Theory and Applications of Categories, Vol. 11, No. 14, 2003, pp. 321–336.

SOME CALCULUS WITH EXTENSIVE QUANTITIES: WAVE EQUATION

ANDERS KOCK AND GONZALO E. REYES

ABSTRACT. We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.

1. Introduction

The aim of this paper is to contribute to a synthetic theory of distributions. The sense in which we understand “synthetic” in this context is that we place ourselves in a setting (category) where everything is smooth (differentiable). Now distributions are sometimes thought of as very non-smooth functions, like the Heaviside function, or the Dirac delta. We take the viewpoint, stressed by Lawvere, that distributions are extensive quantities, where functions are intensive ones. It is only by a spurious comparison with functions that distributions seem non-smooth.

A main assumption about the category in which we work is that it is cartesian closed, meaning that function “spaces”, and hence some of the methods of functional analysis, are available.

This viewpoint also makes it quite natural to formulate the wave equation as an evolution equation, i.e. an ordinary differential equation describing the evolution over time of any initial distribution, so it is an ordinary differential equation with values in the vector space of distributions.

The main construction in the elementary theory of the wave equation is the construction of the fundamental solution: the description of the evolution of a point (Dirac) distribution over time. (Other solutions with other initial states may then by obtained by convolution of the given initial state with the fundamental solution; we shall not go here into this classical technique.)

To say that distributions are extensive quantities implies that they transform covariantly. To say that functions are intensive quantities implies that they transform contravariantly. Distributions are here construed, following Schwartz, as linear functionals on the space of (smooth) functions. But since all functions in the synthetic context are smooth, as well as continuous, there is no distinction between distributions and Radon

Received by the editors 2003-03-24 and, in revised form, 2003-08-28.

Transmitted by F. W. Lawvere. Published on 2003-09-18.

2000 Mathematics Subject Classification: 18F99, 35L05, 46F10.

Key words and phrases: distribution, wave equation, synthetic differential calculus, extensive quantity.

\begin{matrix}c \\ ◯ \end{matrix} Anders Kock and Gonzalo E. Reyes, 2003. Permission to copy for private use granted.

322  ANDERS KOCK AND GONZALO E. REYES

measures.

So we consider a cartesian closed category E with finite limits, in which there is given a commutative ring object R, to be thought of as the real number line.

Already on this basis, one can define the vector space D′_{c}(M) of distributions of compact support on M, for each object M ∈ E, namely the object of R-linear maps R^{M} → R (“vector space” in this context means R-module).

We shall assume that elementary differential calculus for functions R → R is available, as in all models of SDG, cf. [4], [11], [8], etc. We shall also assume some integral calculus, but only in the weakest possible sense, namely we assume

Integration Axiom: For every ψ : R → R, there is a unique Ψ : R → R with Ψ′ = ψ and with Ψ(0) = 0.

Note that we do not assume any order ≤ on R, so that “intervals” [a, b] ⊆ R do not make sense as subsets. “Intervals”, on the contrary, will be construed as distributions: for a, b ∈ R, [a, b] denotes the distribution

ψ ↦ ∫_{a}^{b} ψ(x) dx

The right hand side here of course means Ψ(b)−Ψ(a), where Ψ is the primitive of ψ given by the integration axiom. (This weak form of integration axiom holds in some of the very simple models of SDG, like in the topos classifying the theory of commutative rings.)

Finally, for the specific treatment of the wave equation, we need that the trigonometric functions cos and sin should be present. We assume that they are given as part of the data, and that they satisfy cos^{2} + sin^{2} = 1, and cos′ = − sin, sin′ = cos. Also as part of the data, we need specified an element π ∈ R so that cos π = −1, cos 0 = 1.

Except for the Taylor Series/ Formal Solutions considerations in the end of the paper, the setting does not depend on the “nilpotent infinitesimals” of SDG, but could also be, say, that of Froelicher-Kriegl [2], or Grothendieck’s “Smooth Topos”.

We would also like to remark that one can probably construct such smooth toposes in which no non-trivial distribution of compact support has a density function, or equivalently, no function (other than 0) gives rise to a distribution of compact support; our description of fundamental solutions to the wave equation would not be affected. An example seems to be the topos classifying C^{ω}-algebras, where C^{ω} is the algebraic theory of entire real or complex analytic functions^{1}.

