Generalized Stokes theorem information

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 $\mathbb {R} ^{2}$ or $\mathbb {R} ^{3},$ and the divergence theorem is the case of a volume in $\mathbb {R} ^{3}.$ ^[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.,

\int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.

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 classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850.^[7]^[8]^[9] Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861.^[9]^[10] This classical case relates the surface integral of the curl of a vector field ${\textbf {F}}$ over a surface (that is, the flux of ${\text{curl}}\,{\textbf {F}}$ ) in Euclidean three-space to the line integral of the vector field over the surface boundary.

^ Michel Moisan; Jacques Pelletier. Physics of Collisional Plasmas – Introduction to. Springer.
^ "The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD
^ Spivak, Michael (1965). Calculus on manifolds : a modern approach to classical theorems of advanced calculus. New York: Avalon Publishing. ISBN 0-8053-9021-9. OCLC 187146.
^ Cartan, Élie (1945). Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques. Paris: Hermann.
^ Katz, Victor J. (1979-01-01). "The History of Stokes' Theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.2307/2690275. JSTOR 2690275.
^ Katz, Victor J. (1999). "5. Differential Forms". In James, I. M. (ed.). History of Topology. Amsterdam: Elsevier. pp. 111–122. ISBN 9780444823755.
^
See:
- Katz, Victor J. (May 1979). "The history of Stokes' theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.1080/0025570x.1979.11976770.
- The letter from Thomson to Stokes appears in: Thomson, William; Stokes, George Gabriel (1990). Wilson, David B. (ed.). The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869. Cambridge, England: Cambridge University Press. pp. 96–97. 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.
^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, England: OUP Oxford. p. 146. 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 Edward John Routh. See: Clerk Maxwell, James (1990). Harman, P. M. (ed.). The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862. Cambridge, England: Cambridge University Press. p. 237, footnote 2. ISBN 9780521256254. See also 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".

[1] Michel Moisan; Jacques Pelletier. Physics of Collisional Plasmas – Introduction to. Springer.

[2] "The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD

[3] Spivak, Michael (1965). Calculus on manifolds : a modern approach to classical theorems of advanced calculus. New York: Avalon Publishing. ISBN 0-8053-9021-9. OCLC 187146.

[4] Cartan, Élie (1945). Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques. Paris: Hermann.

[5] Katz, Victor J. (1979-01-01). "The History of Stokes' Theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.2307/2690275. JSTOR 2690275.

[6] Katz, Victor J. (1999). "5. Differential Forms". In James, I. M. (ed.). History of Topology. Amsterdam: Elsevier. pp. 111–122. ISBN 9780444823755.

[7] See:
Katz, Victor J. (May 1979). "The history of Stokes' theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.1080/0025570x.1979.11976770.

The letter from Thomson to Stokes appears in: Thomson, William; Stokes, George Gabriel (1990). Wilson, David B. (ed.). The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869. Cambridge, England: Cambridge University Press. pp. 96–97. 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.

[8] Katz, Victor J. (May 1979). "The history of Stokes' theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.1080/0025570x.1979.11976770.

[9] The letter from Thomson to Stokes appears in: Thomson, William; Stokes, George Gabriel (1990). Wilson, David B. (ed.). The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869. Cambridge, England: Cambridge University Press. pp. 96–97. ISBN 9780521328319.

[10] 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.

[11] 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.

[8] Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, England: OUP Oxford. p. 146. ISBN 0198505930.

[spivak65-9] 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: Clerk Maxwell, James (1990). Harman, P. M. (ed.). The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862. Cambridge, England: Cambridge University Press. p. 237, footnote 2. ISBN 9780521256254. See also 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".

[15] 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: Clerk Maxwell, James (1990). Harman, P. M. (ed.). The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862. Cambridge, England: Cambridge University Press. p. 237, footnote 2. ISBN 9780521256254. See also Smith's prize or the Clerk Maxwell Foundation.

[16] 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".

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