## Real Analysis Exchange

### The Stokes Theorem for the Generalized Riemann Integral

W. F. Pfeffer

#### Abstract

In $\mathbb{R}^m$, we define the generalized Riemann integral over normal $m$-dimensional currents with compact support and bounded multiplicities, and prove the Stokes theorem for continuous $(m-1)$-forms that are pointwise Lipschitz outside sets of $\sigma$-finite $(m-1)$-dimensional Hausdorff measure. In addition, we show that the usual transformation formula holds for local lipeomorphisms, which need not be injective

#### Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 623-636.

Dates
First available in Project Euclid: 27 June 2008

https://projecteuclid.org/euclid.rae/1214571355

Mathematical Reviews number (MathSciNet)
MR1844141

Zentralblatt MATH identifier
1023.26010

#### Citation

Pfeffer, W. F. The Stokes Theorem for the Generalized Riemann Integral. Real Anal. Exchange 26 (2000), no. 2, 623--636. https://projecteuclid.org/euclid.rae/1214571355

#### References

• Z. Buczolich and W. F. Pfeffer, Variations of additive functions, Czechoslovak Math. J., 47 (1997), 525–555.
• Z. Buczolich and W. F. Pfeffer, On absolute continuity, J. Math. Anal. Appl., 222 (1998), 64–78.
• Z. Buczolich, T. De Pauw and W. F. Pfeffer Charges, BV functions, and multipliers for generalized Riemann integrals, Indiana Univ. Math. J., 48 1999, 1471–1511.
• H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
• W. F. Pfeffer, The Gauss-Green theorem, Adv. Math., 87 (1991), 93–147.
• W. F. Pfeffer, The Riemann Approach to Integration, Cambridge Univ. Press, Cambridge, 1993.
• M. Spivak, Calculus on Manifolds, Benjamin, London, 1965.
• W. P. Ziemer, Weakly Differentiable functions, Springer-Verlag, New York, 1989.