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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?

user42912
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    The others are less obvious but in my experience, vector calculus and vector analysis are the same subject. Vector calculus and multivariable calculus are not the same. Multivariable calculus is quite literally one variable calculus generalized; vector calculus does more advanced/abstract things than this (Stokes' theorem in all of its many forms, curls, gradients, divergence, how these things relate in different coordinate system, Frenet frames, etc). – Cameron Williams Jan 19 '16 at 03:38
  • @CameronWilliams You're saying the difference is that when you apply the word "vector," the focus tends to be geometric, but in general, it's more analytic? – walkar Jan 19 '16 at 03:39
  • @walkar This is just the way I've seen it at several universities. Vector analysis/calculus is a bit of a gentle introduction to manifold theory (if you view it the right way), but focuses on the richness of $\mathbb{R}^3$. – Cameron Williams Jan 19 '16 at 03:42
  • @CameronWilliams and what about advanced calculus? – user42912 Jan 19 '16 at 03:42
  • @user42912 That I'm not so sure because I've seen the other three terms to mean different things at different universities. You'll likely get less of a consensus on those, I'd think. – Cameron Williams Jan 19 '16 at 03:42
  • When I took the course at college advanced calculus was souped up multivariable/vector calculus, included some epsilon delta analysis type of stuff and linear algebra (exterior products and such) to put Stoke's theorem on a firmer foundation. – Aydin Gerek Jan 19 '16 at 03:47
  • And for me, advanced calculus was was basically a course for learning analysis-type proofs: that every sequence has a monotone subsequence, characterizations of continuity, I think Heine-Borel popped in for a visit, that sort of thing. Most functions were of one variable, I think. – pjs36 Jan 19 '16 at 04:00
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    Vector analysis seems more problem driven to me, because of its use in electro dynamics, mechanics and fluid dynamics. Calculus of several variables seems more to extend the rigorous approach of calculus to several dimensions. – mvw Jan 19 '16 at 04:02

2 Answers2

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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.

  1. Calc 1 = differential calculus. Roughly a semester of differential calculus (derivatives, emphasis on techniques, support of use in physics).

  2. 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.

  3. 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.

  4. 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.

Fabio Somenzi
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Multi-variable calculus deals with properties of differentiable functions of more than one independent variable, and it can include the study of functions from $\Bbb{R}^n \to \Bbb{R}^mt$. Vector calculus studies the same functions but focuses on objects that have certain properties under linear transformations of variables. (And since it specializes in this way, vector calculus can in a beginning class afford to go deeper into subtle properties; for example, Greene's and Stokes' theorems.)

Vector calculus is in a very real sense a prelude to tensor calculus. Here is an example of an object you might well study in multi-variable calculus, but would not fit well with the methods of vector calculus. Let $f(x,y,z)$ be a sufficiently differentiable function of three real variables. Then let $$ H(x,y,z) \in \Bbb{R}^2 = \bigg( \frac{\partial f}{\partial x},\frac{\partial^2 f}{\partial y \partial z} \bigg) $$ $H$ does not meet any of the transformation properties we assume in vector calculus; yet it is a perfectly good (if boring) functional of $f$ we could study in multivariable calculus.

I agree with the comments that replacing the word "calculus" with "analysis" just implies a tighter degree of rigor; the boundary is a bit fuzzy there.

The term "advanced calculus" is the most interesting in the group, because everybody seems to agree that it means whatever each author or course content or professor says it means. Thus for example, in the Shaum Outline Series "Advanced Calculus" by Spiegel (a truly excellent if not very rigorous book), topics covered include various integral transforms (Fourier, Laplace, but without a real Hilbert-space-inspired slant), differential equations, calculus of variations, and a last chapter tumbling headlong into elliptic integrals. (My memory may be inexact, but I definitely remember that chapter!) But when at MIT I took the grad course entitled "Advanced Calculus for math majors" we did a lot of orthogonal function theory, Bessel functions, complex analysis. and eigenanalysis -- almost no overlap! And in the "Advanced Calculus for engineers" undergrad course, they did mostly methods of mathematical physics.

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Mark Fischler
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