BOOKS

Other Books on Differential Forms:

• Harley Flanders, Differential Forms with Applications to the Physical Sciences,
  Academic Press, New York, 1963; Dover, New York, 1989.
  Excellent, but somewhat dated introduction to differential forms; only covers the positive definite case. An inexpensive paperback.
• Richard L. Bishop & Samuel I. Goldberg, Tensor Analysis on Manifolds,
  Macmillan, New York, 1968; Dover, New York, 1980.
  Another inexpensive paperback. Somewhat more sophisticated; a good reference.
• Robert H. Wasserman, Tensors & Manifolds,
  Oxford University Press, Oxford, 1992.
  A more readable version of Bishop & Goldberg; unfortunately not cheap.
• David Bachman A geometric approach to differential forms,
  Birkhäuser, Boston, 2006.
  Very readable math text.
• David Lovelock and Hanno Rund, Tensors, differential forms, and variational principles,
  Wiley, New York, 1975; Dover, New York, 1989.
  Another inexpensive paperback, covering similar material from a slightly different point of view.
• Hwei Hsu, Vector Analysis (Chapter 10 only),
  Simon & Schuster Technical Outlines, 1969.
  A sort of Schaum outline for differential forms.

Books on Vector Calculus:

• H. M. Schey, Div, Grad, Curl, and All That,
  Norton, New York, 1973 (2nd edition: 1992; 3rd edition: 1997; 4th edition: 2005)
  Excellent informal introduction to the geometry of vector calculus.
• Gabriel Weinreich, Geometrical Vectors,
  University of Chicago Press, Chicago, 1998.
  A nice (if idiosyncratic), geometrical description of differential forms without ever using those words.

Books on Differential Geometry:

• Barrett O'Neill, Elementary Differential Geometry,
  Academic Press, New York, 1966 (2nd edition: 1997).
  Standard, fairly readable introduction to differential geometry in ordinary Euclidean 3-space. Uses differential forms extensively.
• Richard S. Millman & George D. Parker, Elements of Differential Geometry,
  Prentice-Hall, Englewood Cliffs, NJ, 1977 (2nd edition: 1997).
  Good undergraduate text covering similar material to the above. Does not use differential forms.
• Barrett O'Neill, Semi-Riemannian Geometry,
  Academic Press, New York, 1983.
  Graduate math text. The best available treatment of differential geometry without the usual assumption that the metric is positive definite.
• William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry,
  Academic Press, New York, 1986.
  Fairly readable graduate math text. Good but brief treatment of differential forms and integration, but emphasis is on Lie groups.
• Manfredo Perdigão do Carmo, Riemannian Geometry,
  Birkhäuser, Boston, 1992.
  Quite readable math text, but no discussion of differential forms.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, (5 volumes),
  Publish or Perish, Houston, 1970-1975 (2nd edition: 1979).
  Everything you ever wanted to know, and then some, but not easy to read.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation,
  Freeman, San Fransisco, 1973.
  Don't underestimate this classic! Chapter 4 has a nice introduction to differential forms, including great pictures and a discussion of electromagnetism. Part III also discusses the differential geometry needed for relativity.