nLab cogerm differential form

Cogerm differential forms

Cogerm differential forms


A cogerm differential form is a vast generalization of the usual exterior differential forms, which includes also absolute differential forms, as well as “higher” differential forms such as d 2xd^2x and a commutative differential operator (at least in the case of 1-forms).


Let XX be a set. Usually, XX will be a manifold or a generalized smooth space, but the definition does not require this.

By a curve in XX we mean a function c:(ϵ,ϵ)Xc:(-\epsilon,\epsilon)\to X for some real number ϵ>0\epsilon\gt 0. Two curves c:(ϵ,ϵ)Xc:(-\epsilon,\epsilon)\to X and c:(ϵ,ϵ)Xc':(-\epsilon',\epsilon')\to X have the same germ (at 0) if there exists ϵmin(ϵ,ϵ)\epsilon'' \le min(\epsilon,\epsilon') such that cc and cc' agree when restricted to (ϵ,ϵ)(-\epsilon'',\epsilon''). Let CXC X denote the set of germs of curves in XX, i.e. the quotient of the set of curves in XX by the equivalence relation “has the same germ as”.


A (partial) cogerm differential 1-form on XX is a partial function ω:CX\omega : C X \to \mathbb{R}. We write the action of ω\omega on a curve cc (or germ thereof) as ω|c\langle \omega | c \rangle.


  • If f:Xf: X\to \mathbb{R} is a function, then it defines a cogerm differential 1-form, also denoted ff, by evaluation at 00:

    f|c=(fc)(0). \langle f | c \rangle = (f\circ c)(0).

    This is defined on all germs.

  • We also have the differential of ff, denoted df\mathrm{d}f, defined by

    df|c=(fc)(0) \langle \mathrm{d}f | c \rangle = (f\circ c)'(0)

    which is defined on all germs cc having the property that fcf\circ c is differentiable at 00. In particular, if ff is smooth (for XX a smooth space), then df\mathrm{d}f is defined on all smooth germs.

    When XX is a differentiable manifold, df|c \langle \mathrm{d}f | c \rangle depends only on the tangent vector of cc at 00.

  • More generally, any real-valued function on the tangent bundle of XX can be regarded as a cogerm differential 1-form; this includes in particular all exterior differential 1-forms.

  • We also have the second differential d 2f\mathrm{d}^2f, defined by

    d 2f|c=(fc)(0), \langle \mathrm{d}^2f | c \rangle = (f\circ c)''(0),

    which depends only on the 2-jet of cc at 00. We can similarly consider higher differentials which depend on higher jets. A cogerm differential 1-form which depends only on the jet of cc may be called a cojet differential 1-form.

  • For an example of a cogerm differential form that is not a cojet differential form, let ω|c\langle{\omega|c}\rangle be 00 or 11 according as cc is or is not analytic (say for XX the real line). We do not know any more interesting examples.

  • We can also apply arbitrary real functions: if ϕ: n\phi:\mathbb{R}^n\to \mathbb{R} is a function (or even a partial function) and ω 1,,ω n\omega_1,\dots,\omega_n are cogerm differential 1-forms, then we have ϕ(ω 1,,ω n)\phi(\omega_1,\dots,\omega_n) defined by

    ϕ(ω 1,,ω n)|c=ϕ(ω 1|c,,ω n|c) \langle \phi(\omega_1,\dots,\omega_n) | c \rangle = \phi(\langle\omega_1|c\rangle,\dots,\langle\omega_n|c\rangle)
  • For instance, any ω\omega has an absolute value |ω|{|\omega|}.

  • And if X= 2X=\mathbb{R}^2, then we have the “length element” đs=dx 2+dy 2đ s = \sqrt{\mathrm{d}x^2 + \mathrm{d}y^2}.

  • Generalizing đsđs, any absolute differential 1-form can be regarded as a cogerm differential 1-form.

  • Any symmetric bilinear form on tangent vectors can also be regarded as a quadratic covector form and therefore a quadratic cogerm form. In particular, this applies to the metric gg on any (pseudo)-Riemannian manifold. We then have g=đs 2g = đs^2 in the algebra of cogerm forms on a Riemannian manifold.

  • For any function ff, we have a cogerm differential form Δf\Delta f, defined by f(x+dx)f(x)f(x+\mathrm{d}x)-f(x). When ff is the Heaviside function, Δf\Delta f is a candidate for the Dirac delta function.

  • A somewhat better way to represent the Dirac delta function as a cogerm differential form is

    δ={1|dx| x=0anddx0 0 otherwise. \delta = \begin{cases} \frac{1}{|\mathrm{d}x|} &\qquad x=0 \;and\; \mathrm{d}x\neq 0\\ 0 &\qquad otherwise. \end{cases}

    In the section on integration below we will see why this deserves the name “delta function”.

