My instinct was to retag it as "calculus", but now I'm in doubt. "Calculus" is an abbreviation for "the part of real analysis that is usually taught in high school", right?
Okay, "high school" is not completely well defined. Point is "calculus" seems to be a subset of "real analysis". I can imagine something being "real analysis" because it is too advanced for "calculus" (e.g., power series). Can it also be "real analysis" rather than "calculus" simply because it involves neither a derivative nor an integral?
@HenningMakholm To muddy the waters even more... at my university we teach two semesters of "advanced calculus," which, according to my colleagues, is more advanced (duh) than calculus but not as advanced as real analysis.
Advanced calculus course description: "This course is an introduction to advanced analysis. Topics of study include set theory, the topology of Euclidean spaces, functions, continuity, differentiability of functions and mappings, integration, series, uniform convergence, transformation of multiple integrals, differential geometry of curves and surfaces, and vector calculus."
I haven't taught this course yet so I haven't quite figured out how this differs from real analysis.
Sounds like an "overview-of-everything which should be adequate as a general knowledge basis for subjects where you elect not to take the dedicated course" course.
We don't actually teach a course called "real analysis." Whatever real analysis our students get comes from this course. (We only teach undergraduates here.)
@tb It depends on the institution. And, as I said, I haven't quite figured out how "advanced calculus" differs from real analysis, other than it's supposed to be less advanced.
It's my impression that U.S. undergrad math programs (except, I guess, at the best schools, although maybe not even there) aren't as advanced as a lot of the undergrad math programs in, say, Europe. I can't back that up much, other than that it's also my impression that U.S. colleges spend time teaching topics that are often taught in high school in Europe.
Thus, for instance, math majors here often have to take a lot of writing and humanities courses in college, whereas (again, my impression) students in Europe spend more of undergrad focusing on their major subject.
@tb Yes, no measure theory or functional analysis in "advanced calculus" is one of the major distinctions.
@Gortaur That's also true. We don't do much theory in the first calculus sequence, so it gets shoved into advanced calculus.
@Gortaur A large part of the problem is that the high schools in the U.S. aren't all that great. Few of the students we get are ready for theory in the first calculus sequence.
@MikeSpivey In my time the usual rule-of-thumb was that the last year of Danish high school corresponded to the freshman year of an American college. So the level at the end of our 3-year bachelor degrees would be roughly equivalent to a 4-year American degree.
@MikeSpivey I don't know about all of Europe but judging from what I know about central Europe studying at universities is pretty much confined to the specialtie(s): except at "technical universities" where you focus on one subject right away you usually have a main subject and one or two second subjects (like math/physics or history/germanistics+pohilosophy chemistry/biology) for example. General culture and writing are trained marginally at best unless you focus on them.
@Mike: my colleague is from Poland and he told me that there they have in the first semester general topology to get rid of those who isn't abstract enough. I would prefer starting with metric spaces though - quite general and still easy to motivate young students.
