TITLE: What is the difference between advanced calculus, vector calculus, multivariable calculus, multivariable real analysis and vector analysis?
QUESTION [9 upvotes]: What is the difference between advanced calculus, vector calculus, multivariable calculus, multivariable real analysis and vector analysis?
What I think I know

Vector calculus and multivariable calculus are the same.
Multivariable real analysis and vector analysis are the same and both
are the formalization of multivariable/vector calculus.

Am I right? what's the difference between advanced calculus and these other subjects?

REPLY [11 votes]: The issues are terminology of courses.  Someone can technically say that calculus is real analysis.  But it doesn't mean anything in terms of courses you take, books you read, etc.  So the issue is somewhat of a terminology concern, not a formal mathematical one.  
Here is my take of a stereotypical (most common) use of the terms.  Once you know that, you can look at what other people say as deviations from it.

Calc 1 = differential calculus.  Roughly a semester of differential calculus (derivatives, emphasis on techniques, support of use in physics).
Calc 2 = integral calculus.  Same thing as above but for integrals.  Note, you may do a little baby diff EQs or series.  And the border of differential and integral may not be 100% at the semester break.  But close to it.  
Calc 3 = multivariable calculus = vector analysis.  A semester mostly working on partial derivatives, surface integrals, stuff like that.  Introduction of Stokes and Green's thereoms.
Differential equations.  (occasionally jokingly called "Calc 4").  A semester of ordinary differential equations. ($y$ as a function of $x$.  Not multivariable diff EQs.)  

This pretty much finishes the curriculum for a basic science major.  Engineers or physicists may have another semester or two of "math methods", which will be a whirlwind tour of partial differential equations, linear algebra, and perhaps complex analysis.  

"Real analysis" is theoretical calculus.  You prove a lot of the things you already learned in regular calculus.  It's a math major course.  Engineers, physicists, etc. won't bother taking it.  You won't learn many new techniques that are useful to applied problems or following physics derivations (maybe a little in series).

Advanced calculus is another term for real analysis.  Usually used in titles of older books.  Usually a bit less emphasis on proofs and disdain for applications.  But still mostly covering territory that is not that useful for applications.