2. Generalities on distributions

We want to apply parts of the general theory of ordinary differential equations to some of the basic equations of mathematical physics, the wave and heat equations^{2}.

1Added in proof: This has now been verified by the second author, see a forthcoming note.

2We do not discuss the heat equation in the present paper; we hope to return to it and improve the version of [6].

This takes us by necessity to the realm of distributions. Not primarily as a technique, but because of the nature of these equations: they model evolution through time of (say) a heat distribution. A distribution is an extensive quantity, and does not necessarily have a density function, which is an intensive quantity; the most important of all distributions, the point distributions (or Dirac distributions), for instance, do not.

As stressed by Lawvere in [9], functions are intensive quantities, and transform contravariantly; distributions are extensive quantities and transform covariantly. For functions, this is the fact that the “space” of functions on M, R^{M} is contravariant in M, by elementary cartesian-closed category theory. Similarly, the “space” of distributions of compact support on M is a subspace of R^{R^{M}} (carved out by the R linearity condition), and so for similar elementary reasons is covariant in M.

Let us make the formula for covariant functorality D′_{c} explicit. Let f : M → N be a map. The map D′_{c}(f) : D′_{c}(M) → D′_{c}(N)—which may also be denoted f_{∗}—is described by declaring

⟨f_{∗}(T), ϕ⟩ = ⟨T , ϕ ∘ f⟩,  (1)

where T is a distribution on M, and ϕ is a function on N. The brackets denote evaluation of distributions on functions. If we similarly denote the value of the contravariant functor M ↦ R^{M} on a map f by f^{∗}, the defining equation for f_{∗} goes ⟨f_{∗}(T), ϕ⟩ = ⟨T , f^{∗}(ϕ)⟩.

We note that D′_{c}(M) is an R-linear space, and all maps f_{∗} : D′_{c}(M) → D′_{c}(N) are R-linear. Also D′_{c}(M) is a Euclidean vector space V, meaning that the basic differential calculus in available, for instance that the basic axiom of SDG holds; we return to this in Section 2.

For any distribution T of compact support on M, one has its Total, which is just the number ⟨T , 1⟩ ∈ R, where 1 denotes the function on M with constant value 1. Since f^{∗}(1) = 1 for any map f, it follows that f_{∗} preserves Totals. (Alternatively, let 1 denotes the terminal object (= one-point set). Since D′_{c}(1) ≅ R canonically, the Total of T may also be described as !_{∗}(T), where ! : M → 1 is the unique such map. Then preservation of Totals follows from functorality and from uniqueness of maps into 1.)

Recall that a distribution T on M may be multiplied by any function g : M → R, by the recipe

⟨g · T , ϕ⟩ = ⟨T , g · ϕ⟩.  (2)

A basic result in one-variable calculus is “integration by substitution”. We present it here in pure “distribution” form; note that no assumption on monotonicity or even bijectivity of the “substitution” g is made.

2.1. Proposition. Given any function g : R → R, and given a, b ∈ R. Then, as distributions on R,

g_{∗}(g′ · [a, b]) = [g(a), g(b)].

Proof. Let ψ be a test function, and let Ψ be a primitive of it, Ψ′ = ψ. So ⟨[g(a), g(b)], ψ⟩ = Ψ(g(b)) − Ψ(g(a)). On the other hand, by the chain rule, Ψ ∘ g is a primitive of g′ · (ψ ∘ g), and so

Ψ(g(a)) − Ψ(g(b)) = ⟨[a, b], g′ · (ψ ∘ g)⟩ = ⟨g′ · [a, b], ψ ∘ g⟩ = ⟨g_{∗}(g′ · [a, b]), ψ⟩.

The external product of distributions of compact support is defined as follows. If P is a distribution on M, and Q a distribution on N, we get a distribution P × Q on M × N, by

⟨P × Q, ψ⟩ = ⟨P, [m ↦ ⟨Q, ψ(m, −)⟩]⟩.

In general, the external product construction × will not be the same as the external product construction \overline{×} given by^{3}

⟨P \overline{×} Q, ψ⟩ = ⟨Q, [n ↦ ⟨P, ψ(−, n)⟩]⟩.