The cogerm differential

For any curve cc and real number hh, let c hc_h be the curve defined by c h(t)=c(t+h)c_h(t) = c(t+h). If ω\omega is any cogerm differential 1-form, then we define its differential dω\mathrm{d}\omega by

dω|c=lim h0ω|c hω|ch. \langle \mathrm{d}\omega | c \rangle = \lim_{h\to 0} \frac{\langle \omega | c_h \rangle - \langle \omega | c\rangle}{h}.

Note that while the fraction on the right-hand side only makes sense for a specific representative curve cc and a sufficiently small hh, the limit depends only on the germ of cc.

Of course, in general, dω\mathrm{d}\omega may not be defined on as many curves as ω\omega is.


If XX is a smooth space, so that we have a notion of smooth curve, then we say that a cogerm differential 1-form ω\omega is smooth if

  1. ω\omega is defined on all germs of smooth curves, and
  2. dω\mathrm{d}\omega is smooth (coinductively).

For example, if f:Xf:X\to \mathbb{R} is a smooth function, then it is also a smooth cogerm differential 1-form, and its differentials d nf\mathrm{d}^n f are those mentioned above. By stopping the coinduction at an appropriate point and requiring only continuity, we obtain an analogous definition of kk-times continuously differentiable function C kC^k; for example, ω\omega is C 2C^2 if ω\omega, dω\mathrm{d}\omega, and d 2ω\mathrm{d}^2\omega are defined on all smooth curves and the last of these is continuous. (But the obvious definition of differentiable function without continuity is too weak, even if we require dω\mathrm{d}\omega to be defined on all differentiable germs, as the classic example of xy 2/(x 2+y 2)x y^2/(x^2 + y^2) shows; extended continuously to the origin, this and its differential are defined on all smooth curves, although the differential is not continuous, but this function is not differentiable, which we know because its differential is not linear as a covector form.)

Note that the cogerm differential d\mathrm{d} is not the same as the exterior differential, except when applied to 00-forms. In particular, d 20\mathrm{d}^2 \neq 0.

We have the expected multivariable chain rule:


If ϕ: n\phi:\mathbb{R}^n\to \mathbb{R} is a differentiable function and ω 1,,ω n\omega_1,\dots,\omega_n are cogerm differential 1-forms, then

d(ϕ(ω 1,,ω n))= 1ϕ(ω 1,,ω n)dω 1++ nϕ(ω 1,,ω n)dω n \mathrm{d}(\phi(\omega_1,\dots,\omega_n)) = \partial_1\phi(\omega_1,\dots,\omega_n) \mathrm{d}\omega_1 + \cdots + \partial_n\phi(\omega_1,\dots,\omega_n) \mathrm{d}\omega_n

where iϕ\partial_i\phi denotes the partial derivative with respect to the i thi^{th} variable.

In particular, taking ϕ(x,y)=xy\phi(x,y) = x y, we have the product rule:

d(ωη)=ωdη+ηdω. \mathrm{d}(\omega\eta) = \omega \mathrm{d}\eta + \eta \mathrm{d}\omega.

This also enables us to calculate the iterated cogerm differentials of functions. If x:Xx:X\to \mathbb{R} is a “coordinate” and f:f:\mathbb{R}\to \mathbb{R} is a function, so that f(x):Xf(x) : X\to \mathbb{R} is a function of that coordinate, then by the theorem we have

df(x)=f(x)dx.\mathrm{d}f(x) = f'(x) \mathrm{d}x.

Thus, by the product rule, we have

d 2f(x)=f(x)dx 2+f(x)d 2x \mathrm{d}^2 f(x) = f''(x) \mathrm{d}x^2 + f'(x) \mathrm{d}^2x

and so on. Note that the first formula justifies the common notation df(x)dx\frac{\mathrm{d}f(x)}{\mathrm{d}x} for the derivative f(x)f'(x), while the second almost justifies the common notation d 2f(x)dx 2\frac{\mathrm{d}^2f(x)}{\mathrm{d}x^2} for the second derivative f(x)f''(x) — it would be correct only if d 2x=0\mathrm{d}^2x=0, which is not generally the case. Instead it would be better to write f(x)= 2f(x)x 2f''(x) = \frac{\partial^2f(x)}{\partial x^2}, indicating that f(x)f''(x) is the coefficient of dx 2\mathrm{d}x^2 in a canonical expansion of d 2f(x)\mathrm{d}^2 f(x).


Let c:[a,b]Xc:[a,b] \to X be a curve and ω\omega a cogerm differential 1-form; we would like to integrate ω\omega over cc. There are at least two possible definitions.

Naive integration

The naive integral of ω\omega over cc is defined to be

cω= t=a bω|c tdt.\int_c \omega = \int_{t=a}^b \langle \omega | c_t \rangle \mathrm{d}t.

if this exists. If ω\omega is an exterior differential 11-form or an absolute differential 11-form, then this agrees with its usual line integral over cc.

