# Difference between Advanced Calculus and Calculus on Manifolds?

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?

• 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. – user204299 Feb 26 '16 at 0:35
• 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. – David Feb 26 '16 at 3:40
• 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. – Alfred Yerger Apr 7 '17 at 4:24