There isn't complete consensus on what topics should be included. Multivariable differentiation and integration for sure. Vector analysis (with or without differential forms) up to Stokes' theorem. In the older books there might be a single chapter for each of complex variables and ODEs. Also the treatment would usually be rigorous (epsilons and deltas). The problem is that in the early '70s the standard one-year course in "advanced calculus", as taught to mathematicians, physicists, and engineers, was done away with and replaced with the more fragmented courses of today.
Maybe 'calculus' is a common name in USA. In Europe calculus is what we learn(ed() at school. More accurate names imo are math methods (calculus on steroids, working things out like Stokes stuff) and real/complex analysis (epsilon/delta, hard core stuf). There again, I haven't looked in 47 years.
At least in the U.S., advanced calculus tends to act as a bridge between traditional multivariable calculus — calculus of scalar valued functions defined on R^2, R^3 — and differential geometry. That is, advanced calculus tends to function as a true vector calculus course; in my experience, multivariable calculus didn’t spend enough time treating the calculus of vector-valued maps. Spivak has a great little book titled “Calculus on Manifolds”; Marsden and Tromba’s “Vector Calculus” is also terrific; finally, Loomis and Sternberg’s “Advanced Calculus” treats some even more advanced topics like Lie derivatives and Exterior calculus.
In the States, the main difference between calculus and analysis is that the former is calculation-based and the latter is proof-based.
I should have been more careful. I was meaning to say the main differences in *how calculus and analysis are taught*; I was not meaning to make any claim about what the two meant historically. In this sense, analysis (real, complex, Fourier, and functional) is most certainly about proving theorems. At the undergraduate level, numerical analysis doesn’t seem to be a proper subcamp of analysis, though this could simply be because of how it is taught at my university where only calculus, linear algebra and ODEs are prerequisite courses.
To stop with Newton and Leibniz is an injustice, however. Cauchy and Weierstrass and others of the 19th century are equally worth mention in the development of modern analysis. In my opinion, the 1800's is when analysis (using this as a pure mathematician would) got underway.
My initial response was simply to indicate that “advanced calculus”, as a university level course, typically functions as a bridge to learning differential geometry and treating vector calculus techniques more generally.