Genuine integration

The (genuine) integral of ω\omega over cc is defined as follows. Given a tagged partition a=t 0<t 1<t n1<t n=ba = t_0 \lt t_1 \cdots \lt t_{n-1} \lt t_n = b with tags t i *[t i1,t i]t^*_i \in [t_{i-1},t_i], we define the corresponding Riemann sum? to be

i=1 nω|Δt ic t i * \sum_{i=1}^n \langle \omega | \Delta t_i \cdot c_{t^*_i} \rangle

Here Δt i=t it i1\Delta t_i = t_i - t_{i-1}, and for a curve cc and a number hh the curve hch\cdot c is defined by (hc)(t)=c(ht)(h\cdot c)(t) = c(h t). Now we take the limit as the tagged partitions shrink.

It is convenient to do this in the manner of the Henstock integral. That is, we consider gauges δ:[a,b] >0\delta : [a,b] \to \mathbb{R}_{\gt 0} and say that a tagged partition is δ\delta-fine if [t i1,t i](t i *δ(t i *),t i *+δ(t i *))[t_{i-1},t_i] \subset (t^*_i - \delta(t^*_i), t^*_i + \delta(t^*_i)) for all ii. Then we say that a number II is the integral cω\int_c \omega if for all ϵ>0\epsilon\gt 0 there exists a gauge δ\delta such that the Riemann sum of any δ\delta-fine tagged partition is within ϵ\epsilon of II.

The genuine integral also agrees with the usual integral for exterior 1-forms, and probably also for absolute differential forms. However, in other cases it disagrees with the naive integral. In particular, it “detects only the degree-1 part” of a form, in a way that we can make precise as follows.

Integration of negligible forms

Let us say that ω\omega is o(dx)o(dx) if lim h0ω|hch=0\lim_{h\to 0} \frac{\langle \omega | h\cdot c \rangle}{h} = 0 for any curve cc. Some examples of forms that are o(dx)o(dx) are dx 2\mathrm{d}x^2 and d 2x\mathrm{d}^2x.


If ω\omega is o(dx)o(dx), then cω=0\int_c \omega = 0 for any curve cc.


Suppose ϵ>0\epsilon\gt 0; then for any t[a,b]t\in [a,b] there is a δ(t)\delta(t) such that ω|hc t<ϵbah{\langle \omega {|} h \cdot c_t\rangle} \lt \frac{\epsilon}{b-a} h for any 0<h<δ(t)0\lt h\lt \delta(t). This defines a gauge δ\delta on [a,b][a,b]. Now suppose we have a δ\delta-fine tagged partition a=x 0<<x n=ba = x_0 \lt \cdots \lt x_n = b with tags t i[x i1,x i]t_i \in [x_{i-1},x_i], so that Δx i=x ix i1<δ(t i)\Delta x_i = x_i - x_{i-1} \lt \delta(t_i). Then the corresponding Riemann sum is, by definition, iω|Δx ic t i\sum_{i} \langle \omega {|} \Delta x_i \cdot c_{t_i}\rangle . Since Δx i<δ(t i)\Delta x_i \lt \delta(t_i), each ω|Δx ic t i<ϵbaΔx i\langle \omega {|} \Delta x_i \cdot c_{t_i}\rangle \lt \frac{\epsilon}{b-a} \Delta x_i. Thus, when we sum them up, we get something less than ϵba iΔx i=ϵba(ba)=ϵ\frac{\epsilon}{b-a} \sum_i \Delta x_i = \frac{\epsilon}{b-a} (b-a) = \epsilon. Thus, for any ϵ\epsilon there is a gauge δ\delta such that the Riemann sum over any δ\delta-fine tagged partition is <ϵ\lt\epsilon; so the integral is zero.

Thus, for instance, cd 2x=0\int_c \mathrm{d}^2 x = 0 according to the genuine integral, while according to the naive integral it would be (by the fundamental theorem of calculus) dx|c bdx|c a=(xc)(b)(xc)(a)\langle \mathrm{d}x | c_b \rangle - \langle \mathrm{d}x|c_a \rangle = (x\circ c)'(b) - (x\circ c)'(a).

The answer 00 given by the genuine integral is preferable if we regard dx\mathrm{d}x as a “first-order change in xx”, so that dx 2\mathrm{d}x^2 or d 2x\mathrm{d}^2x are “second-order” quantities and hence ought to be negligible. For instance, we usually compute the area under a curve y=f(x)y=f(x) between x=ax=a and x=bx=b by dividing the interval [a,b][a,b] into subintervals of width Δx\Delta x and approximating the area for each subinterval using a rectangle, obtaining f(x)Δxf(x) \Delta x. This leads us to integrate the differential form f(x)dxf(x) \mathrm{d}x.

