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Well, this question may seem silly and I fear it's even out of topic here. My motivation to ask that is to know the correct terminology when talking about that here in Math.SE. The point is, here in Brazil the topics covered in Spivak's Calculus on Manifolds and Munkres Analysis on Manifolds is called "Analysis in $\Bbb R^n$", but it seems that in english people just call this "analysis on manifolds". I feel that there should be another name, because "manifolds" is much more general than $\Bbb R^n$.

So, how do people call in english the set of topics covered in those books? Analysis in $\Bbb R^n$, analysis on manifolds, multivariable analysis or what?

Thanks very much in advance!

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Mutivariable calculus and vector calculus, although (I was told) the latter can be something else. Them native english speakers usually mean Functional Analysis and Measure Theory when they say analysis, I believe. – Git Gud Jun 20 '13 at 22:28
Real analysis...? – DonAntonio Jun 20 '13 at 22:28
Real analysis. The $n$ is implied. – copper.hat Jun 20 '13 at 22:41
Thanks very much! – user1620696 Jun 20 '13 at 22:45
See also…. – lhf Jun 20 '13 at 23:07
up vote 2 down vote accepted

Contexts vary, naturally, but in the U.S. a common convention is that "real analysis" refers, somewhat perversely, to measure theory... sometimes with an emphasis on $\mathbb R^n$ or limitation to that case, but rarely exclusively so.

One could suggest "analysis on Euclidean spaces", which makes sense in many ways, but this doesn't exclude products of $\mathbb R^n$'s and ("flat") multi-tori. In fact, it's not at all a bad thing to include tori and multi-tori, offering the option to look at Fourier series in contrast to Fourier transforms, which are less often considered in "real analysis" courses, because the difficulties are best met with ideas that fit even less well into a "measure theory" context than the Hilbert-space and Banach-space ideas relevant to basic Fourier series.

In the end, although some people will misunderstand, and it's actually slightly broader, I think "analysis on Euclidean spaces" is the best reference to analysis on $\mathbb R^n$.

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The easiest sufficiently precise term to use in this case is "real analysis". So far as I've heard it used, this refers only to analysis over $\mathbb R^n$.

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I fear that this is not so, admittedly perversely. – paul garrett Jun 20 '13 at 23:08

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