|
In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e., : This modern form of Stokes' theorem is a vast generalization of a classical result. Lord Kelvin communicated it to George Stokes in a letter dated July 2, 1850.〔See: *Victor J. Katz (May 1979) ("The history of Stokes' theorem," ) ''Mathematics Magazine'', 52 (3): 146–156. *The letter from Thomson to Stokes appears in: William Thomson and George Gabriel Stokes with David B. Wilson, 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, 1990), (pages 96–97. ) *Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in 1861 in: Hermann Hankel, ''Zur allgemeinen Theorie der Bewegung der Flüssigkeiten'' (the general theory of the movement of fluids ) (Göttingen, (Germany): Dieterische University Buchdruckerei, 1861); see (pages 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: George G. Stokes with Sir Joseph Larmor and John Wm. Strutt (Baron Rayleigh), ed.s, ''Mathematical and Physical Papers by the late Sir George Gabriel Stokes'', ... (Cambridge, England: University of Cambridge Press, 1905), vol. 5, (pages 320–321 ).〕〔Olivier Darrigol,''Electrodynamics from Ampere to Einstein'', p. 146,ISBN 0198505930 Oxford (2000)〕〔Spivak (1965), p. vii, Preface.〕 Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name, even though it was actually first published by Hermann Hankel in 1861.〔〔See: *The 1854 Smith's Prize Examination is available on-line at: (Clerk Maxwell Foundation ). Maxwell took this examination and tied for first place with Edward John Routh in the Smith's Prize examination of 1854. See footnote 2 on page 237 of: James Clerk Maxwell with P. M. Harman, ed., ''The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862'' (Cambridge, England: Cambridge University Press, 1990), (page 237 ); see also Wikipedia's article "Smith's prize" or the (Clerk Maxwell Foundation ). *James Clerk Maxwell, ''A Treatise on Electricity and Magnetism'' (Oxford, England: Clarendon Press,1873), volume 1, (pages 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".〕 This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ: : This classical statement, along with the classical divergence theorem, fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above. == Introduction == The fundamental theorem of calculus states that the integral of a function ''f'' over the interval (''b'' ) can be calculated by finding an antiderivative ''F'' of ''f'' : : Stokes' theorem is a vast generalization of this theorem in the following sense. * By the choice of ''F'', . In the parlance of differential forms, this is saying that ''f''(''x'') d''x'' is the exterior derivative of the 0-form, i.e. function, ''F'': in other words, that d''F'' = ''f'' d''x''. The general Stokes theorem applies to higher differential forms ω instead of just 0-forms such as ''F''. * A closed interval (''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''). In even simpler terms, one can consider that points can be thought of as the boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral (''f'' d''x'' = d''F'') over a 1-dimensional manifolds ((''b'' )) by considering the anti-derivative (''F'') at the 0-dimensional boundaries ((''b'' )), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (dω) over ''n''-dimensional manifolds (Ω) by considering the anti-derivative (ω) at the (''n'' − 1)-dimensional boundaries (dΩ) of the manifold. So the fundamental theorem reads: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「stokes' theorem」の詳細全文を読む スポンサード リンク
|