See also differentiable function.
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Differentiation is the process that assigns to a function $f : X \to Y$ its derivative, sometimes denoted $d f$. The derivative is a function that, roughly speaking, assigns to each point $x \in X$ the linear transformation $d f_x$ that maps infinitesimal differences $y - x$ (for points $y$ infinitesimally close to $x$) to infinitesimal differences $f(y) - f(x)$.
Classically, if $f: \mathbb{R}^n \to \mathbb{R}$ is a function, the derivative $d f_x$ at a point $x \in \mathbb{R}^n$ is defined by a limiting process: it is the unique (when it exists) linear transformation $L: \mathbb{R}^n \to \mathbb{R}$ such that
or in other words such that
where $g(y)$ is $o({|y - x|})$ (see o-notation?). We say in that case that $f$ is differentiable at $x$, with derivative $d f_x = L$. When $n = 1$, such a linear transformation $L: \mathbb{R} \to \mathbb{R}$ is multiplication by a scalar $\lambda \in \mathbb{R}$. This is often denoted $f'(x)$, so that the derivative may be regarded as another function $f': \mathbb{R} \to \mathbb{R}$.
We say $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable if it is differentiable at every point $x$ of its domain. In that case, we get a global derivative function $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ which takes a pair $(x, h)$ to $d f_x(h)$. We can then ask whether this function is itself differentiable, i.e., whether the differentiation process can be iterated. The $n$-th derivative is the result of differentiating $n$ times (the $0$-th derivative of $f$ is just $f$). We say that $f: \mathbb{R}^n \to \mathbb{R}$ is infinitely differentiable (or more briefly, that $f$ is smooth) if it has an $n$-th derivative for each natural number $n$.
Notice that a differentiable function is necessarily continuous (indeed, it is locally a Lipschitz function). One denotes the class of functions having a continuous^{1} $n^{th}$ derivative by $C^n(\mathbb{R}^n)$. Thus continuous functions $f: \mathbb{R}^n \to \mathbb{R}$ form a filtration
and the intersection $C^\infty(\mathbb{R}^k) \coloneqq \bigcap_{n \geq 0} C^n(\mathbb{R}^k)$ coincides with the class of smooth functions.
Evidently differentiation gives a function $D: C^{n+1}(\mathbb{R}^k) \to C^n(\mathbb{R}^k \times \mathbb{R}^k)$. The convenient setting for developing the differential calculus is to go directly to the smooth case, where we have $D: C^\infty(\mathbb{R}^k) \to C^\infty(\mathbb{R}^k \times \mathbb{R}^k)$ without having to keep track of the degree of differentiability.
This development carries over more generally to smooth manifolds $X$, which are locally (smoothly) isomorphic to Euclidean spaces $\mathbb{R}^k$.
Differentiation in more conceptual terms is a functor $d : Diff \to Diff$ on the category Diff of smooth manifolds that sends
a manifold $X$ to its tangent bundle $T X$; recall that the points of $T X$ are ordered pairs $(x, v)$ where $x \in X$ and $v$ is a tangent vector at $x$, i.e., an (augmented) derivation $v: C^\infty(X) \to \mathbb{R}$ on the algebra of smooth functions, augmented by evaluation $ev_x: C^\infty(X) \to \mathbb{R}$ at $x$;
a smooth function $f : X \to Y$ to its differential $d f : T X \to T Y$:
if $\gamma : [-1,1] \to X$ is a path in $X$ that represents a vector $v \in T_x X$, then $(d f)(v) \in T_{f(x)} Y$ is the vector represented by the path $[-1,1] \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y$.
Equivalently, a smooth map $f: X \to Y$ induces an algebra map $f^\ast = - \circ f: C^\infty(Y) \to C^\infty(X)$, and hence by composition sends a derivation $v: C^\infty(X) \to \mathbb{R}$ to a derivation $v \circ f^\ast: C^\infty(Y) \to \mathbb{R}$ augmented by $ev_{f(x)}: C^\infty(Y) \to \mathbb{R}$). Concretely: $v \circ f^\ast(\phi) = v(\phi \circ f)$ for $\phi \in C^\infty(Y)$. Then the differential $d f: T X \to T Y$ is defined by $d f(x, v) = (f(x), v \circ f^\ast)$.
According to either of these descriptions, differentiation is manifestly functorial. In the English literature, this fact is known as the “chain rule”. In more traditional form:
In a context $H$ of synthetic differential geometry (SDG) the functor $d$ is simply the internal hom out of the infinitesimal interval $D$
where $D \hookrightarrow \mathbb{R}$ is the smooth locus defined by $x^2 = 0$. (Classically, i.e., in the topos $Set$, this locus consists of just a single point^{2}, but in toposes that model SDG it should be regarded as an infinitesimal neighborhood about a point.) This description as internally representable functor underlies the following classical fact:
Differentiation is a product-preserving functor.
