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Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
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How do I make the conceptual transition from multivariable calculus to differential forms?
There is an amazing book called Foundations of Classical Electrodynamics: Charge, Flux, and Metric written by Friedrich Hehl and Yuri Obukhov. The book is completely metric-free and heavily uses diffe …