Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T01:01:58.207Z Has data issue: false hasContentIssue false
  • English
  • Français

The multidimensional fundamental theorem of calculus

Part of: Calculus on manifolds; nonlinear operators Real functions Functions of several variables

Published online by Cambridge University Press:  09 April 2009

Washek F. Pfeffer
Washek F. Pfeffer
Affiliation:
Department of MathematicsUniversity of California Davis, California 95616, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

On compact oriented differentiable manifolds, we define a well behaved Riemann type integral which coincides with the Lebesgue integral on nonnegative functions, and such that the exterior derivative of a differentiable (not necessarily continuously) exterior form is always integrable and the Stokes formula holds.

MSC classification

Secondary: 26B20: Integral formulas (Stokes, Gauss, Green, etc.) 58C35: Integration on manifolds; measures on manifolds 26A39: Denjoy and Perron integrals, other special integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 2 , October 1987 , pp. 143 - 170
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Federer, H., Geometric measure theory (Springer, New York, 1969).Google Scholar
[2]Greub, W., Halperin, S. and Vanstone, R., Connections, curvature, and cohomology (Academic Press, New York, 1972).Google Scholar
[3]Henstock, R., ‘A Riemann-type integral of Lebesgue power’, Canad. J. Math. 20 (1968), 79–87.CrossRefGoogle Scholar
[4]Jarnik, J. and Kurzweil, J., ‘A non-absolutely convergent integral which admits C1-transformations’, Časopis Pěst. Mat. 109 (1984), 157–167.CrossRefGoogle Scholar
[5]Jarnik, J. and Kurzweil, J., ‘A non absolutely convergent integral which admits transformation and can be used for integration on manifolds’, Czechoslovak Math. J. 35 (1985), 116–139.CrossRefGoogle Scholar
[6]Kurzweil, J., ‘Generalized ordinary differential equations and continuous dependence on a parameter’, Czechoslovak Math. J. 7 (1957), 418–466.CrossRefGoogle Scholar
[7]Yee, Lee Peng and Naak-In, Wittaya, ‘A direct proof that Henstock and Denjoy integrals are equivalent’, Bull. Malaysian Math. Soc. (2) 5 (1982), 43–47.Google Scholar
[8]Loomis, L. H. and Witney, H., ‘An inequality related to isoperimetric inequality’, Bull. Amer. Math. Soc. 55 (1949), 961–962.CrossRefGoogle Scholar
[9]Mařik, J., ‘The surface integral’, Czechoslovak Math. J. 6 (1956), 522–558.CrossRefGoogle Scholar
[10]McLeod, R. M., The generalized Riemann integral (Carus Math. Monographs 20 (1980), MAA).CrossRefGoogle Scholar
[11]Mawhin, J., Introduction à l'analyse (Cabay, Louvain-la-Neuve, 1979).Google Scholar
[12]Mawhin, J., ‘Generalized Riemann integrals and the divergence theorem for differentiable vector fields’, pp. 704–714 (E. B. Christoffel, Birkhäuser Verlag, Basel, 1981).CrossRefGoogle Scholar
[13]Mawhin, J., ‘Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields’, Czechoslovak Math. J. 31 (1981), 614–632.CrossRefGoogle Scholar
[14]Mařik, J. and Matyska, J., ‘On a generalization of the Lebesgue integral in Em’, Czechoslovak Math. J. 15 (1965), 261–269.CrossRefGoogle Scholar
[15]Pfeffer, W. F., The Riemann-Stieltjes approach to integration, (TWISK 187, NRIMS:CSIR, Pretoria, 1980).Google Scholar
[16]Pfeffer, W. F., ‘Une intégrale riemannienne et le théorème de divergence’, Analyse Math., C. R. Acad. Sci. Paris Sér. I 299 (1984), 299–301.Google Scholar
[17]Pfeffer, W. F., ‘The divergence theorem’, Trans. Amer. Math. Soc. 295 (1986), 665–685.CrossRefGoogle Scholar
[18]Rudin, W., Principles of mathematical analysis (McGraw-Hill, New York, 1976).Google Scholar
[19]Saks, S., Theory of the integral (Dover, New York, 1964).Google Scholar
[20]Warner, F. W., Foundations of differentiable manifolds and Lie groups (Scott-Foresman, London, 1971).Google Scholar
[21]Whitney, H., Geometric integration theory (Princeton University Press, Princeton, N. J., 1957).CrossRefGoogle Scholar