For example, the product of two smooth functions $f, g: X \to \mathbb{R}$ is a composition
and the product rule for differentiation derives by applying the functor $d$ which results in the composite
which reduces to computing $d mult$.
We attempt to give an exposition of differentiation not via taking limits in the sense of analysis, but rather, by working with infinitesimal spaces as in synthetic differential geometry.
For that purpose we regard objects such as the real line $\mathbb{R}$ and more generally any smooth manifold as being objects of a well adapted smooth topos of smooth spaces. But in fact much less is required for most of the discussion. For the basic theory, it is sufficient to work in the canonical site of such a topos, namely, the category $SmoothLoc$ of smooth loci. This site is defined via a categorial equivalence with the opposite category of the category of smooth algebras:
Apart from the real line $\mathbb{R}$, the other object involved in our discussion of the basics of differentiation is the first order infinitesimal line segment $D$ (an infinitesimally thickened point), a subobject of $\mathbb{R}$. This subobject collects those points of $\mathbb{R}$ that square to zero under the canonical multiplicative structure of $\mathbb{R}$
Although this is just $\{0\}$ when interpreted in the usual context of Set, here we are to interpret this in the ambient smooth topos, or equivalently just in our site $SmoothLoc$ for it. That is to say, $D$ satisfies a pullback square in $SmoothLoc$:
Dually, the categorial equivalence $C^\infty$ from $SmoothLoc$ to the opposite category of smooth algebras carries the above pullback square to a pushout square of smooth algebras:
Here along the bottom and on the right are the ordinary smooth algebras of smooth functions on the smooth manifolds $\mathbb{R}^1 = \mathbb{R}$ and $\mathbb{R}^0 = *$, whereas $C^\infty(D)$ is new notation, suggestive and justified, for whatever the algebra of functions on the object $D$ is, where $D$ is defined by the fact that it has this algebra of functions. The bottom morphism sends a smooth function $f : \mathbb{R} \to \mathbb{R}$ to the function $f((-)^2) : \mathbb{R} \to \mathbb{R}$. If we write $x = Id : \mathbb{R} \to \mathbb{R}$ for the canonical coordinate function, then this sends $x$ to the function $x^2$.
Notice that the commutativity of this pushout square means that the image in $C^\infty(D)$ of the squared coordinate function $x^2 \in C^\infty(\mathbb{R})$ is zero. This is because the former is the image under the bottom morphism of the coordinate function $x$ and that maps under the right vertical morphism to $0 \in C^\infty(*) \simeq \mathbb{R}$.
In fact one can show that the pushout $C^\infty(D)$ is generated by the coordinate function $x \in C^\infty(\mathbb{R}))$ subject to the relation $x^2 = 0$: it is the ring of dual numbers
(Here in the second step we have used Hadamard's lemma.)
This computation of the smooth algebra $C^\infty(D)$ of smooth functions on $D$ expresses the intuitive idea that $D$ is such a small neighbourhood of $0$ in $\mathbb{R}$ that while the canonical coordinate function of $\mathbb{R}$ restricted to $D$ is non-vanishing (it would vanish only when restricted entirely to the point 0) it is “so small that its square is always 0”. Readers who like this intuitive statement should keep it in mind, as it is accurate and useful, whereas readers who do not like it could stick with the above precise definition, which unwinds this slogan in the internal logic of the ambient smooth topos that we keep alluding to.
In order to remind us about the infinitesimal nature of the generator $x$ in $C^\infty(D)$ we shall usually call this “$dx$” and write
We discussion of differential 1-forms are functions on the space of infinitesimal paths. For more comprehensive such discussion see (Stel 13).
Define the smooth locus $\mathbb{R}^{(\Delta^1_{inf})}$ as the product $\mathbb{R} \times D$ of the real line with the first order infinitesimal line segment. We think of this as the space of infinitesimal paths in $\mathbb{R}$ (see Spaces of infinitesimal k-simplicies ). We do so by thinking of a generalized element of $\mathbb{R}^{(\Delta^1_{inf})}$ in the category $SmoothLoc$, namely a pair $(x, \epsilon) \in \mathbb{R} \times D$, as the linear path in $\mathbb{R}$ stretching from $x$ to $x + \epsilon$ (both regarded as generalized elements of $\mathbb{R}$).
