We first start with a tree structure on the set of real coefficient polynomials with domain and codomain , and call it the polynomial tree. We will say iff for some
Proposition 1. This gives us a partial order on .
Proof. since If and then and thus so And lastly if and then and Thus which implies and thus so
The fact that differentiation yields a unique function gives us the tree structure: the chain of derivatives (predecessors) is well ordered with minimal element By the Weierstrass approximation theorem we can attempt to extend this to the closure of the class of smooth functions with compact support. We extend our indexing set to the ordinal and define the th (and th) derivative linearly by the rule
Recall integration is a set valued operation–sending a function to its set of antiderivatives which all differ by some constant.
Example 2. Let We have
Hence for finite since the “last term” is and If we begin to integrate, we will add polynomials trailing after Continuing this denumerably we then obtain
where is the conumber of in the sense that it satisfies in the th term of the th partial sum. The denumerable integration converges if the sequence of constants chosen satisfies
Suppose the sequence satisfies this condition and let us assume the sum converges to some then we can write
In particular we thus have since
We thus have and We also define the degree of the functions which are not polynomials as Let be a polynomial with degree and have infinite degree. Then we clearly have that and have degree
Definition 3. A function is cyclic if for some nonzero finite number The number is called the differential order of and is denoted If a function is not cyclic we say (note in this definition ).
Example 4. The functions and are cyclic with orders and respectively. The only function with finite degree which is cyclic is with order
In fact being cyclic occurs iff
for at least some where and Thus we have
With this presentation it is clear that the order of the th term in the above term is if Otherwise the order is
Let us now define the equivalence relation where iff and are cyclic with same order and where such that Reflexivity is clear, and symmetry/transitivity involve simple arithmetic.
Example 5. Hence some equivalence classes are and for any polynomial
Proposition 6. gives a tree structure on which we call the calculus tree.
Proof. This is trivial considering the relation we modded out by was precisely loops in the poset–together with the fact that the poset has a unique minimal element.
Also see update.