Advertisement

Introduction to Discrete Calculus

Chapter

Abstract

In this chapter we review conventional vector calculus from the standpoint of a generalized exposition in terms of exterior calculus and the theory of forms. This generalization allows us to distill the important elements necessary to operate the basic machinery of conventional vector calculus. This basic machinery is then redefined in a discrete setting to produce appropriate definitions of the domain, boundary, functions, integrals, metric and derivative. These definitions are then employed to demonstrate how the structure of the discrete calculus behaves analogously to the conventional vector calculus in many different ways.

Keywords

Boundary Operator Differential Form Fundamental Theorem Incidence Matrix Laplacian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Agarwal, S., Branson, K., Belongie, S.: Higher order learning with graphs. In: Proc. of the 23rd Int. Conf. on Mach. Learn., vol. 148, pp. 17–24 (2006) Google Scholar
  2. 10.
    Apostol, T.M.: Calculus, vol. 1, 2nd edn. Wiley, New York (1967) zbMATHGoogle Scholar
  3. 23.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, vol. 14, pp. 585–591. MIT Press, Cambridge (2001) Google Scholar
  4. 24.
    Bell, N., Hirani, A.N.: PyDEC: A Python library for Discrete Exterior Calculus. http://code.google.com/p/pydec/ (2008)
  5. 25.
    Bell, N., Hirani, A.N.: PyDEC: Algorithms and software for Discretization of Exterior Calculus (2010, in preparation) Google Scholar
  6. 31.
    Besag, J.: Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B 36(2), 192–236 (1974) MathSciNetzbMATHGoogle Scholar
  7. 34.
    Biggs, N.: Algebraic Graph Theory. Cambridge Tracts in Mathematics, vol. 67. Cambridge University Press, Cambridge (1974) zbMATHCrossRefGoogle Scholar
  8. 36.
    Biggs, N.: Algebraic potential theory on graphs. Bulletin of the London Mathematical Society 29, 641–682 (1997) MathSciNetCrossRefGoogle Scholar
  9. 40.
    Bochev, P.B., Hyman, J.M.: Principles of mimetic discretizations of differential operators. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds.) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and Its Applications, vol. 142, pp. 89–119. Springer, New York (2006) CrossRefGoogle Scholar
  10. 47.
    Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998) zbMATHGoogle Scholar
  11. 48.
    Bossavit, A.: Applied differential geometry—a compendium. http://butler.cc.tut.fi/~bossavit/BackupICM/Compendium.html (2005)
  12. 54.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of International Conference on Computer Vision, vol. 1 (2003) Google Scholar
  13. 59.
    Branin, F.H. Jr.: The algebraic-topological basis for network analogies and the vector calculus. In: Proc. of Conf. on Generalized Networks, pp. 453–491, Brooklyn, NY (1966) Google Scholar
  14. 74.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. (ed.) Problems in Analysis, pp. 195–199. Princeton University Press, Princeton (1970) Google Scholar
  15. 79.
    Chung, F.: Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9(1), 1–19 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 80.
    Chung, F.R.K.: The Laplacian of a hypergraph. In: Proc. of a DIMACS Workshop, Discrete Math. Theoret. Comput. Sci., vol. 10, pp. 21–36. Am. Math. Soc., Providence (1993) Google Scholar
  17. 81.
    Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997) zbMATHGoogle Scholar
  18. 99.
    Darling, R.: Differential Forms and Connections. Cambridge University Press, Cambridge (1994) zbMATHCrossRefGoogle Scholar
  19. 102.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. arXiv:math.DG/0508341 (2005)
  20. 103.
    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: Proc. of SIGGRAPH (2005) Google Scholar
  21. 105.
    Desbrun, M., Polthier, K.: Discrete differential geometry. Computer Aided Geometric Design 24(8–9), 427 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 111.
    Dodziuk, J.: Difference equations, isoperimetric inequality and the transience of certain random walks. Transactions of the American Mathematical Society 284, 787–794 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 112.
    Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: Ellworthy, K.D. (ed.) From Local Times to Global Geometry, Control and Physics. Pitman Research Notes in Mathematics Series, vol. 150, pp. 68–74. Longman, Harlow (1986) Google Scholar
  24. 115.
    Doyle, P., Snell, L.: Random Walks and Electric Networks. Carus Mathematical Monographs, vol. 22. Math. Assoc. Am., Washington (1984) zbMATHGoogle Scholar
  25. 118.
    Eckmann, B.: Harmonische Funktionen und Randwertaufgaben in einem Komplex. Commentarii Mathematici Helvetici 17, 240–245 (1945) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 127.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2000) Google Scholar
  27. 139.
    Flanders, H.: Differential Forms. Academic Press, New York (1963) zbMATHGoogle Scholar
  28. 142.
    Frankel, T.: The Geometry of Physics: An Introduction, 2nd edn. Cambridge University Press, Cambridge (2004) zbMATHGoogle Scholar
  29. 161.
    Grady, L.: Random walks for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(11), 1768–1783 (2006) CrossRefGoogle Scholar
  30. 178.
    Gross, P.W., Kotiuga, P.R.: Electromagnetic Theory and Computation: A Topological Approach. Cambridge University Press, Cambridge (2004) CrossRefGoogle Scholar
  31. 189.
