Differential Forms

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter introduces differential forms, exterior differentiation, and multi-vectors in a concrete and explicit way by restricting the discussion to ℝn. This is extended to more general settings later. Roughly speaking, differential forms generalize and unify the concepts of the contour integral, curl, element of surface area, divergence, and volume element that are used in statements of Stokes’ theorem and the divergence theorem. At first it may seem unnecessary to learn yet another new mathematical construction. The trouble is that without an appropriate extension of the concept of the cross product, it is difficult and messy to try to extend the theorems of vector calculus to higher dimensions, and to non-Euclidean spaces. As was illustrated in Chapter 1 in the context of heat and fluid flow problems, these theorems play a central role. Likewise, in probability flow problems involving stochastic differential equations and their associated Fokker–Planck equations, these theorems play a role in assessing how much probability density flows past a given surface. Since the problems of interest (such as the stochastic cart in Figure 1.1) will involve stochastic flows on Lie groups, understanding how to extend Stokes’ theorem and the divergence theorem to these generalized settings will be useful. The first step in achieving this goal is to understand differential forms in ℝn.


Tensor Product Scalar Multiplication Divergence Theorem Exterior Derivative Wedge Product 
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  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T., Manifolds, Tensor Analysis, and Applications, 2nd ed., Applied Mathematical Sciences 75, Springer, New York, 1988.Google Scholar
  2. 2.
    Cartan, H., Differential Forms, Hermann, Paris; Houghton Mifflin, Boston, 1970.Google Scholar
  3. 3.
    Darling, R.W.R., Differential Forms and Connections, Cambridge University Press, London, 1994.Google Scholar
  4. 4.
    do Carmo, M.P., Differential Forms and Applications, Springer-Verlag, Berlin, 1994.Google Scholar
  5. 5.
    Flanders, H., Differential Forms with Applications to the Physical Sciences, Dover, New York, 1989.Google Scholar
  6. 6.
    Greub, W.H., Multilinear Algebra, 2nd ed., Universitext, Springer-Verlag, Berlin, 1987.Google Scholar
  7. 7.
    Guggenheimer, H.W., Differential Geometry, Dover, New York, 1977.Google Scholar
  8. 8.
    Guillemin, V., Pollack, A., Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.Google Scholar
  9. 9.
    Lang, S., Fundamentals of Differential Geometry, Springer, New York, 1999.Google Scholar
  10. 10.
    Lindell, I.V., Differential Forms in Electromagnetics, IEEE Press/Wiley-Interscience, New York, 2004.Google Scholar
  11. 11.
    Lovelock, D., Rund, H., Tensors, Differential Forms, and Variational Principles, Dover, New York, 1989.Google Scholar
  12. 12.
    Mukherjee, A., Topics in Differential Topology, Hindustan Book Agency, New Delhi, 2005.Google Scholar
  13. 13.
    Schreiber, M., Differential Forms: A Heuristic Introduction, Universitext, Springer-Verlag, New York, 1977.Google Scholar
  14. 14.
    Tu, L.W., An Introduction to Manifolds, Springer, New York, 2008.Google Scholar
  15. 15.
    Warner, F.W., Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, 1983.Google Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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