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170 CHAPTER 7. CALCULUS
Proposition 52 If R is any ring, we can make R[x] into a differential ring by
defining r ′ = 0 ∀r ∈ R and x ′ = 1.
Definition 81 In any differential ring, an element α with α ′ = 0 is referred to
as a constant. For any ring R, we write R const for the constants of R, i.e.
R const = {r ∈ R | r ′ = 0}.
We will use C to stand for any appropriate field of constants.
We will revisit this definition in section 8.4.
Proposition 53 In any differential field
( ) ′ f
= f ′ g − fg ′
g g 2 . (7.1)
( )
The proof is by differentiating f = g f
g
to get
( ) ( ) ′
f f
f ′ = g ′ + g
g g
and solving for
(
f
g
) ′.
(7.1) is generally referred to as the quotient rule and can
be given a fairly tedious analytic proof in terms of ɛ/δ, but from our present
point of view it is an algebraic corollary of the product rule. It therefore follows
that K(x) can be made into a differential field.
Notation 22 (Fundamental Theorem of Calculus) (Indefinite) integration
is the inverse of differentiation, i.e.
∫
F = f ⇔ F ′ = f. (7.2)
The reader may be surprised to see a “Fundamental Theorem” reduced to the
status of a piece of notation, but from the present point of view, that is what
it is. We shall return to this point in section 8.3. The reader may also wonder
“where did dx go?”, but x is, from this point of view, merely that object such
that x ′ = 1, i.e. x = ∫ 1.
We should also note that we have no choice over the derivative of algebraic
expressions 1 .
Proposition 54 Let K be a differential field, and θ be algebraic over K with
p(θ) = 0 for some polynomial p = ∑ n
i=0 a iz i ∈ K[z]. Then K(θ) can be made
into a differential field in only one way: by defining
θ ′ = −
∑ n
i=0 a′ i θi
∑ n
i=0 ia . (7.3)
i−1
iθ
In particular, if the coefficients of p are all constants, so is θ.
The proof is by formal differentiation of p(θ) = 0.
1 It would be more normal to write “algebraic functions” instead of “algebraic expressions”,
but for reasons described in section 8.1 we reserve ‘function’ for specific mappings (e.g. C →
C), and the proposition is a property of differentiation applied to formulae.