    Harrison, J.: Geometric Hodge star operator with applications to the theorems of Gauss and Green. Mathematical Proceedings of the Cambridge Philosophical Society 140(1), 135–155 (2006). doi: 10.1017/S0305004105008716 MathSciNetzbMATHCrossRefGoogle Scholar
  32. 191.
    Hassin, R.: Maximum flow in (s, t) planar networks. Information Processing Letters 13(3), 107 (1981) MathSciNetCrossRefGoogle Scholar
  33. 197.
    Hestenes, D.: Space-Time Algebra. Gordon and Breach, New York (1966) zbMATHGoogle Scholar
  34. 199.
    Hiptmair, R.: Discrete Hodge-operators: an algebraic perspective. In: Teixeira, F.L. (ed.) Geometric Methods for Computational Electromagnetics. Progress in Electromagnetics Research, vol. 32, pp. 247–269. EMW Publishing, Cambridge (2001). Chap. 10 Google Scholar
  35. 200.
    Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003) Google Scholar
  36. 218.
    Jancewicz, B.: The extended Grassmann algebra of ℝ3. In: Baylis, W.E. (ed.) Clifford (Geometric) Algebras with Applications to Physics, Mathematics, and Engineering, pp. 389–421. Birkhäuser, Boston (1996). Chap. 28. Google Scholar
  37. 219.
    Jiang, X., Lim, L.H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Mathematical Programming (to appear) Google Scholar
  38. 232.
    Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie 72, 497–508 (1847) Google Scholar
  39. 248.
    Kotiuga, P.: Theoretical limitations of discrete exterior calculus in the context of computational electromagnetics. IEEE Transactions on Magnetics 44(6), 1162–1165 (2008). doi: 10.1109/TMAG.2007.915998 CrossRefGoogle Scholar
  40. 260.
    Lichnerowicz, A.: Théorie globale des connexions et des groupes d’holonomie. Edizioni Cremonese, Roma (1962) Google Scholar
  41. 275.
    Mattiusi, C.: The finite volume, finite difference, and finite elements methods as numerical methods for physical field problems. Advances in Imaging and Electron Physics 113, 1–146 (2000) CrossRefGoogle Scholar
  42. 276.
    Matveev, S.V.: Lectures on Algebraic Topology. EMS Series of Lectures in Mathematics. European Mathematical Society, Zürich (2006) zbMATHCrossRefGoogle Scholar
  43. 284.
    Milnor, J.: On the relationship between differentiable manifolds and combinatorial manifolds. In: Collected Papers of John Milnor, vol. III: Differential Topology, pp. 19–28. Am. Math. Soc., Providence (1956) Google Scholar
  44. 288.
    Muhammad, A., Egerstedt, M.: Control using higher order Laplacians in network topologies. In: Proc. of the 17th Int. Symp. on Math. Theory of Networks and Systems, pp. 1024–1038 (2006) Google Scholar
  45. 290.
    Munkres, J.R.: Elements of Algebraic Topology. Perseus Books, Cambridge (1986) Google Scholar
  46. 291.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1999) zbMATHGoogle Scholar
  47. 304.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization. Dover, New York (1998) zbMATHGoogle Scholar
  48. 307.
    Poincaré, H.: Analysis situs. Journal de l’École Polytechnique 2(1), 1–123 (1895) Google Scholar
  49. 323.
    Roth, J.P.: An application of algebraic topology to numerical analysis: On the existence of a solution to the network problem. Proceedings of the National Academy of Sciences of the United States of America 41(7), 518–521 (1955) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 335.
    Schutz, B.F.: Geometrical Methods of Mathematical Physics. Cambridge University Press, Cambridge (1980) zbMATHGoogle Scholar
  51. 340.
    Seymour, P.: Sums of circuits. In: Bondy, J.A., Murty, U.R.S. (eds.) Graph Theory and Related Topics, pp. 341–355. Academic Press, New York (1979) Google Scholar
  52. 345.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000) CrossRefGoogle Scholar
  53. 353.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 1, 3rd edn. Publish or Perish, Houston (2005) Google Scholar
  54. 359.
    Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley (1986) zbMATHGoogle Scholar
  55. 360.
    Strang, G.: Computational Science and Engineering. Wellesley-Cambridge Press, Wellesley (2007) zbMATHGoogle Scholar
  56. 364.
    Sullivan, J.M.: A crystalline approximation theorem for hypersurfaces. Ph.D. thesis, Princeton University, Princeton, NJ (1990) Google Scholar
  57. 379.
    Tonti, E.: On the geometrical structure of the electromagnetism. In: Gravitation, Electromagnetism and Geometrical Structures, for the 80th Birthday of A. Lichnerowicz, pp. 281–308. Pitagora Editrice, Bologna (1995) Google Scholar
  58. 380.
    Tonti, E.: A direct discrete formulation of field laws: The cell method. Computer Modeling in Engineering and Sciences 2(2), 237–258 (2001) Google Scholar
  59. 388.
    Veblen, O., Whitehead, J.: The Foundations of Differential Geometry. Cambridge University Press, Cambridge (1932) Google Scholar
  60. 395.
    Warnick, K.F., Selfridge, R.H., Arnold, D.V.: Teaching electromagnetic field theory using differential forms. IEEE Transactions on Education 40(1), 53–68 (1997) CrossRefGoogle Scholar
  61. 399.
    Weinreich, G.: Geometrical Vectors. University of Chicago Press, Chicago (1998) zbMATHGoogle Scholar
  62. 400.
    Weyl, H.: Repartición de corriente en una red conductora. Revista Matemática Hispano-Americana 5(6), 153–164 (1923) Google Scholar
  63. 402.
    Whitney, H.: Non-separable and planar graphs. Transactions of the American Mathematical Society 34, 339–362 (1932) MathSciNetCrossRefGoogle Scholar
  64. 403.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

Personalised recommendations