However, a better approximation would be to replace the rectangle with a trapezoid, having an area of 12(f(x+Δx)+f(x))Δx\frac{1}{2}(f(x+\Delta x) + f(x))\Delta x. At least if ff is differentiable, we can approximate f(x+Δx)f(x+\Delta x) by f(x)+f(x)Δxf(x) + f'(x) \Delta x, and thus approximate the area by

12(f(x)+f(x)Δx+f(x))Δx=f(x)Δx+12f(x)(Δx) 2. \frac{1}{2}(f(x) + f'(x)\Delta x + f(x))\Delta x = f(x) \Delta x + \frac{1}{2} f'(x) (\Delta x)^2.

This would lead us to integrate the form f(x)dx+12f(x)dx 2f(x) \mathrm{d}x + \frac{1}{2} f'(x) \mathrm{d}x^2. Since this is based on a better approximation, we should expect it to give at least as good an answer. And with the genuine integral it does give the same answer, since the additional term 12f(x)dx 2\frac{1}{2} f'(x) \mathrm{d}x^2 is o(dx)o(dx) and hence has vanishing integral. But with the naive integral, it does not.

The fundamental theorem of calculus

The naive integral satisfies a fundamental theorem of calculus for all cogerm forms: for any ω\omega we have

cdω=ω|c bω|c a.\int_c \mathrm{d}\omega = \langle \omega| c_b\rangle - \langle \omega | c_a \rangle.

This follows directly from the definition of the naive integral and the cogerm differential d\mathrm{d}.

The genuine integral does not satisfy as general a fundamental theorem of calculus as the naive integral. It does, however, satisfy FTC for differentials of functions:

cdf=f(c(b))f(c(a)) \int_c \mathrm{d}f = f(c(b)) - f(c(a))

This can be proven in exactly the same way as the usual FTC for (line) integrals.

The restriction of FTC to differentials of functions is fairly natural if we recognize that FTC is a special case of the generalized Stokes' theorem, which is about exterior differentials of forms, not the commutative cogerm differential. It just so happens that if ff is a function, then the exterior differential of ff regarded as a 0-form agrees with the cogerm differential of ff regarded as a cogerm 1-form.

Existence of integrals

The existence of naive integrals is fairly easy to verify, since they are just defined in terms of ordinary 1-variable integrals. But the existence of genuine integrals is rather less obvious. Here we prove that they exist for at least one reasonably general class of forms.

Recall that a 1-cojet differential 1-form is a cogerm 1-form that depends only on the 1-jet of a curve, i.e. its value and its tangent vector. A 1-cojet form is equivalently just a function (not necessarily linear) on the tangent bundle TXT X; we may write it as ω(x,dx)\omega(\mathbf{x},\mathrm{d}\mathbf{x}), where x\mathbf{x} is a point of XX (with coordinates xx, yy, zz, …) and dxT xX\mathrm{d}\mathbf{x}\in T_{\mathbf{x}}X a tangent vector at x\mathbf{x} (with coordinates dx\mathrm{d}x, dy\mathrm{d}y, dz\mathrm{d}z, …).


Suppose ω\omega is a 1-cojet differential form and that

  1. ω(x,0)=0\omega(\mathbf{x},0) = 0 for all x\mathbf{x}.

  2. For each xX\mathbf{x}\in X and dxT xX\mathrm{d}\mathbf{x}\in T_{\mathbf{x}}X the function λh.ω(x,hdx):\lambda h.\omega(\mathbf{x},h \, \mathrm{d}\mathbf{x}):\mathbb{R} \to \mathbb{R} is differentiable from the right at h=0h=0, and the derivative of the above function is continuous as a function on TXT X.

Then the genuine integral of ω\omega over any differentiable curve exists.


The Riemann sums in the definition of the genuine integral simplify in the case of a 1-cojet form to

n=1 ω(c(t i *),Δt ic(t i *)). \sum_{n=1}^\infty \omega(c(t^*_i), \Delta t_i \cdot c'(t^*_i)).

Note that Δt i=t it i1\Delta t_i = t_i - t_{i-1} is always positive. By the assumptions, for any positive hh we can write

ω(x,hdx)=f(x,dx)h+g(x,dx,h)h\omega(\mathbf{x}, h\cdot \mathrm{d}\mathbf{x}) = f(\mathbf{x}, \mathrm{d}\mathbf{x})\, h + g(\mathbf{x}, \mathrm{d}\mathbf{x},h)\,h

where ff is continuous and lim h0g(x,dx,h)=0\lim_{h\to 0} g(\mathbf{x},\mathrm{d}\mathbf{x},h) = 0. Therefore the Riemann sum is

n=1 f(c(t i *),c(t i *))Δt i+g(c(t i *),c(t i *),Δt i)Δt i. \sum_{n=1}^\infty f(c(t^*_i), c'(t^*_i))\,\Delta t_i + g(c(t^*_i), c'(t^*_i),\Delta t_i)\,\Delta t_i.

