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This question might be a little soft as it does not have a definite answer, so I hope I do not break the conventions of this forum by posting it here.

I have now come across the term "calculus of operators" several times in various books, as well as the "calculus of several variables", "calculus of variations", calculus of residues", "calculus of pseudodifferential operators", and so on.

Each time I don't really understand what is meant by this - my guess is it involves an algebraic system that contains operations broadly to be understood as differentiation and integration - but I am not sure at all, in fact I don't think it fits the bill.

I would be really grateful if I could get some better explanation than mine for the term "calculus of [something]". Many thanks !

harlekin
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    "calculus of [...]" means "systematic computational procedure related to [...]". For example, residue calculus is the ideas related to using residues to compute integrals. A functional calculus (in functional analysis) is a process of making sense of non-polynomial functions (continuous, holomorphic, measurable) of linear operators. Historically, differential and integral calculus were such a substantial advance that the undecorated label "calculus" was understood to refer to them. The effect is that you think any other calculus is somehow connected to it, but that's wrong. – KCd Apr 02 '12 at 00:24
  • Rules (okay, system) of calculation. That's why you see process calculus, lambda calculus. –  Apr 02 '12 at 01:39
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    Note that calculus means small stone. So if you have calculi in the kidneys, or the gall bladder, it can be very painful. It all comes from the time that calculi were used on counting boards, or variants of the abacus. By the way, the abacus was known and used very early in the Mediterranean world, and then disappeared. – André Nicolas Apr 02 '12 at 05:19
  • Dentists who find calculus in your teeth need to remove it. To them it still has the original meaning: a small stone. – Michael Hardy Apr 02 '12 at 17:23
  • @André The abacus did not disappear from Europe. It was widely used in various forms up until modern times. The English department of the Exchequer is named for the checkered cloth that was used to turn the counting table ("counter") into an abacus. See this picture for an example of a Medieval "rechentisch" = "counting table", on which small tokens ("counters") were moved about in a manner identical to the movements of beads on an abacus. – MJD May 30 '12 at 16:42
  • Related: http://math.stackexchange.com/questions/54763/what-do-algebra-and-calculus-mean and http://mathoverflow.net/questions/36758/difference-between-a-calculus-and-an-algebra – anon May 30 '12 at 16:52
  • @MarkDominus: I guess it depends on how one defines modern. Effectively, counting board methods had disappeared in Western and Central Europe by the 17th century. But it is true that the abacus either persisted in Russia, or was reintroduced, I don't know which. – André Nicolas May 30 '12 at 17:08

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While it's often used to refer to various areas of...calculus I guess...in general it's "a system of rules and procedures for performing calculations". While many of these are related to ideas in differential/integral calculus (matrix, vector, stochiastic, residue, etc.) some are not (or at least much less so), such as lambda calculus, or even propositional calculus.

Robert Mastragostino
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"Calculus of" usually refers to the methods of differential and integral calculus in conjunction with the topic at hand. It often excludes many forms of calculus, which is a common misconception.

afedder
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  • "Calculus" often excludes many forms of calculus? What does that mean? –  May 30 '12 at 18:10