However, if [a, b] and [c, d] are intervals (viewed as distributions on R, as described above), [a, b] × [c, d] = [a, b] \overline{×} [c, d], as distributions on R^{2}, by an application of Fubini’s Theorem, (which holds in the context here—it is a consequence of equality of mixed partial dervatives). Distributions arising in this way on R^{2}, we call rectangles. The evident generalization to higher dimensions, we call boxes. We have

⟨[a, b] × [c, d], ψ⟩ = ∫_{a}^{b} ∫_{c}^{d} ψ(x, y) dy dx,

in traditional notation. Notice that we can define the boundary of the box [a, b] × [c, d] as the obvious distribution on R^{2},

(p_{c}^{2})_{∗}[a, b] + (p_{b}^{1})_{∗}[c, d] − (p_{d}^{2})_{∗}[a, b] − (p_{a}^{1})_{∗}[c, d]

where p_{c}^{2}(x) = (x, c), p_{b}^{1}(y) = (b, y), etc.

By a singular box in an object M, we understand the data of a map γ : R^{2} → M and a box [a, b] × [c, d] in R^{2}, and similarly for singular intervals and singular rectangles. Such a singular box gives rise to a distribution on M, namely g_{∗}([a, b] × [c, d]).

By “differential operator” on an object M, we here understand just an R-linear map D : R^{M} → R^{M}. If D is such an operator, and T is a distribution on M, we define D(T) by

⟨D(T), ψ⟩ := ⟨T , D(ψ)⟩,

and in this way, D becomes a linear operator D′_{c}(M) → D′_{c}(M).

In particular, if X is a vector field on M, one defines the directional derivative D_{X}(T) of a distribution T on M by the formula

⟨D_{X}(T), ψ⟩ = ⟨T , D_{X}(ψ)⟩.  (3)

3In fact, the two external product formations described here provide the covariant functor D′_{c}(−) with two structures of monoidal functor E → E, in fact, they are the monoidal structures that arise because D′_{c}(−) is a strong functor on E with a monad structure, [3], [5].

This in particular applies to the vector field ∂/∂x on R, and reads here ⟨T′, ψ⟩ = ⟨T , ψ′⟩ (ψ′ denoting the ordinary derivative of the function ψ). (This is at odds with the minus sign which is usually put in into the definition of T′, but it will cause no confusion—we are anyway considering second order operators, where there is no discrepancy.)

The following proposition is an application of the covariant functorality of the functor D′_{c}, which will be used in connection with the wave equation in dimension 2. We consider the (orthogonal) projection p : R^{3} → R^{2} onto the xy-plane; Δ denotes the Laplace operator in the relevant R^{n}, so for R^{3}, Δ is ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} + ∂^{2}/∂z^{2}.

2.2. Proposition. For any distribution S (of compact support) on R^{3},

p_{∗}(Δ(S)) = Δ(p_{∗}(S)).

(The same result holds for any orthogonal projection p of R^{n} onto any linear subspace; the proof is virtually the same, if one uses invariance of Δ under orthogonal transformations.)

Proof. For any ψ : R^{2} → R,

Δ(p^{∗}ψ)) = p^{∗}(Δ(ψ)),

namely ∂^{2}ψ/∂x^{2} + ∂^{2}ψ/∂y^{2}. From this, the Proposition follows purely formally.

3. Calculus in Euclidean vector spaces

Recall that a vector space in the present context just means an R-module. A vector space E is called Euclidean if differential and integral calculus for functions R → E is available. An axiomatic account is given in [4], [11], [8] and other places. The coordinate vector spaces are Euclidean, but so are also the vector spaces R^{M}, and D′_{c}(M) for any M. To describe for instance the (“time”) derivative ḟ of a function f : R → D′_{c}(M), we put

⟨ḟ(t), ψ⟩ = \frac{d}{dt}⟨f(t), ψ⟩.

Similarly, from the integration axiom for R, one immediately proves that D′_{c}(M) satisfies the integration axiom, in the sense that for any h : R → D′_{c}(M), there exists a unique H : R → D′_{c}(M) satisfying H(0) = 0 and H′(t) = h(t) for all t. In particular, if h : R → D′_{c}(M), the “integral” ∫_{a}^{b} h(u) du makes sense (as H(b) − H(a)), and the Fundamental Theorem of Calculus holds, almost by definition.

