Multivariable calc “second course” that does differential forms

I’ve worked through a computation-heavy, “standard” but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) “second course”? In particular, I’m looking for a book that

and leads well into an intro manifolds book like Tu’s one. (Bonus points if it actually talks a bit about manifolds itself or works with differential forms defined on manifolds instead of only $\Bbb R^n$!)

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Possible recommendations – I’m not 100% sure if what you are looking for exists but these are all well-written and worth investigating.

  • Edwards, Advanced calculus: a differential forms approach
  • Bloch, A First Course in Geometric Topology and Differential Geometry
  • Bachmann, A Geometric Approach to Differential Forms

The last one is particularly nice, in my opinion.