Now, essentially the same argument as in the proof that o(dx)o(dx) forms have zero integral shows that the second term can be neglected; thus we may as well consider only

n=1 f(c(t i *),c(t i *))Δt i \sum_{n=1}^\infty f(c(t^*_i), c'(t^*_i))\,\Delta t_i

However, this is just an ordinary Riemann sum for computing the integral

t=a bf(c(t),c(t))dt. \int_{t=a}^b f(c(t),c'(t)) \, \mathrm{d}t.

Since ff is assumed continuous, this integral exists.

The hypotheses of this theorem are a bit restrictive, but they include both the usual exterior 1-forms and absolute differential forms such as |dx|{|\mathrm{d}x|} and đl=dx 2+dy 2&#273;l = \sqrt{\mathrm{d}x^2+\mathrm{d}y^2}. The latter are not differentiable at dx=0\mathrm{d}\mathbf{x}=0, but they do have one-sided derivatives along any line approaching the origin. It is also necessary to exclude forms such as the following:

  • A function on XX, regarded as a cogerm differential form. If such a function is nonzero on any region, then its genuine integral will diverge. These are excluded by requirement (1).

  • Forms such as dx\sqrt{\mathrm{d}x}, whose genuine integral also diverges. These are excluded by requirement (2).

There are, however, some cogerm forms that are not 1-cojet forms, nor are they o(dx)o(dx), yet their geniune integrals exist. An example is d 2x\sqrt{\mathrm{d}^2x}, whose genuine integral over a curve c:[a,b]Xc:[a,b]\to X is essentially t=a b(xc)(t)dt\int_{t=a}^b \sqrt{(x\circ c)''(t)} \,\mathrm{d}t. There is probably a generalization of the above existence theorem to kk-cojet forms, involving somewhat more complicated vanishing and differentiability conditions.

Parametrization invariance

Since both naive and genuine integrals agree with ordinary line integrals when restricted to exterior 1-forms, such integrals are invariant under orientation-preserving reparametrization. However, for general cogerm 1-forms, neither integral is so invariant. Nevertheless, we may ask for conditions under which they are.

The naive integral is very much not invariant under reparametrization. This is most obvious when ω\omega is just a function, in which case its integral is just its ordinary integral. Similarly, a naive integral like cd 2x\int_c \mathrm{d}^2 x, which equals (xc)(b)(xc)(a)(x\circ c)'(b) - (x\circ c)'(a) by FTC, is not invariant under reparametrization since it involves the derivative of the coordinate function of cc.

The genuine integral is somewhat more invariant under reparametrization. For instance,

  • Since the genuine integral cω=0\int_c \omega = 0 whenever ω\omega is o(dx)o(dx), independently of whatever cc might be, any such integral is parametrization-independent.

  • The genuine integral of any form along any curve is invariant under orientation-preserving affine reparametrization. This follows fairly directly from its definition.

There are forms whose genuine integral exists, but is not parametrization-invariant, such as d 2x\sqrt{\mathrm{d}^2x}. However, we can isolate a useful class of forms, including both exterior forms and absolute forms, whose genuine integral is so invariant. This is most easily done by reformulating the integral, as follows.

Affine integration

For 1-cojet differential 1-forms satisfying a weak sublinearity condition, there is an equivalent definition of the genuine integral that is more obviously parametrization-invariant.

A 1-cojet form can be considered as a function on tangent vectors (x,v)(x,v) where vT xXv\in T_x X. Let us say that such a form ω\omega is tangent-Lipschitz if for each xXx\in X there is a constant L>0L\gt 0 such that

|ω(x,v)ω(x,w)|<Lvw {|\omega(x,v) - \omega(x,w)|} \lt L \,\Vert v-w\Vert

for all v,wVv,w\in V. Here we are using a norm on the tangent space T xXT_x X. Since this vector space is finite-dimensional, the notion of tangent-Lipschitz is independent of the norm chosen (although the numerical value of LL will vary with the norm).

Note that this says nothing at all about the continuity of ω\omega as a function of xx. In particular, if each ω(x,)\omega(x,-) is separately either linear or absolute, then ω\omega is tangent-Lipschitz. Thus, tangent-Lipschitz is a weak replacement for linearity in the tangent variable.

To define our new version of the integral, suppose first that XX is a finite-dimensional real affine space, regarded as a smooth manifold. Then we can canonically identify all its tangent spaces with the same vector space VV.

