Intuition for chains and cochains I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level.  In particular, it would be nice to see some pictures and "physical" applications of these guys.
Anyone have any good references or other resources that might help?
 A: Allen Hatcher's Algebraic Topology is excellent in this respect (intuition). It's also available for free on the author's home page: http://www.math.cornell.edu/~hatcher/AT/ATpage.html
A: I recommend the works by Alain Bossavit on this: he presents electromagnetism in terms of chains, cochains, and integration theory, and gives a very physical and intuitive, pictorial understanding of this topic. Moreover, he also uses it for numerical computations of electromagnetic problems, combined with a discretization of these objects. I recommend these works to start with:


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*Computational electromagnetism and geometry: building a finite-dimensional "Maxwell's house" (2004) https://www.researchgate.net/publication/242462763_Computational_electromagnetism_and_geometry_Building_a_finite-dimensional_Maxwell%27s_house (originally published in a journal 1999–2000);

*Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (2nd ed., Academic Press 2004);

*Discretization of electromagnetic problems: The "generalized finite differences" approach (2005) https://doi.org/10.1016/S1570-8659(04)13002-0;

*A uniform rationale for Whitney forms on various supporting shapes (2010) https://doi.org/10.1016/j.matcom.2008.11.005.
You can check other works by the same author and the references in them. Hope this helps!
