 3 := P of the sample, one hasfd totalUg =

P3i=1

@U@i

di; and the circle is really nec-essary, e.g. if one considers the dierential consequencesof the integral postulate

HW

fd totalUg !=0:[6] J.D. Jackson, Classical Electrodynamics, 2nd Ed (Wiley,

New York, 1975).[7] M. Born and E. Wolf, Principles of Optics, 6th Ed. (Cam-

bridge University Press, Cambridge, 1980).

7 Further reading Joos, Georg. Theoretische Physik. 13th ed.Akademische Verlagsgesellschaft Wiesbaden 1980.ISBN 3-400-00013-2

Katz, Victor J. (May 1979), The History of StokesTheorem,Mathematics Magazine 52 (3): 146156,doi:10.2307/2690275

Marsden, Jerrold E., Anthony Tromba. Vector Cal-culus. 5th edition W. H. Freeman: 2003.

Lee, John. Introduction to Smooth Manifolds.Springer-Verlag 2003. ISBN 978-0-387-95448-6

Rudin, Walter (1976), Principles of MathematicalAnalysis, New York: McGrawHill, ISBN 0-07-054235-X

Spivak, Michael (1965), Calculus on Manifolds:A Modern Approach to Classical Theorems of Ad-vanced Calculus, HarperCollins, ISBN 978-0-8053-9021-6

Stewart, James. Calculus: Concepts and Contexts.2nd ed. Pacic Grove, CA: Brooks/Cole, 2001.

Stewart, James. Calculus: Early TranscendentalFunctions. 5th ed. Brooks/Cole, 2003.

8 External links Hazewinkel, Michiel, ed. (2001), Stokes formula,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Proof of the Divergence Theorem and Stokes The-orem

Calculus 3 Stokes Theorem from lamar.edu anexpository explanation

6 9 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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Introduction General formulationTopological preliminaries; integration over chainsUnderlying principle Special casesKelvinStokes theoremGreens theoremIn electromagnetism

Divergence theorem

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