Let c:[a,b]Xc:[a,b]\to X be a curve in such an affine space XX, and ω\omega a 1-cojet differential 1-form on some open subset UXU\subseteq X containing the image of cc. Then we can regard ω\omega as a function U×VU\times V\to \mathbb{R}. Now for any tagged partition a=t 0<t 1<t n1<t n=ba = t_0 \lt t_1 \cdots \lt t_{n-1} \lt t_n = b with tags t i *[t i1,t i]t^*_i \in [t_{i-1},t_i], we define the corresponding Riemann sum? to be

i=1 nω(c(t i *),Δc i) \sum_{i=1}^n \omega(c(t^*_i), \Delta c_i)

where Δc i=c(t i)c(t i1)\Delta c_i = c(t_i) - c(t_{i-1}), the subtraction being the usual way to subtract points in an affine space and obtain a vector. We then take the limit of such Riemann sums in the Henstock way, as before; let us call this the affine integral.

Note that this Riemann sum manifestly depends only on the points c(t i)c(t_i) and c(t i *)c(t^*_i) (and the order in which they occur). Thus, any increasing bijection ϕ:[a,b][a,b]\phi:[a',b'] \to [a,b] induces a bijection between tagged partitions which respects their Riemann sums. It also maps intervals to intervals, so any gauge on one interval induces one on the other interval. Thus the limits also coincide (in the strong sense that one exists if and only if the other does, and in that case they are equal), and so the affine integral is completely parametrization-independent.

We now show that the affine integral agrees with the genuine integral.


If ω\omega is a tangent-Lipschitz 1-cojet form on an open subset of an affine space XX, then its genuine and affine integrals over any differentiable curve agree, in the strong sense that each exists if and only if the other does and in that case they are equal.


Let c:[a,b]Xc:[a,b]\to X be differentiable. Then by definition of differentiability, for any t 1<t *<t 2t_1 \lt t^*\lt t_2 in [a,b][a,b] we can write

c(t 2)c(t 1)c(t *)(t 2t 1)=g(t *,t 1,t 2)(t 2t 1) c(t_2) - c(t_1) - c'(t^*)(t_2-t_1) = g(t^*,t_1,t_2)(t_2-t_1)

where g(t *,t 1,t 2)0g(t^*,t_1,t_2) \to 0 as t 2t 10t_2-t_1 \to 0. Thus, since ω\omega is tangent-Lipschitz, for any tagged partition we have

|ω(c(t i *),Δc i)ω(c(t i *),Δt ic(t *))|<Lg(t *,t 1,t 2)Δt i. {|\omega(c(t^*_i), \Delta c_i) - \omega(c(t^*_i),\Delta t_i \cdot c'(t^*))|} \lt L \Vert g(t^*,t_1,t_2) \Vert \cdot \Delta t_i.

Since g(t *,t 1,t 2)0g(t^*,t_1,t_2) \to 0 as t 2t 10t_2-t_1 \to 0, we can choose gauges to make this difference vanish in the limit, just as we did for integrating o(dx)o(dx) forms. But ω(c(t i *),Δc i)\omega(c(t^*_i), \Delta c_i) and ω(c(t i *),Δt ic(t *))\omega(c(t^*_i),\Delta t_i \cdot c'(t^*)) are exactly the terms in the Riemann sums for the affine and genuine integrals.

Finally, a similar argument shows that the affine integral can be extended to all finite-dimensional smooth manifolds by local charts.


Let UXU\subseteq X and VYV\subseteq Y be open subsets of affine spaces and ϕ:UV\phi:U\to V be differentiable. Let ω\omega be a tangent-Lipschitz 1-cojet form on VV and c:[0,1]Uc:[0,1]\to U a differentiable curve. Then for the affine integrals we have

cϕ *ω= ϕcω. \int_c \phi^*\omega = \int_{\phi c} \omega.

By definition, ϕ *ω(x,v)=ω(ϕ(x),d xϕ(v))\phi^*\omega(x,v) = \omega(\phi(x),d_x\phi(v)). Thus, the LHS is a limit of Riemann sums of the form

iω(ϕ(c(t i *)),d c(t i *)ϕ(Δc i)). \sum_i \omega(\phi(c(t^*_i)),d_{c(t^*_i)}\phi(\Delta c_i)).

while the RHS is a limit of sums of the form

iω(ϕ(c(t i *)),Δ(ϕc) i). \sum_i \omega(\phi(c(t^*_i)),\Delta (\phi c)_i).

Since ϕ\phi is differentiable, for any point c(t i *)c(t^*_i) we have

Δ(ϕc) i=ϕ(c(t i))ϕ(c(t i1))=d c(t i *)ϕ(Δc i)+E(c(t i *),Δc i)Δc i \Delta (\phi c)_i = \phi(c(t_{i})) - \phi(c(t_{i-1})) = d_{c(t^*_i)}\phi(\Delta c_i) + E(c(t^*_i),\Delta c_i)\cdot {\Vert \Delta c_i \Vert }

where E(c(t i *),Δc i)0E(c(t^*_i),\Delta c_i)\to 0 as t it i10t_i - t_{i-1} \to 0. Since ω\omega is tangent-Lipschitz, this gives

ω(ϕ(c(t i *)),Δ(ϕc) i)ω(ϕ(c(t i *)),d c(t i *)ϕ(Δc i))<LE(c(t i *),Δc i)Δc i. \Big\Vert \omega(\phi(c(t^*_i)),\Delta (\phi c)_i) - \omega(\phi(c(t^*_i)),d_{c(t^*_i)}\phi(\Delta c_i)) \Big\Vert \lt L \cdot E(c(t^*_i),\Delta c_i) \cdot {\Vert \Delta c_i \Vert }.

