Difference between calculus and analysis

Solution 1:

Remember, that the term "calculus" really stands for a method of calculation, especially one of several highly systematic methods of treating problems by a special system of ... notations. Think of vector and tensor calculus, calculus of variations, lambda calculus etc. Binding this term to integrals and derivatives is probably rather a pedagogical, than scientifical tradition.

Solution 2:

The way I see it, "calculus" is just shorthand for "differential calculus" and "integral calculus". So it's mostly "calculus" when you are concerned with finding derivatives and integrals, and solving (easy) differential equations, with emphasis on the mechanical part.

"Analysis" is a much broader term, that includes the concepts and proofs concerning calculus (continuity, differentiation, integration), but many others, including for example Measure Theory and Functional Analysis.

Solution 3:

As (I believe) the words are commonly used, calculus is the art of calculating, and analysis is the art of analysis.

The focus of a calculus course is on computation. Infinitesimal* calculus deals with methods for computing limits, derivatives and integrals symbolically, or by numerical approximations complete with error analysis. Even some/most/all of the proofs can be seen as exercises in manipulating approximations towards a goal.

*: I include differential and integral calculus in this category. Although the methods don't include 'true' infinitesimals as in the hyperreal numbers, the focus of these subjects is still usually along the lines of manipulating things by breaking them down into infinitesimal parts, or recombining the infinitesimal parts to yield an object of interest.

The focus of a real analysis course, on the other hand, is more about analysis -- breaking the subject matter apart into useful ideas. Methods of topology and measure theory are developed and applied, the theory and properties of derivatives and integrals are developed, and the ideas and structures involved are generalized beyond the 'simple' case of multivariate real functions.

Solution 4:

In line with @ArthurFischer's comment: IIRC, in French "un calculus" is a small calculation. I think that broadly speaking, calculus courses and texts are (by tradition if not by intrinsic definition) about the rules and procedures needed to do calculations with derivatives and integrals: hence "integral calculus" and "differential calculus" (not to mention "Ito calculus" and "Malliavin calculus" if you want to get really fancy).

I tend to agree with Martin that "analysis", broadly speaking, tends to refer more to the body of work that looks at the proofs of these results, and how they assemble into a coherent whole that can be derived from basic foundations.