[jacobolus]

> for someone studying vector calculus properly

If you can’t invert vectors, you aren’t studying vector calculus properly. ;-)

Differential forms are a half-baked formalism.

Unfortunately I don’t know of any great undergraduate level geometric calculus textbooks. Ideally there would be something like Hubbard & Hubbard’s book (http://matrixeditions.com/5thUnifiedApproach.html) written using GA as a formalism.

Hestenes & Sobczyk’s book (http://geocalc.clas.asu.edu/html/CA_to_GC.html) is a hard slog, and not appropriate for an undergraduate audience.

[spekcular]

> Differential forms are a half-baked formalism.

I can't emphasize enough how wrong this is. It's the standard formalism in research-level physics and math for good reasons.