Is there a branch of linguistics using Calculus as a mathematical tool? I mean, can we use differential or integrate in linguistic study?
Calculus can be a useful tool in quantitative linguistics. One simple example would be the deduction of the theoretical equation for the Piotrowski law modeling language change by Altmann et al.
The increase of new forms
p' is proportional to the product of the proportions
p of new and
1-p old forms:
p' ∝ p(1 - p). Introducing the factor of proportionality
b then yields the differential equation
p' = bp(1-p), which can be solved for
p as a function of time:
p = 1/(1 + a exp(-bt)). (Skipping a few details here, like Piotrowski's earlier fitting of a tangens hyperbolicus curve, which is compatible with the above, since
1/(1 + a exp(-bt)) = 1/2 tanh w(t-t₁) + 1/2, where
1/2 tanh w(t-t₁) was one of Piotrowski's original proposals. Also, this can be (and was) generalized for partial change, and there are other viable approaches besides the interactionistic model.)
If you are interested in quantitative linguistics, I'd suggest the works of Prof. Gabriel Altmann and Prof. Reinhard Köhler.
Definitely. This is a little obscure, probably to obscure to be of any real use, but there is a chapter from the book Aspects of Automatic Text Analysis, The Mathematics of Semantic Spaces (PDF), that models semantics as a metric space. It's general calculus on metric spaces, and this is probably the limit of mathematical abstraction in applied linguistics that I've seen. Unless you've been introduced to some very high level math, this material won't make any sense.
Other then that, I know acoustic phonetics has derivatives, integrals, limits, series, etc. For example, one way to describe a sound wave quantitatively is take it's Fourier transform which involves the evaluation of an integral. Then there's statistical analysis of corpora, where calculus isn't directly involved, but the formulas won't make any sense unless your familiar with the concepts of basic calculus. Things like weighting, infinite sums, neighbourhoods, continuity, finding volumes and lengths, optimization etc. are implicit in a lot of statistical measurements. Including vector calculus. A lot of linguistic models employ continuous sets, not just discrete sets, and require a metric or a topology to define the set as a mathematical space.
Lambda calculus is often used in formal semantics to express the meaning of a sentence compositionally as a series of successive function applications. On Barbara Partee's website there's a nice lesson on how lambda calculus (or, as some say, the lambda calculus) can be used to model English semantics. Lesson 7 introduces lambda calculus, but it might be worth looking at earlier lessons to see why it is useful.
For sure, lambda calculus is not "calculus" in the typical sense of differential calculus that we learn in our high school math classes, but it is a kind of calculus in that it involves an operator (lambda) that operates on a function and returns another function, binding a variable in the process. In that sense the lambda operator gives you something like the "derivative" of the original function.
There's a mathematical physicist who related graphs grammars and Feynman graphs. She comes up with the, perhaps unwarranted, conclusion that any context free graph grammar determines an insertion Lie algebra and commutative Hopf algebra.
For calculus to be useful, you first have to have some kind of continuous numeric function. This not being the case with grammar, calculus won't be illuminating for the study of unbounded dependencies or association relations: grammar is based on simple "does / doesn't" logic. However, it is in principle applicable to the study of linguistic behavior, e.g formant values of pronounced vowels, or perhaps a sociolinguistic study of the "Yeah-no" construction. In the former case I would say that the phenomenon itself (vocal tract resonances and physical movements) are fundamentally continuous. In the latter case, even though sociolinguistic changes reduce to binary do / don't choices, the relevant system of relations is so complex as to be intractable, if treated binarily, so numeric functions are the only practical way to discuss many social phenomena.
This question of using calculus or a calculus-like approach has been on my mind for some time. It's not calculus such that you'd use on a continuous data set, such as the set of real or imaginary numbers, like you'd find in a Real Analysis course, but rather as a philosophical concept.
So let's define: Calculus: An analysis of functions and how such functions change.
Underlying Assumption: There's an underlying function that can describe the mechanism of language change in the more general sense (versus language-specific).
So for example, given a phonological system in a particular semi-isolated linguistic community, the phonemes will change or shift with respect to time (this is the hypothesis, and by "semi-isolated" I mean that contact with other languages won't be a major factor to language-change). For a graphic on this, see the entry on "The Great Vowel Shift" in English, by Princeton: https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Great_Vowel_Shift.html
This would be the English-centric example to where a wider typological example would have to be substantiated as universal in all languages. Supposing The Great Vowel Shift shows us the underlying mechanism or function, then if we're given the Vowel system of the English language in the 1400's, we could predict the dipthongization over the next 150 - 200 years and further shifts thereafter.
The point here is to discover some underlying phenomenon that appears linguistically universal whereby we can model linguistic change and through the model we can make a prediction as to what a language will 'look' and 'sound' like via extrapolation of such a model.
My intent with providing my example is not necessarily to answer the question, but to add to the thought process that may help lead to an answer, or better, a linguistic calculus.