Which of these things is not like the others? What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I am now trying to learn mathematics properly, from the ground up, mostly for the love of it.
For functions of a single variable, "analysis" seems to be used almost exclusively for the theory, while "calculus" is the tip of that iceberg, or "how to compute stuff". However, my impression is that when it comes to functions of several variables, the nomenclature is rather more messy. Gibbs seems to have provided the first systematic treatment of the subject in his "Vector Analysis" book, but I would appreciate if someone could explain to me the differences between the following names (and which, if any, are used interchangeably):


*

*vector analysis

*vector calculus

*multi-variable calculus

*topological vector spaces

*functional analysis


Say I have finished a course on real analysis (e.g. Rudin). Where should I look next if I want to move on to functions several variables?
 A: Two common books you may want to look at are "Analysis on Manifolds" by Munkres and "Calculus on Manifolds" by Spivak. 
Multivariable calculus and vector calculus and vector analysis are usually terms reserved for the end of a calculus sequence (usually first 3 or 4 terms in the US, not proof based, basic introduction to stokes/green/divergence theorem, partial derivatives in $\mathbb{R}^3$). Essentially calculus on vector fields. These are often taught from books like Stewart's Calculus, or for smarter kids, Marsden & Tromba's Vector calculus. In Munkres or Spivak's book, you'll see more general and rigorous versions of this (and a few other topics, like de Rham Cohomology if you have time). 
Not all books called "calculus" are necessarily non-rigorous (Spivak's Calculus is reasonably rigorous). 
Topological vector spaces are simply vector spaces where addition and scalar multiplication are continuous. They are basic structures needed to study functional analysis (see, for example, Chapter 1 of Rudin's Functional Analysis). Functional analysis can be thought of as infinite dimensional linear algebra (say on function spaces) in some sense. You study the geometry of infinite dimensional vector spaces among other things [ often function spaces ], how things converge, etc. Typically, you should have a background in measure and integration (e.g. at the level of Royden's real analysis text or Folland's Real Analysis text) before approaching this subject (unless you're following an applied book for undergrads like Kreyszig). Famous results are Hahn-Banach (related to extending functionals), open mapping theorem, closed graph theorem, Banach-Steinhaus Theorem, Spectral theorems, etc. This is nothing like the prior categories mentioned.
