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This is an interesting distinction that I don't fully grasp yet. There's quite some books on the topic of the so-called "Advanced Calculus". Some of the most famous of these are the books by Edwards, by Widder, by Friedman, by Sternberg and Loomis, and a number of other authors.

Then there's the books on the so-called "Calculus on Manifolds" - famous books like Spivak's or Munkres' come to mind.

Now my question is: what's the difference here? This misunderstanding is strengthened by the fact that there also seem to be books out there that deal with both; Hubbard & Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" is an example of such a book.

So what's going on here?

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    $\begingroup$ Advanced Calculus tends to just be another name for undergrad Analysis, which is only done in ${\mathbb{R}^n}$. However Calculus on Manifolds, is as the names implies, done on general manifolds. $\endgroup$
    – user204299
    Feb 26, 2016 at 0:35
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    $\begingroup$ I don't know Munkres' book, but Spivak's deals only with submanifolds of $\mathbf{R}^n$, not manifolds in general. It's hard to give a uniform definition of the phrase "advanced calculus" other than by reference to where courses by that name typically fit into an undergraduate curriculum. The best thing is to look at the table of contents of any particular book you might be interested in. $\endgroup$
    – David
    Feb 26, 2016 at 3:40
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    $\begingroup$ Related: My answer to this similar question here. math.stackexchange.com/a/1734136/112357 The long story short is that it's kind of a spectrum. Calculus on manifolds is certainly on the high end of that spectrum. Hubbard's book is probably more in the middle. $\endgroup$
    – A. Thomas Yerger
    Apr 7, 2017 at 4:24

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Based on the various texts I've browsed, the term "advanced calculus" can refer to anything beyond a one-year "mechanical" (i.e., the focus is computational facility rather than theoretical understanding) course in one-variable calculus. Thus "advanced calculus" could refer to (A) one-variable real analysis, (B) a computational course dealing with partial derivative, multiple integrals, and vector calculus, (C) a mathematically rigorous course on multivariable calculus (generally using differential forms), or (D) a first course in complex variables or Lebesgue theory. Thus the term is incredibly vague. On one extreme, I've seen an advanced calculus book with nary a proof. (For example, see the Schaum's study guide on advanced calculus.) On the other extreme, Loomis and Sternberg covers calculus in the setting of Banach spaces, and introduces a large amount of abstract, linear, and multilinear algebra, in order to rigorously describe differentiable manifolds embedded in Banach spaces.

As far as I'm aware, there are really only three undergrad books involving "calculus on manifolds": Spivak's Calculus on Manifolds, Munkres's Analysis on Manifolds, and Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. In order of increasing difficulty, they are Hubbard and Hubbard, Munkres, and Spivak. Each of them really only talk about manifolds in R^n, although Munkres, and especially Spivak give clues as to how to generalize them. Hubbard and Hubbard hide some of the more difficult proofs out of gentleness, while Spivak hides some proofs and writes some other very unclear, opaque, or just plain wrong ones out of lack of pedagogical experience (Calculus on Manifolds is his first book, written when he was 25). Thus, the best one out of the three is probably Munkres, who goes out of his way to help students develop intuition. Hilariously, Loomis and Sternberg was written for freshmen (at Harvard...), but its difficulty makes it suitable for few undergrads. It is a beautiful book, but really more appropriate for grad students (but then again, they insist on using Jordan measure instead of Lebesgue in the name of keeping things elementary...)

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    $\begingroup$ Yeah, the decision to not use Lebesgue made it quite "elementary" $\endgroup$
    – timur
    Apr 25, 2020 at 23:15

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