As a particular case of special importance, we consider a linear vector field on a Euclidean R-module V. To say that the vector field is linear is to say that its principalpart formation V → V is a linear map, Γ, say. We have then the following version of a classical result. By a formal solution for an ordinary differential equation, we mean a solution defined on the set D_{∞} of nilpotent elements in R (these form a subgroup of (R, +)).

3.1. Proposition. Let a linear vector field on a Euclidean vector space V be given by the linear map Γ : V → V. Then the unique formal solution of the corresponding differential equation, i.e., the equation Ḟ (t) = Γ(F (t)) with initial position v, is the map D_{∞} × V → V given by

(t, v) ↦ e^{t·Γ}(v),  (4)

where the right hand side here means the sum of the following “series” (which has only finitely many non-vanishing terms, since t is assumed nilpotent):

v + tΓ(v) + \frac{t^{2}}{2!}Γ^{2}(v) + \frac{t^{3}}{3!}Γ^{3}(v) + …

(Here of course Γ^{2}(v) means Γ(Γ(v)), etc.)

Proof. We have to prove that Ḟ (t) = Γ(F (t)). We calculate the left hand side by differentiating the series term by term (there are only finitely many non-zero terms):

Γ(v) + \frac{2t}{2!} · Γ^{2}(v) + \frac{3t^{2}}{3!} Γ^{3}(v) + ... = Γ(v + t · Γ(v) + \frac{t^{2}}{2!} · Γ^{2}(v) + ...)

using linearity of Γ. But this is just Γ applied to F (t).

There is an analogous result for second order differential equations of the form F^{··} (t) = Γ(F (t)) (with Γ linear); the proof is similar and we omit it:

3.2. Proposition. The formal solution of this second order differential equation F^{··}= ΓF, with initial position v and initial velocity w, is given by

F (t) = v + t · w + \frac{t^{2}}{2!}Γ(v) + \frac{t^{3}}{3!}Γ(w) + \frac{t^{4}}{4!}Γ^{2}(v) + \frac{t^{5}}{5!}Γ^{2}(w) + ....

We shall need the following result (“change-of-variable Lemma”); for V = R, it is identical to Proposition 2.1, and the proof is in any case the same.

3.3. Proposition. Given f : R → V, where V is a Euclidean vector space, and given g : R → R. Then for any a, b ∈ R,

∫_{a}^{b} f(g(x)) · g′(x) dx = ∫_{g(a)}^{g(b)} f(u) du.

Linear maps between Euclidean vector spaces preserve differentiation and integration of functions R → V; we shall explicitly need the following particular assertion 3.4. Proposition. Let F : V → W be a linear map between Euclidean vector spaces. Then for any f : R → V,

F (∫_{a}^{b} f(t) dt) = ∫_{a}^{b} F (f(t)) dt

.

4. Spheres and balls as distributions

Let S be a distribution in R^{n}; ultimately, it will be the unit sphere, see below. We describe some families of distributions derived from it. Let t ∈ R (not necessarily t > 0—we haven’t even assumed an order relation on R). We then have the homothety “multiplying by t from R^{n} to R^{n}”, which we denote H^{t}, so

H^{t}(\underline{x}) = t · \underline{x},

for any \underline{x} ∈ R^{n}.

We are going to use the covariant functorality of D′_{c} with respect to these maps H^{t}. Note that for any distribution T on R^{n},

H_{∗}^{0}(T) = T otal(T) · δ(\underline{0}),  (5)

where δ(\underline{0}) denotes the Dirac distribution at \underline{0} ∈ R^{n}, given by ⟨δ(\underline{0}), ψ⟩ = ψ(\underline{0}) . We put

S^{t} := H_{∗}^{t}(S).

It has the same Total as S, but its support^{4} is larger (e.g. for t = 2, it is 2^{n−1} times as big as that of S). So if S is the unit sphere, S^{t} is “the diluted sphere of radius t”. We also want an undiluted sphere of radius t; we put

S_{t} := t^{n−1} · S^{t}.

Note that in dimension 1, S_{t} = S^{t}.

The ball of radius 1 is made up from shells (undiluted spheres) “of radius u (0 ≤ u ≤ 1)” (heuristically!), motivating us to put

B := ∫_{0}^{1} S_{u} du,

using integration in D′_{c}(R^{n}). Let t ∈ R. We put

B^{t} := H_{∗}^{t}(B).

