Different types of Calculi [closed]

We can define a derivation over a field $K$ to be a map $D:K\rightarrow K$ such that

$D(u+v) = Du + Dv$ and $D(uv) = uDv + vDu$,

for $u,v\in K$, and the constant subfield

$Const_D(K) = \lbrace u\in K : Du=0 \rbrace$

of $K$ with respect to $D$.

From this, we can get the familiar analytic properties of differentiation.

My question is: are there any other operators similar to this which we can define over a field, from which we then get some interesting relation to the analytic world.

pbs

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$\lambda$-calculus is a method devised by Church to define recursive/computable/effective functions. Maybe I shouldn't say "method". It is one way to define the notion of recursion, rather. In any case, it has little to nothing to do with the calculus you're talking of. – Malte Mar 29 2012 at 9:32
You could be more precise about what you mean by "Calculus". Do you mean derivations on the algebra $\mathcal{C}^{\infty}(\mathbb{R})$? Or do you mean an $\mathbb{R}$-algebra with some operators $D$, $I$ that somehow verify some axioms? Or...? – Qfwfq Mar 29 2012 at 13:34
Schubert calculus en.wikipedia.org/wiki/Schubert_calculus ... also not what you mean. In fact, to get a meaningful question, you will have to say what you mean by "calculus". – Gerald Edgar Mar 29 2012 at 15:05