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For the equation governing viscous drag in fluids, see Stokes' law.

In differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in July 1850.^[1] Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.

[edit] Introduction

The 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:

$\int_a^b f(x)\,\mathrm dx = F(b) - F(a).$

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

By the choice of F, $\scriptstyle \frac{dF}{dx}=f(x)$ . 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: dF = f dx. The general Stokes theorem applies to higher differential forms $ω$ instead of 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 open interval. More generally, Stokes' theorem applies to oriented manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientation from that of the manifold. 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).

So the fundamental theorem reads:

$\int_{[a, b]} f(x)\,dx = \int_{[a, b]} dF = \int_{\{a\}^- \cup \{b\}^+} F = F(b) - F(a).$

[edit] General formulation

Let M be an oriented smooth manifold of dimension n and let $\scriptstyle\alpha$ be an n-differential form that is compactly supported on M. The integral of $\scriptstyle\alpha$ over M is defined as follows: Let {f_i} be a partition of unity associated with a locally finite cover {U_i} of (consistently oriented) coordinate neighborhoods, then define the integral

$\int_M \alpha \equiv \sum_i \int_{U_i} f_i \, \alpha\, ,$

where each term in the sum is evaluated by pulling back to Rⁿ. This is well-defined.

Stokes' theorem reads: If $\scriptstyle\omega$ is an (n − 1)-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then

An integration manifold known as "normal" (here instead of M called D) for the special case n=2

$\int_M \mathrm {d}\omega = \oint_{\partial M} \omega.\!\,$

Here $\,\mathrm d$ is the exterior derivative, which is defined using the manifold structure only. ^[2] The r.h.s. of the equation is often used to formulate integral laws; the l.h.s. then leads to equivalent differential formulations (see below).

The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form $\scriptstyle\omega$ is defined:
A proof becomes particularly simple if the submanifold M is a so-called "normal manifold", as in the figure on the r.h.s., which can be segmented into vertical stripes (e.g. parallel to the x_n direction), such that after a partial integration concerning this variable, nontrivial contributions come only from the upper and lower boundary surfaces (coloured in yellow and red, repectively), where the complementary mutual orientations are visible through the arrows.

[edit] Topological reading; integration over chains

Let M be a smooth manifold. A smooth singular k-simplex of M is a smooth map from the standard simplex in R^k to M. The free abelian group, S_k, generated by singular k-simplices is said to consist of singular k-chains of M. These groups, together with boundary map, ∂, define a chain complex. The corresponding homology (resp. cohomology) is called the smooth singular homology (resp. cohomology) of M.

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

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 k-th group in the singular cochain, S_k*, the linear functionals on S_k. In other words, a k-form $\scriptstyle\omega$ defines a functional

$I(\omega)(c) = \oint_c \omega \,$

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

closed forms, i.e. $\mathrm d\omega=0\,$ , have zero integral over boundaries, i.e. for manifolds that can be written as $\partial \,\sum_c M_c$ , and
exact forms, i.e. $\omega=\mathrm d\sigma\,$ , have zero integral over cycles, i.e. if the boundaries sum up to the empty set: $\sum_c\partial M_c=\varnothing$ .

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 k-th homology group, then for any corresponding real numbers, {a_i}, there exist a closed form, $\scriptstyle\omega$ , such that:

$\oint_{c_i} \omega = a_i ,$

and this form is unique up to exact forms.

[edit] Underlying principle

To simplify these topological reasonings, it is worthwhile to consider the underlying principle by a drawing for d=2 dimensions: In the most simple form the essential idea can be understood by the diagram on the left, which says that by "oriented tiling of a manifold" the interior paths compensate each other pairwise because of "opposite directions". As a consequence, only the edge contributions of a closed path remain.

In the left hand side of the sketch one sees four small, identically oriented tiles. The "interior paths" shown run in opposite directions; their contributions to the path integral thus compensate each other pairwise. As a consequence, only the contribution from the edge curve remains.

It thus suffices to prove Stokes' theorem for sufficiently fine tilings, which usually is not difficult. Of course, simplices (triangles) can be replaced by tiles.

[edit] Special cases

The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. Because in Cartesian coordinates the traditional versions can be formulated without the machinery of differential geometry they are more accessible, older and have familiar names. 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.

[edit] Kelvin-Stokes theorem

An illustration of the Kelvin-Stokes theorem, with surface

Σ

, its boundary $\scriptstyle{\partial \Sigma,}$ and the "normal" vector n.

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 the Stokes' theorem in many introductory university vector calculus courses. It is also sometimes known as the curl theorem.

The classical Kelvin-Stokes theorem:

$\int_{\Sigma} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot d \mathbf{r},$

which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary, 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 three-space. The curve of the line integral, ∂Σ, must have positive orientation, meaning that dr points counterclockwise when the surface normal, dΣ, points toward the viewer, following the right-hand rule.

It can be rewritten for the student acquainted with forms as

$\iint\limits_{\Sigma}\left\{\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\,dy\,dz +\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\,dz\,dx +\left (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy\right\}$ $=\oint\limits_{\partial\Sigma}\left \{P\,dx+Q\,dy+R\,dz\right\}$

where P, Q and R are the components of F.

