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I am looking for a nice illustration of how vector calculus relates to differential forms. A demonstration that employs physics is appreciable (e.g. electromagnetism).

In particular, while dualizing gradient, divergence and curl gives the same set of operators again, this is not the same with the differential forms perspective. How to appropriately interpret that is usually left blank in physics books. Mathematics books often don't bother at all about classical vector calculus.

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I found the exposition in Dubrovin-Fomenko-Novikov quite nice --- look up the hodge star operation (towards the end of the Wikipedia page the three-dimensional case is spelled out). If I remember correctly it's in volume 3 of D-F-N. Some basic stuff is also explained at the beginning of Bott-Tu. –  t.b. Sep 4 '11 at 18:46
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Try

  • John Baez, Javier P. Muniain: "Gauge Fields, Knots and Gravity"

The first part, "Electromagnetism", explains Maxwell's equations first using classical vector calculus and how to rewrite them using differential forms.

Another classical book that explains differential forms and their use in both classical electromagnetism and in general relativity is

  • Misner, Thorne, Wheeler: "Gravitation"

The latter is a +1000 pages tome, but it is possible to select the chapters that you are interested in and start there, you don't need to read it cover to cover.

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