I'm not directly answering your question, but I would like to suggest to great books for you to learn the meaning of exterior calculus, integration and de Rham cohomology.

1. "Vector Analysis on Manifolds" by Janich. - I learned that stuff from this book and for me is still the best.

2. "Mathematical Methods in Classical Mechanics" by Arnold. - Well known for its great insight on the meaning of exterior derivative, Lie derivative, Stokes theorem, paralell transport, riemannian curvature, among other concepts.

I'm not directly answering your question, but I would like to suggest two great books for you to learn the meaning of exterior calculus, integration and de Rham cohomology.

1. "Vector Analysis on Manifolds" by Janich. - I learned that stuff from this book and for me is still the best.

2. "Mathematical Methods in Classical Mechanics" by Arnold. - Well known for its great insight on the meaning of exterior derivative, Lie derivative, Stokes theorem, paralell transport, riemannian curvature, among other concepts.