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.