一般化されたストークスの定理

ウィキペディア

索引トップランキングカテゴリー

出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2024/04/01 07:19 UTC 版)

出典

^ (英語) Physics of Collisional Plasmas – Introduction to | Michel Moisan | Springer
^ "The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD
^ 本間利久; 五十嵐一，川口秀樹『数値電磁力学』森北出版、2002年、59頁。ISBN 4-627-71641-9。
^ 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.. History of Topology. Amsterdam: Elsevier. pp. 111–122. ISBN 9780444823755
^
以下を参照:
- 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, Baron Rayleigh. eds. Mathematical and Physical Papers by the late Sir George Gabriel Stokes. 5. Cambridge, England: University of Cambridge Press. pp. 320–321
^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, England. p. 146. ISBN 0198505930
^ ^a ^b Spivak (1965), p. vii, Preface.
^
以下を参照：
- 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. 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".
^ Renteln, Paul (2014). Manifolds, Tensors, and Forms. Cambridge, UK: Cambridge University Press. pp. 158–175. ISBN 9781107324893
^ Lee, John M. (2013). Introduction to Smooth Manifolds. New York: Springer. pp. 481. 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”. 2022年5月9日閲覧。
^ Whitney, Geometric Integration Theory, III.14.
^ Harrison, J. (October 1993). “Stokes' theorem for nonsmooth chains”. Bulletin of the American Mathematical Society. New Series 29 (2): 235–243. arXiv:math/9310231. Bibcode: 1993math.....10231H. doi:10.1090/S0273-0979-1993-00429-4.

脚注

^ 数学者にとってこの事実は既知であるため、周回積分の円の記号は冗長とされしばしば省略される。しかし、熱力学では $\oint _{W}\mathrm {d} _{\text{total}}U$ がよく現れる（ここで全微分を外微分と混同しないこと）に注意する。積分経路 $W$ は高次元多様体上の1次元の閉曲線である。つまり、熱力学での応用では、 $U$ はサンプルの温度 $α 1 := T$ 、体積 $α 2 := V$ 、および電気分極 $α 3 := P$ の関数であり、 $\mathrm {d} _{\text{total}}U=\sum _{i=1}^{3}{\frac {\partial U}{\partial \alpha _{i}}}\mathrm {d} \alpha _{i}$ であり、円の記号は必要である。たとえば「積分」仮定の異なる「微分」結果を考慮する場合
$\oint _{W}\mathrm {d} _{\text{total}}U{\stackrel {!}{=}}0$
^ $γ$ と $Γ$ はどちらも閉曲線だが、 $Γ$ は必ずしもジョルダン曲線とは限らない。