I've been trying to grasp the concept of differential forms, which I have been encountering while studying the text "Geometric Measure Theory" by Frank Morgan. Unfortunately the explanation is very sparse and while the internet contains many definitions, I have a hard time getting the bigger picture from just reading them. Are there any lists of exercises that I could do to assist my understanding by just working with them?
Thank you.
Maybe you will find useful the notes by Sjamaar - "Manifolds and Differential Forms" which can be downloaded for free at his website. The explanation is geometrically motivated and straightforward from the ground up, and it contains lots of doable exercises and explicit detailed examples which may help you grasp everything you need to know and more. Donu Arapura has a nice elementary summary of the concepts and uses of differential forms in his notes Arapura - "Introduction to differential forms" freely downloadable too.
The most explicit, introductory but detailed, and full of exercises references for an elementary introduction to all of this, are the books:
(The second one has a draft old version available online, but the second edition of the book has been very improved).
As a conceptual complement, a very interesting book geared toward theoretical physics applications is Baez/Muniain - "Gauge Fields, Knots and Gravity" where the meaning and extensive use of covectors and differential forms in general is used as a primary tool to formulate physical theories in geometric terms.
Using the general framework to rederive classical results should be a good exercise and let you see how the general framework relates to what you already know. Here are two exercises I think will be enlightening.
Show that the de Rham complexes of $\mathbb{R}^2$ and $\mathbb{R}^3$ are isomorphic to $$0\rightarrow C^\infty(\mathbb{R}^2,\mathbb{R})\stackrel{\text{grad}}{\longrightarrow}C^\infty(\mathbb{R}^2,\mathbb{R}^2)\stackrel{\text{rot}}{\longrightarrow}C^\infty(\mathbb{R}^2,\mathbb{R})\rightarrow 0$$ and $$0\rightarrow C^\infty(\mathbb{R}^3,\mathbb{R})\stackrel{\text{grad}}{\longrightarrow}C^\infty(\mathbb{R}^3,\mathbb{R}^3)\stackrel{\text{curl}}{\longrightarrow}C^\infty(\mathbb{R}^3,\mathbb{R}^3)\stackrel{\text{div}}{\longrightarrow}C^\infty(\mathbb{R}^3,\mathbb{R})\rightarrow 0$$ respectively.
Use the generalized Stokes' theorem to rederive Green's theorem, the divergence theorem and the classical stokes' theorem from classical multivariable analysis.
I would recommend Do Carmo's "Differential Forms and Applications". That's where I first learned differential form. It explains differential form very clearly and it contains exercises for each chapter. I benefit a lot by doing the exercises in the book.
