What's the difference between Rigorous Calculus and Analysis?

press

I dont understand the difference between Rigorous Calculus books (Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approch", Loomis and Sternberg's "Advanced Calculus", Spivak's "Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus") and Analysis books (Pugh's "Real Mathematical Analysis" and Rudin's "Principles of Mathematical Analysis")?

Casually looking through Sternberg's book, it looks not a thing like a regular Calculus book, but very much like a regular Analysis text (Pugh's), except it contains a lot of seemingly unrelated content, too. Seems like a lot of topics in these Calculus and Analysis books overlap.So, would you say Sternberg's Advanced Calculus book= Pugh's Analysis book?

jbunniii

There is no clear distinction between rigorous calculus vs. analysis, except that the latter is a much broader topic, encompassing not just rigorous calculus but also functional analysis, measure theory, harmonic analysis, and so forth.

If we restrict our attention to books covering primarily the topics associated with classical calculus: limits, continuity, differentiation, integration, series expansions and the like, then some will have analysis in the title and others will say calculus. But this does not imply anything about the level of sophistication. Loomis and Sternberg's Advanced Calculus is at least as challenging as Pugh's Real Mathematical Analysis or Rudin's Principles of Mathematical Analysis.

I personally, and rather arbitrarily, consider a book to fall in the "calculus" category if it doesn't do much if any topology. Spivak's Calculus and Apostol's Calculus are the prime examples of this.