These variants are frequently used:

$\int_{\Sigma} \left( g \left(\nabla \times \mathbf{F}\right) + \left( \nabla g \right) \times \mathbf{F} \right)$ $\cdot d\mathbf{\Sigma}$ $\ = \oint_{\partial\Sigma} g \mathbf{F} \cdot d \mathbf{r},$

$\int_{\Sigma} \left( \mathbf{F} \left(\nabla \cdot \mathbf{G} \right) - \mathbf{G}\left(\nabla \cdot \mathbf{F} \right) + \left( \mathbf{G} \cdot \nabla \right) \mathbf{F} - \left(\mathbf{F} \cdot \nabla \right) \mathbf{G} \right) \cdot d\mathbf{\Sigma}$ $\ = \oint_{\partial\Sigma} \left( \mathbf{F} \times \mathbf{G}\right) \cdot d \mathbf{r}.$

[edit] 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 Kelvin-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:

Name	Differential form	Integral form (using Kelvin-Stokes theorem plus relativistic invariance, $\scriptstyle\int \frac{\partial}{\partial t} ...\to \frac{\mathrm d}{\mathrm dt} \int ...$ )
Maxwell-Faraday equation Faraday's law of induction:	$\ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\$	$\ \oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S \nabla \times \mathbf{E} \cdot d\mathbf{A} = -\,{\mathrm d \over {\mathrm d t}} \int_S \mathbf{B} \cdot d\mathbf{A}\$ (with C and S not necessarily stationary)
Ampère's law (with Maxwell's extension):	$\ \ \nabla \times \mathbf{H} = \mathbf{j} + \frac{\partial \mathbf{D}} {\partial t}\ \$	$\,\ \begin{align}\oint_C \mathbf{H} \cdot d\mathbf{l} &= \int_S \nabla \times \mathbf{H} \cdot d \mathbf{A}\\ &= \int_S \mathbf{j} \cdot d \mathbf{A} + {\mathrm d \over {\mathrm dt}} \int_S \mathbf{D} \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^[3]^[4]

$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t},$

$\nabla \times \mathbf{H} = \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{j},$

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

[edit] Divergence theorem

Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem)

$\int_{\mathrm{Vol}} \nabla \cdot \mathbf{F} \ d_\mathrm{Vol} = \oint_{\partial \mathrm{Vol}} \mathbf{F} \cdot d \mathbf{\Sigma}$

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.

[edit] 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.

[edit] Notes

^ Olivier Darrigol,Electrodynamics from Ampere to Einstein, p. 146,ISBN 0198505930 Oxford (2000)
^ On the r.h.s., the circle within the integral sign stresses the fact that the (n-1)-manifold $\partial M$ is closed. For mathematicians this fact is known, therefore the circle is redundant and often left away. However, one should keep in mind here that in thermodynamics, where frequently expressions as $\oint_W \,\{d_{\,total\,}U\}$ appear (wherein the total derivative, see below, should not be mixed-up with the exterior one), the integration path W is a one-dimensional closed line on a much higher-dimensional manifold. I.e. in a thermodynamic application, where U is a function of the temperature $α 1 : = T$ , the volume $\alpha_2:=V\,,$ and the electrical polarization $α 3 : = P$ of the sample, one has $\{d_{\,total\,}U\}=\sum_{i=1}^3\,\frac{\partial U}{\partial\alpha_i}\mathrm d\alpha_i\,,$ and the circle is really necessary, e.g. if one considers the differential consequences of the integral postulate $\oint_W\,\{d_{\,total\,}U\}\, \stackrel{!}{=}\,0\,.$
^ J.D. Jackson, Classical Electrodynamics, 2nd Ed (Wiley, New York, 1975).
^ M. Born and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, Cambridge, 1980).

[edit] 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 Stokes' Theorem", Mathematics Magazine 52 (3): 146–156, http://links.jstor.org/sici?sici=0025-570X(197905)52%3A3%3C146%3ATHOST%3E2.0.CO%3B2-O
Marsden, Jerrold E., Anthony Tromba. Vector Calculus. 5th edition W. H. Freeman: 2003.
Spivak, Michael (1965), Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, HarperCollins, ISBN 978-0-8053-9021-6
Stewart, James. Calculus: Concepts and Contexts. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001.
Stewart, James. Calculus: Early Transcendental Functions. 5th ed. Brooks/Cole, 2003.

[edit] External links

[0] Olivier Darrigol,Electrodynamics from Ampere to Einstein, p. 146,ISBN 0198505930 Oxford (2000)

[1] On the r.h.s., the circle within the integral sign stresses the fact that the (n-1)-manifold $\partial M$ is closed. For mathematicians this fact is known, therefore the circle is redundant and often left away. However, one should keep in mind here that in thermodynamics, where frequently expressions as $\oint_W \,\{d_{\,total\,}U\}$ appear (wherein the total derivative, see below, should not be mixed-up with the exterior one), the integration path W is a one-dimensional closed line on a much higher-dimensional manifold. I.e. in a thermodynamic application, where U is a function of the temperature $α 1 : = T$ , the volume $\alpha_2:=V\,,$ and the electrical polarization $α 3 : = P$ of the sample, one has $\{d_{\,total\,}U\}=\sum_{i=1}^3\,\frac{\partial U}{\partial\alpha_i}\mathrm d\alpha_i\,,$ and the circle is really necessary, e.g. if one considers the differential consequences of the integral postulate $\oint_W\,\{d_{\,total\,}U\}\, \stackrel{!}{=}\,0\,.$

[2] J.D. Jackson, Classical Electrodynamics, 2nd Ed (Wiley, New York, 1975).

[3] M. Born and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, Cambridge, 1980).

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