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The Kelvin–Stokes theorem, named after Lord Kelvin and George Stokes, also known as the Stokes' theorem, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on [math]\displaystyle{ \mathbb{R}^3 }[/math]. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. If a vector field [math]\displaystyle{ \mathbf{A} = (P(x, y, z), Q(x, y, z), R(x, y, z)) }[/math] is defined in a region with smooth oriented surface [math]\displaystyle{ \Sigma }[/math] and has first order continuous partial derivatives then: where [math]\displaystyle{ \partial \Sigma }[/math] is boundary of region with smooth surface [math]\displaystyle{ \Sigma }[/math]. The above classical Kelvin-Stokes theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. The Kelvin–Stokes theorem is a special case of the "generalized Stokes' theorem." In particular, a vector field on [math]\displaystyle{ \mathbb{R}^3 }[/math] can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.
References
- Γ may not be a Jordan curve, if the loop γ interacts poorly with ψ. Nonetheless, Γ is always a loop, and topologically a connected sum of countably-many Jordan curves, so that the integrals are well-defined.
- Stewart, James (2010). Essential Calculus: Early Transcendentals. Cole. https://books.google.com/books?id=btIhvKZCkTsC&pg=PA786.
- Robert Scheichl, lecture notes for University of Bath mathematics course [3]
- Colley, Susan Jane (2002). Vector Calculus (4th ed.). Boston: Pearson.
- Edwards, Harold M. (1994). Advanced Calculus: A Differential Forms Approach. Birkhäuser. ISBN 0-8176-3707-9.
- Conlon, Lawrence (2008). Differentiable Manifolds. Modern Birkhauser Classics. Boston: Birkhaeuser. https://books.google.com/books?id=r2K31Pz5EGcC&pg=PA194.
- Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" Bai-Fu-Kan(jp)(1979/01) ISBN:978-4563004415 [2] (Written in Japanese)
- There do exist textbooks that use the terms "homotopy" and "homotopic" in the sense of Theorem 2-1.[5] Indeed, this is very convenient for the specific problem of conservative forces. However, both uses of homotopy appear sufficiently frequently that some sort of terminology is necessary to disambiguate, and the term "tubular homotopy" adopted here serves well enough for that end.
- Lee, John M. (2002). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218. Springer. https://books.google.com/books?id=xygVcKGPsNwC&pg=PA421.
- L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American Mathematical Society Translations, Ser. 2, Vol. 11, American Mathematical Society, Providence, R.I., 1959, pp. 1–114. MR0115178 (22 #5980 [4]). See theorems 7 & 8. https://mathscinet.ams.org/mathscinet-getitem?mr=0115178
