I am a (now) math major who has taken a standard undergraduate course in engineering multivariable calculus. I say "engineering" since the course was directed at engineers, and mainly focused on computation. Since then, I have actually taken real analysis too, covering chapters 1-9 of Rudin. It covers a little bit about multivariable calculus rigorously, but it doesn't really cover the vector calculus part. I have heard chapter 10 of Rudin is bad for this. Some suggestions I have gotten for it is "Calculus on Manifolds" by Spivak and "Analysis on Manifolds" by Munkres. However, since I have taken plenty of higher level math courses (including topology) and generally feel mathematically mature, perhaps I could take a course in differential geometry at this point instead? From what I have understood, it covers vector calculus and generalises it to smooth manifolds, provided you know the inverse/implicit function theorems, differentiation in multiple dimensions and the chain rule.
Am I correct, or would I do myself a disservice by skipping rigorous vector calculus? If so, what would I miss if I were to skip it?