Now LL and EE depend on the point chosen, but nevertheless, we can choose gauges as before to make this term vanish in the limit. Thus, the two integrals agree.

Stieltjes integrals, δ\delta-functions, and distributions

An alternative approach to proving FTC would be to observe that essentially by definition of df\mathrm{d}f, the form

f(x+dx)f(x)df f(x+\mathrm{d}x) - f(x) - \mathrm{d}f

is o(dx)o(dx). Therefore, cdf= c(f(x+dx)f(x))\int_c \mathrm{d}f = \int_c (f(x+\mathrm{d}x)-f(x)) if either exists. If the latter integral can be calculated using only partitions tagged by their left endpoint, then it is obviously f(c(b))f(c(a))f(c(b))-f(c(a)) — but it is not clear that such partitions suffice.

In the list of examples above, we denoted f(x+dx)f(x)f(x+\mathrm{d}x)-f(x) by Δf\Delta f. More generally, we might expect that integration of Δf\Delta f (perhaps multiplied by another function gg) is a sort of Stieltjes integration?. However, it is again not clear whether left endpoints suffice.

We can, at least, show that the second definition of the Dirac delta function has its expected properties. Recall that this was

δ={1|dx| x=0anddx0 0 otherwise. \delta = \begin{cases} \frac{1}{|\mathrm{d}x|} &\qquad x=0 \;and\; \mathrm{d}x\neq 0\\ 0 &\qquad otherwise. \end{cases}

For any function f:f:\mathbb{R}\to \mathbb{R}, if a0ba\le 0 \le b then we have a bf(x)δdx=f(0)\int_a^b f(x)\, \delta \,\mathrm{d}x = f(0) for the affine integral.


We define a gauge, denoted η\eta (since the letter δ\delta is already in use) as follows:

η(x)={ x=0 |x| x0.\eta(x) = \begin{cases} \infty &\qquad x=0\\ {|x|} &\qquad x\neq 0. \end{cases}

Then in any η\eta-fine tagged partition, 00 must be the tag of the subinterval containing it. Therefore, the Riemann sum of f(x)δdxf(x)\, \delta \,\mathrm{d}x over any such tagged partition is simply

f(0)1|dx|dx f(0) \frac{1}{|\mathrm{d}x|} \mathrm{d}x

where dx\mathrm{d}x is the width of the subinterval containing 00. But this is positive, so the Riemann sum is simply f(0)f(0). Thus, the integral equals f(0)f(0).

One might instead define δ\delta to be 11 if x=0x=0, so that we would integrate f(x)δf(x)\, \delta rather than f(x)δdxf(x) \,\delta \,\mathrm{d}x. Our choice matches more closely the common informal notation “f(x)δ(x)dx\int f(x)\, \delta(x)\,\mathrm{d}x”, although of course here δ\delta is a function of dx\mathrm{d}x as well as xx. Our choice also gives δ\delta the usual scaling and precomposition properties, usually written as

δ(αx)=δ(x)|α| \delta(\alpha x) = \frac{\delta(x)}{|\alpha|}
δ(g(x))= g(c)=0δ(xc)|g(c)| \delta(g(x)) = \sum_{g(c)=0} \frac{\delta(x-c)}{|g'(c)|}

In our notation, instead of δ(g(x))\delta(g(x)) we would write δ(g(x),dg)\delta(g(x),\mathrm{d}g), which is how we precompose a differential form with a function; the above formulas then follow from dg=g(x)dx\mathrm{d}g = g'(x)\,\mathrm{d}x.

Note that δ\delta-functions are usually defined as measures or distributions, which are integrated over unoriented regions. Since our integrals are oriented, we have to specify that [a,b][a,b] is traversed left-to-right (as in the notation a b\int_a^b) in order to get f(0)f(0) as the answer; if we integrated b a\int_b^a instead we would get f(0)-f(0).

We can, of course, avoid the minus sign by integrating against δ|dx|\delta\,|\mathrm{d}x| instead of δdx\delta \,\mathrm{d}x. (We could also omit the absolute value in the definition of δ\delta, but this would break the usual scaling properties.)

