Discrete Versions of Stokes’ Theorem Based on Families of Weights on Hypercubes

  • Gilbert Labelle
  • Annie Lacasse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


This paper generalizes to higher dimensions some algorithms that we developed in [1,2,3] using a discrete version of Green’s theorem. More precisely, we present discrete versions of Stokes’ theorem and Poincaré lemma based on families of weights on hypercubes. Our approach is similar to that of Mansfield and Hydon [4] where they start with a concept of difference forms to develop their discrete version of Stokes’ theorem. Various applications are also given.


Discrete Stokes’ theorem Poincaré lemma hypercube 


  1. 1.
    Brlek, S., Labelle, G., Lacasse, A.: Incremental algorithms based on discrete Green theorem. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 277–287. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Brlek, S., Labelle, G., Lacasse, A.: The discrete Green theorem and some applications in discrete geometry. Theoret. Comput. Sci. 346(2), 200–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lacasse, A.: Contributions à l’analyse de figures discrètes en dimension quelconque. PhD thesis, Université du Québec à Montréal (September 2008)Google Scholar
  4. 4.
    Mansfield, E.L., Hydon, P.E.: Difference forms. Found. Comp. Math.(2007),
  5. 5.
    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: Discrete differential geometry. Oberwolfach Semin, vol. 38, pp. 287–324. Birkhäuser, Basel (2008)CrossRefGoogle Scholar
  6. 6.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computing homology. Homology Homotopy Appl 5(2), 233–256 (2003); (electronic) Algebraic topological methods in computer science (Stanford, CA, 2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2-3), 245–263 (2005)Google Scholar
  8. 8.
    Spivak, M.: Calculus on manifolds. A modern approach to classical theorems of advanced calculus. W. A. Benjamin, Inc., New York (1965)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Gilbert Labelle
    • 1
  • Annie Lacasse
    • 2
  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUQAMMontréalCanada
  2. 2.Laboratoire d’Informatique, de Robotique et de Microélectronique de MontpellierMontpellier Cedex 5France

Personalised recommendations