It has the same Total as B, but its support is larger (“if t > 1”—heuristically), so if S is the unit sphere, B is “the diluted ball of radius t” (think of the expanding universe). We also want an undiluted ball of radius t; we put

B_{t} := t^{n} · B^{t}.

We then have 4.1. Proposition. For all t ∈ R,

4We haven’t here introduced the notion of support of a distribution, and only use the word here for motivating the word “diluted”.

B_{t} = ∫_{0}^{t} S_{v} dv.

Proof.

B_{t} = t^{n} · B^{t} = t^{n} · H_{∗}^{t}(B)
= t^{n} · H_{∗}^{t}(∫_{0}^{1} S_{u} du)

= t^{n} · ∫_{0}^{1} H_{∗}^{t}(S_{u}) du

(by Proposition 3.4)

= t^{n} ∫_{0}^{1} H_{∗}^{t}(u^{n−1} · H_{∗}^{u}(S)) du

= t^{n} · ∫_{0}^{1} u^{n−1}H_{∗}^{t·u}(S) du

= ∫_{0}^{1}(t · u)^{n−1} · H_{∗}^{t·u}(S) t du

= ∫_{0}^{t} v^{n−1}H_{∗}^{v}(S) dv

(by change-of-variable Lemma (Proposition 3.3), with v := t · u), which is ∫_{0}^{t} S_{v} dv, as claimed.

We now give explicit defining formulae for S in dimensions 1, 2 and 3. These are of course standard integral formulae in disguise—explicit integral formulae come by applying the definitions, and then integral formulae for S^{t}, S_{t} and B_{t} may be derived (using Proposition 4.1 and related arguments)—we give some of these formulae below.

Dimension 1 S := δ(1) + δ(−1)

Dimension 2 S := cis_{∗}([0, 2π]), where cis : R → R^{2} is the map θ ↦ (cos θ, sin θ).

Dimension 3 S := s·sph_{∗}([0, 2π]×[0, π]), where s : R^{2} → R is the function (θ, ϕ) ↦ sin ϕ, and where sph is “the spherical coordinates map” R^{2} → R^{3} given by

(θ, ϕ) ↦ (cos θ sin ϕ, sin θ sin ϕ, cos ϕ).  (6)

In dimension 2, for instance, we have

⟨S^{t}, ψ⟩ = ∫_{0}^{2π} ψ(t · cos θ, t · sin θ) dθ,

⟨S_{t}, ψ⟩ = ∫_{0}^{2π} t · ψ(t · cos θ, t · sin θ) dθ,

and so by Proposition 4.1,

⟨B_{t}, ψ⟩ = ∫_{0}^{t} [∫_{0}^{2π} u · ψ(u · cos θ, u · sin θ) dθ] du,

which the reader may want to rearrange, using Fubini, into the standard formula for integration in polar coordinates over the disk of radius t; but note we have no assumptions like “t > 0”.

Note also that B_{0} = 0, whereas S^{0} and B^{0} are constants times the Dirac distribution at the origin \underline{0} (use (5)). The constants are the “area” of the unit sphere, or the “volume” of the unit ball, in the appropriate dimension. Explicitly,

S^{0} = 2 · δ(\underline{0}), 2π · δ(\underline{0}), 4π · δ(\underline{0}),  (7)

and

B^{0} = 2 · δ(\underline{0}), π · δ(\underline{0}), \frac{4π}{3} · δ(\underline{0})  (8)

in dimensions 1,2, and 3, respectively.

We shall also have occasion to consider the distribution (of compact support) t · S^{t} on R^{3} as well as its projection p_{∗}(t · S^{t}) on the xy-plane (using functorality of D′_{c} with respect to the projection map p : R^{3} → R^{2}).