Since we have successfully represented the δ\delta-distribution as a cogerm differential form, one may wonder whether other distributions can also be so represented. It is unclear to me whether this is possible. For instance, if ϕ ϵ\phi_\epsilon are test functions converging to a distribution θ\theta as ϵ0\epsilon\to 0, we could consider a differential form such as θ(x,dx)=ϕ dx(x)\theta(x,\mathrm{d}x) = \phi_{\mathrm{d}x}(x), but it’s not clear whether this would have the right behavior under integration.

Higher forms

One can define a cogerm differential kk-form to be a function on germs of kk-dimensional hypersurfaces in XX. These include in particular the usual exterior differential kk-forms as well as absolute differential kk-forms. And they can be integrated over kk-dimensional hypersurfaces, by a similar formula as above.

There is a sort of “exterior differential” acting from cogerm kk-forms to cogerm (k+1)(k+1)-forms, which we denote by (d)(\mathrm{d}\wedge -) to avoid confusion with the commutative cogerm differential d\mathrm{d}. When k=1k=1, the definition is

dω|c=lim A01A c(A)ω. \langle \mathrm{d}\wedge \omega {|} c \rangle = \lim_{A \to 0} \frac{1}{\Vert A \Vert} \oint_{c(\partial A)} \omega.

Here the limit is over some neighborhoods AA of 0 20\in\mathbb{R}^2 whose areas A\Vert A \Vert shrink to zero, and the integral over the boundary is defined as above. The definition for general kk is similar.

This limit can only exist if the integral c(A)ω\oint_{c(\partial A)} \omega is invariant under oriented reparametrization of A\partial A. Moreover, if ω\omega is an absolute form, then usually there will not be enough cancellation to make the limit above finite (it would be analogous to defining a 1-dimensional “derivative” as lim h0f(x+h)+f(x)h\lim_{h\to 0} \frac{f(x+h)+f(x)}{h} rather than lim h0f(x+h)f(x)h\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}). Thus, this “exterior differential” seems barely (if at all) more general than the usual one acting on exterior kk-forms.

It is unclear whether there is a notion of exterior differential which is significantly more general. Similarly, it is unclear whether the wedge product of exterior forms can be sensibly extended to cogerm ones, or whether there is a sensible commutative cogerm differential d\mathrm{d} acting on cogerm kk-forms for k>1k\gt 1.

Nonsymmetrized version

More general than cojet forms (but analytic), we have the coflare differential forms, in which there are any number of differentials (or higher differentials) d 0x\mathrm{d}_{0}x, d 1x\mathrm{d}_{1}x, d 1,0x=d 1d 0x\mathrm{d}_{1,0}x = \mathrm{d}_{1}\mathrm{d}_{0}x, etc. Whereas jets are based on the tangent bundle TXT X, flares (of rank, say, 33) are based on the iterated tangent bundle T(T(TX))T (T (T X)). Both cojet forms and exterior differential forms are included in the calculus of coflare forms, as are nonlinear versions of exterior forms such as absolute differential forms. The differentials in cojet forms always have the subscript 00 (which we may take to be the default), while the exterior form dxdy\mathrm{d}x \wedge \mathrm{d}y is really d 0xd 1yd 1xd 0y\mathrm{d}_{0}x \mathrm{d}_{1}y - \mathrm{d}_{1}x \mathrm{d}_{0}y (possibly divided by 2!2!, depending on your conventions).

A more abstract version of coflare forms (including both cogerm forms and the coflare forms of the previous paragraph) may be based on germs of maps with domain p\mathbb{R}^p instead of only curves (with domain \mathbb{R}). To fix notation, number the coordinates on p\mathbb{R}^p from 00 to p1p - 1; if ϕ\phi is a partial function from p\mathbb{R}^p to \mathbb{R} and UU is the subset of domϕ\dom \phi on which ϕ\phi is (say) 33-times differentiable, then write D i,j,kϕ:U\mathrm{D}_{i,j,k}\phi\colon U \to \mathbb{R} for the third partial derivative of ϕ\phi with respect to variable ii, variable jj, and variable kk (in any order). Then if cc is a thrice-differentiable function to XX from a neighbourhood of the origin in p\mathbb{R}^p and ff is a thrice-differentiable function to \mathbb{R} from a neighbourhood of c(0)c(0) in XX, then we have d i,j,kf|cD i,j,k(fc)(0)\langle \mathrm{d}_{i,j,k}f | c \rangle \coloneqq \mathrm{D}_{i,j,k}(f \circ c)(0), etc.

Even more abstractly, there is no reason to limit the domain of cc to p\mathbb{R}^p, although it's not clear if the forms on more general classes of germs correspond to anything of independent interest.


Historical reference saved for later reading:

  • Henk Bos (1973). Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus. Archive for History of Exact Sciences 14:1–90. DOI. PDF hosted by Leo Corry.

Last revised on August 29, 2021 at 17:21:58. See the history of this page for a list of all contributions to it.