Related but not duplicate: What courses require multivariable analysis?
By multivariable analysis I mean the rigorous version of multivariable calculus (something equivalent to Ch.9-10 in baby Rudin or topics cover in Analysis on Manifolds by Munkres).
My question is,
Why do advance analysis courses(at least those I took) emphasize so little in topics related to multivariable analysis, compare to topics like measure theory and Lebesgue integral? Is there any reason undergraduate/beginning graduate math education is designed this way?
Aren't they equally important prerequisite for different advanced topics(e.g. Measure Theory to Probability/Stochastic Process, Multivariable Analysis to Differential Geometry, etc.)?
My experience and observation,
In my undergraduate institution, 1st semester real analysis is first few chapters of baby Rudin, 2nd semester is introduction to measure theory. 1st semester graduate level is point set topology/measure theory, 2nd semester graduate level(which I didn't finish) is closer to functional analysis.
In the graduate program I'm currently in, 1 year sequence of graduate real analysis and qualifying exam focus almost exclusively on Measure theory and Lebesgue Integral. Topics in multivariable analysis are included in 3rd quarter of honor undergraduate/intro. graduate course where graduate students are only require to take it if not passing (preliminary) assessment exam.
The focus of real analysis qualifying exam in many places seem to be more on Measure theory and Lebesgue integral, too.