Differential Forms: Integration on Manifolds and Stokes's Theorem Review

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Differential Forms: Integration on Manifolds and Stokes's Theorem ReviewFortunately there are several books, at an introductory level suitable for undergraduate students, on how differential forms constitute a "new" powerful mathematical technique that surpasses the outdated vector calculus. This book by Steven H. Weintraub is a very good example among others -- such as: (i) "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards (Birkhäuser, Boston, 1994); (ii) "Vector Calculus, Linear Algebra, and Differential Forms" by John H. Hubbard and Barbara Burke Hubbard (Prentice Hall, NJ, 2nd ed., 2002).
As far as I know, it was in "Gravitation" -- by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Freeman, San Francisco, 1973) -- that a pictorial representation of forms was clearly presented to physicists for the first time. These authors went even further, explaining how "forms illuminate electromagnetism, and electromagnetism illuminates forms" (p. 105).
However, until now, it seems that in engineering forms have been disregarded -- despite early attempts by George A. Deschamps (see, e.g., his paper "Electromagnetism and differential forms", Proc. IEEE, Vol. 69, pp. 676-679, 1981), not to mention Harley Flanders's book ("Differential Forms with Applications to the Physical Sciences", Dover, NY, 1989). Perhaps the book by Ismo V. Lindell ("Differential Forms in Electromagnetics", IEEE Press/Wiley, NJ, 2004) will be able to change this sad scenario.
It seems that the difficulty lies mainly in the fact that a proper understanding of k-forms, as antisymmetric (0,k) tensors in differentiable manifolds, requires the study of technical demanding subjects such as de Rham cohomology. However, this book shows that it is possible to make an introduction to forms without mastering such concepts in topological and smooth manifolds -- although there is an extensive bibliography on this subject out there (the books by John M. Lee on manifolds are my favorite).
For more advanced readers, the book by Friedrich H. Hehl and Yuri N. Obukhov on the "Foundations of Classical Electrodynamics" (Birkhäuser, Boston, 2003) is, in my opinion, the most elegant exposition on the relation between electromagnetism and forms.Differential Forms: Integration on Manifolds and Stokes's Theorem Overview

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