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Douglas Lind
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Harold Edwards' book Advanced Calculus: A Differential Forms Approach starts with forms as the basic objects, and gives really nice intuitive explanations. It was written in 1969, as an undergraduate text from an unconventional point of view, and is still available from Birkhauser. But Edwards told me a few years ago that it was probably too hard for today's students (everything is done quite rigorously).

Douglas Lind
  • 2.8k
  • 26
  • 19