We say a smooth differential 1-form on $\mathbb{R}$ is a function (meaning: a morphism in $SmoothLoc$)
with the property that its restriction to $\mathbb{R}$ vanishes, hence so that the composite morphism
vanishes, where the first morphism is the inclusion of $\mathbb{R}$ as the constant infinitesimal paths in $\mathbb{R}$, hence $(Id, 0) : \mathbb{R} \to \mathbb{R} \times D$.
We call the coproduct of smooth algebras their tensor product (classically it is the smoothly completed algebraic tensor product). For each smooth loci $X$ and $Y$, we have
Using this property, we can dualize the above definition of a diferential 1-form. Namely, in the opposite category of smooth algebras, a 1-form on the line is a morphism
hence an assignment $x \mapsto g(x) + h(x) dx$ such that $g(x) = 0$. In other words, it is any element in $C^\infty(\mathrm{R}) \otimes \mathbb{R}[dx]/((dx)^2)$ that is proportional to the generator $dx$. We write $\omega = h dx$ with $h \in C^\infty(\mathbb{R})$.
We can now “synthetically” describe differentiation as the process of taking infinitesimal differences of functions.
Consider the two morphisms of smooth loci
and
where the first is the restriction of the canonical addition operation $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ of real numbers, and where the second is the projection on the first factor out of the product.
Given an element $f \in C^\infty(\mathbb{R})$, hence a smooth function $f : \mathbb{R} \to \mathbb{R}$, we can then form two functions on infinitesimal paths by precomposing with these two morphisms:
first we get
This is the function on infinitesimal paths in the line that only depends on the starting point of a path, not on the length of the path, and which sends that starting point to the value of $f$ on that point. Therefore by slight abuse of notation we just keep writing “$f$” for this function.
The second combination is
This is the function on infinitesimal paths that sends any path $(x) \to (x + \epsilon)$ to the value of the function on the endpoint $(x + \epsilon)$ of the paths: in terms of these generalized elements $\tilde f$ is the assignment
For instance consider the function $f : x \mapsto x^2$. Then
This is evident, if maybe still somewhat mysterious, in the internal language of the ambient smooth topos that we keep alluding to, but to make it entirely explicit and concrete notice that the functions here are dually morphisms of smooth algebras
and the equation
holds in $C^\infty(\mathbb{R}) \otimes (\mathbb{R} \oplus dx \mathbb{R})$ by the above discussion, where now $x \in C^\infty(\mathbb{R})$ and $dx \in C^\infty(D) = \mathbb{R} \oplus dx \mathbb{R}$ are the two canonical generators.
But notice that the expression $x^2 + 2 x dx$ is not a dfifferential 1-form by the above definition, equivalently the function $\tilde f$ does not vanish when restricted to constant infinitesimal paths.
To get such, we form the derivative or infinitesimal difference, the function
that acts on generalized elements of the space of infinitesimal paths in the line as
By construction on the right this now is something linear in the infinitesimal length $\epsilon$ of the path. Write $f' \in C^\infty(\mathbb{R})$ for the coefficient:
This function $f'$ is the differential of $f$. It is the function that gives the change of the function $f$ along a “unit infinitesimal path”.
Dually, we have a morphism
that sends
The 1-form on the right is the de Rham differential of $f$, usually written
Equivalently we may rephrase this differential $d f$ as a morphism of tangent bundles $d f \colon T \mathbb{R} \longrightarrow T \mathbb{R}$ as follows.
First, by the Kock-Lawvere axiom of SDG we have the standard fact that $T \mathbb{R} := \mathbb{R}^D \simeq \mathbb{R} \times \mathbb{R}$. Therefore given a function
this is by the internal hom-adjunction equivalently a function of the form
The condition on the original function makes this adjunct be a section of the tangent bundle of $\mathbb{R}$. This section is $x \mapsto f'(x)$, hence is the derivative of $f$ regarded as a tangent vector on $\mathbb{R}$. Moreover, since $\mathbb{R}$ is a microlinear space, this induces by rescaling a function
This is the differential of $f$ regarded as a map of tangent bundles.
Discussion of this history of the concept, with emphasis on its roots all the way back in Zeno's paradoxes of motion is in
Discussion of differential forms as functions on infinitesimal simplices is in
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
Without the continuity assumption on derivatives, the class of (once-)differentiable functions exhibits non-trivial descriptive set-theoretic complexity, as may be surmised by this paper by Kechris and Woodin. Thus the continuity assumption eliminates troublesome pathologies. ↩
According to John Bell, the fact that classically this is merely a point bears comparison to Bishop Berkeley’s objections to the differential calculus he put forth in The Analyst. ↩