Generalized Stokes theorem
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Calculus |
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In
Stokes' theorem says that the integral of a differential form over the
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]
This modern form of Stokes' theorem is a vast generalization of a
Introduction
The second fundamental theorem of calculus states that the integral of a function over the interval can be calculated by finding an antiderivative of :
Stokes' theorem is a vast generalization of this theorem in the following sense.
- By the choice of , . In the parlance of differential forms, this is saying that is the exterior derivative of the 0-form, i.e. function, : in other words, that . The general Stokes theorem applies to higher degree differential forms instead of just 0-forms such as .
- A closed interval is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points and . Integrating over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to becompactly supportedin order to give a well-defined integral.
- The two points and form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds with boundary. The boundary of is itself a manifold and inherits a natural orientation from that of . For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, inherits the opposite orientation as , as they are at opposite ends of the interval. So, "integrating" over two boundary points , is taking the difference .
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 () over a 1-dimensional manifold () by considering the anti-derivative () at the 0-dimensional boundaries (), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals () over -dimensional manifolds () by considering the antiderivative () at the -dimensional boundaries () of the manifold.
So the fundamental theorem reads:
Formulation for smooth manifolds with boundary
Let be an
More generally, the integral of over is defined as follows: Let be a partition of unity associated with a locally finite cover of (consistently oriented) coordinate charts, then define the integral
The generalized Stokes theorem reads:
Theorem (Stokes–Cartan) — Let be a
Here is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as to stress the fact that the -manifold has no boundary.[note 1] (This fact is also an implication of Stokes' theorem, since for a given smooth -dimensional manifold , application of the theorem twice gives for any -form , which implies that .) 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 is an embedded oriented submanifold of some bigger manifold, often , on which the form is defined.
Topological preliminaries; integration over chains
Let M be a
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 .
Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to Rk. 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, Ck(M, Z), the linear functionals on Ck(M, Z). In other words, a k-form ω defines a functional
- closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over manifolds that can be written as ∂Σc Mc, and
- exact forms, i.e., ω = dσ, have zero integral over cycles, i.e. if the boundaries sum up to the empty set: ∂Σc Mc = ∅.
Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa.[11] Formally stated, the latter reads:[12]
Theorem (Stokes' theorem for chains) — If c is a smooth k-chain in a smooth manifold M, and ω is a smooth (k − 1)-form on M, then
Underlying principle
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 be a
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
Generalization to rough sets
The formulation above, in which 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 -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 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 and its boundary are well-behaved away from a small set of points (a
A version of Stokes' theorem that allows for roughness was proved by Whitney.[16] Assume that is a connected bounded open subset of . Call a standard domain if it satisfies the following property: there exists a subset of , open in , whose complement in has Hausdorff -measure zero; and such that every point of has a generalized normal vector. This is a vector such that, if a coordinate system is chosen so that is the first basis vector, then, in an open neighborhood around , there exists a smooth function such that is the graph and is the region . Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff -measure and a finite or countable union of smooth -manifolds, each of which has the domain on only one side. He then proves that if is a standard domain in , is an -form which is defined, continuous, and bounded on , smooth on , integrable on , and such that is integrable on , then Stokes' theorem holds, that is,
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.[17]
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
Classical (vector calculus) case
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 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 ) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, , must have positive orientation, meaning that points counterclockwise when the surface normal, , 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:
Theorem — Suppose is defined in a region with smooth surface and has continuous first-order
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
Name | Differential form | Integral form (using three-dimensional Stokes theorem plus relativistic invariance, ) |
---|---|---|
Maxwell–Faraday equation Faraday's law of induction: |
(with C and S not necessarily stationary) | |
Ampère's law (with Maxwell's extension): |
(with C and S not necessarily stationary) |
The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in
Divergence theorem
Likewise, the divergence theorem
Volume integral of gradient of scalar field
Let be a scalar field. Then
Proof: Let be a vector. Then
See also
Footnotes
- ^ 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 appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where is a function of the temperature , the volume , and the electrical polarization of the sample, one has
and the circle is really necessary, e.g. if one considers the differential consequences of the integral postulate
- ^ and are both loops, however, is not necessarily a Jordan curve
References
- ^ Michel Moisan; Jacques Pelletier. Physics of Collisional Plasmas – Introduction to. Springer.
- ^ "The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD
- OCLC 187146.
- ^ Cartan, Élie (1945). Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques. Paris: Hermann.
- JSTOR 2690275.
- ISBN 9780444823755.
- ^ See:
- Katz, Victor J. (May 1979). "The history of Stokes' theorem". Mathematics Magazine. 52 (3): 146–156. .
- The letter from Thomson to Stokes appears in: ISBN 9780521328319.
- Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in 1861 in: Hankel, Hermann (1861). Zur allgemeinen Theorie der Bewegung der Flüssigkeiten [On the general theory of the movement of fluids]. Göttingen, Germany: Dieterische University Buchdruckerei. pp. 34–37. 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: Stokes, George Gabriel (1905). Larmor, Joseph; Strutt, John William (eds.). Mathematical and Physical Papers by the late Sir George Gabriel Stokes. Vol. 5. Cambridge, England: University of Cambridge Press. pp. 320–321.
- ISBN 0198505930.
- ^ a b Spivak (1965), p. vii, Preface.
- ^ See:
- The 1854 Smith's Prize Examination is available online at: Clerk Maxwell Foundation. Maxwell took this examination and tied for first place with Smith's prize or the Clerk Maxwell Foundation.
- Clerk Maxwell, James (1873). A Treatise on Electricity and Magnetism. Vol. 1. Oxford, England: Clarendon Press. pp. 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".
- The 1854 Smith's Prize Examination is available online at: Clerk Maxwell Foundation. Maxwell took this examination and tied for first place with
- ISBN 9781107324893.
- ISBN 9781441999818.
- ^ Stewart, James (2010). Essential Calculus: Early Transcendentals. Cole.
- ^ This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) [1], please refer the [2]
- ^ "This proof is also same to the proof shown in".
- ^ Whitney, Geometric Integration Theory, III.14.
- S2CID 17436511.
- ISBN 9780471431329.
- ^ Born, M.; Wolf, E. (1980). Principles of Optics (6th ed.). Cambridge, England: Cambridge University Press.
Further reading
- ISBN 0-273-08510-7.
- Katz, Victor J. (May 1979). "The History of Stokes' Theorem". Mathematics Magazine. 52 (3): 146–156. JSTOR 2690275.
- ISBN 978-981-4583-93-0.
- ISBN 0-521-58956-8.
- Marsden, Jerrold E.; Anthony, Tromba (2003). Vector Calculus (5th ed.). W. H. Freeman.
- Lee, John (2003). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
- ISBN 0-07-054235-X.
- ISBN 0-8053-9021-9.
- Stewart, James (2009). Calculus: Concepts and Contexts. Cengage Learning. pp. 960–967. ISBN 978-0-495-55742-5.
- Stewart, James (2003). Calculus: Early Transcendental Functions (5th ed.). Brooks/Cole.
- ISBN 978-1-4419-7399-3.
External links
- "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