We insert for reference two obvious “change of variables” equations. Recall that H_{t} : R^{n} → R^{n} is the homothetic transformation “multiplying by t”. We have, for any vector field F on R^{n} (viewed, via principal part, as a map R^{n} → R^{n}):

div(F ∘ H_{t}) = t · (div F) ∘ H_{t},  (9)

and

t^{n} ∫_{B_{1}} ϕ ∘ H_{t} = ∫_{B_{t}} ϕ.  (10)

5. Divergence Theorem for Unit Sphere

The Main Theorem of vector calculus is Stokes’ Theorem: ∫_{∂γ} ω = ∫_{γ} dω, for ω an (n − 1)-form, γ a suitable n-dimensional figure (with appropriate measure on it) and ∂γ its geometric boundary. In the synthetic context, the theorem holds at least for any singular cubical chain γ : I^{n} → M (I^{n} the n-dimensional coordinate cube), because the theorem may then be reduced to the fundamental theorem of calculus, which is the only way integration enters in the elementary synthetic context; measure theory not being available therein. For an account of Stokes’ Theorem in this context, see [11] p.139. Below, we shall apply the result not only for singular cubes as in loc.cit., but also for singular boxes, like the usual (γ : R^{2} → R^{2}, [0, 2π] × [0, 1]), “parametrizing the unit disk B by polar coordinates”,

γ(θ, r) = (r cos θ, r sin θ).  (11)

We shall need from vector calculus the Gauss-Ostrogradsky “Divergence Theorem”

flux of F over ∂γ = ∫_{γ}(divergence of F),

with F a vector field, for the geometric “figure” γ = the unit ball in R^{n}. For the case of the unit ball in R^{n}, the reduction of the Divergence Theorem to Stokes’ Theorem is a matter of the differential calculus of vector fields, differential forms, inner products etc. (See e.g. [7] p. 204). For the convenience of the reader, we recall the case n = 2.

Given a vector field F(x, y) = (F (x, y), G(x, y)) in R^{2}, apply Stokes’ Theorem to the differential form

ω := −G(x, y) dx + F (x, y) dy

for the singular rectangle γ given by (11) above. Then, using the equational assumptions on cos, sin and their derivatives, we have

γ^{∗}(dx) = cos θ dr − r sin θ dθ
γ^{∗}(dy) = sin θ dr + r cos θ dθ
γ^{∗}(dx ∧ dy) = r (dr ∧ dθ)

Since dω = (∂G/∂y + ∂F/∂x) dx ∧ dy = div(F) dx ∧ dy, then

γ^{∗}(dω) = div(F)r (dr ∧ dθ)

On the other hand,

γ^{∗}ω = (F sin θ − G cos θ) dr + (F r cos θ + Gr sin θ) dθ,  (12)

(all F, G, and F to be evaluated at (r cos θ, r sin θ)). Therefore

∫_{γ} dω = ∫_{0}^{2π} ∫_{0}^{1} div(F)r dr dθ;

this is ∫_{B_{1}} div(F) dA. On the other hand by Stokes’ Theorem ∫_{γ} dω = ∫_{∂γ} ω which is a curve integral of the 1-form (12) around the boundary of the rectangle [0, 2π]×[0, 1]. This curve integral is a sum of four terms corresponding to the four sides of the rectangle. Two of these (corresponding to the sides θ = 0 and θ = 2π) cancel, and the term corresponding to the side where r = 0 vanishes because of the r in r (dr ∧ dθ), so only the side with r = 1 remains, and its contribution is, with the correct orientation,

∫_{0}^{2π}(F (cos θ, sin θ) cos θ + G(cos θ, sin θ) sin θ) dθ = ∫_{S_{1}} F · n ds

where n is the outward unit normal of the unit circle. This expression is the flux of F over the unit circle, which thus equals the divergence integral calculated above.

6. Time Derivatives of Expanding Spheres and Balls

We now combine vector calculus with the calculus of the basic ball and sphere distributions, as introduced in Section 3, to prove the following result:

6.1. Theorem. In R^{n} (for any n), we have, for any t,

\frac{d}{dt}S^{t} = t · Δ(B^{t}),

(Δ = the Laplace operator).

Proof. We consider the effect of the two expressions on an arbitrary function ψ. We have

⟨\frac{d}{dt}S^{t}, ψ⟩ = ⟨S, \frac{d}{dt}(ψ ∘ H^{t})⟩ by various definitions

= ⟨S, u ↦ (∇ψ(H^{t}(u)) · u⟩ by differential calculus

= flux over S of (∇ψ ∘ H^{t}) by special property of S

= ⟨B, div(∇ψ ∘ H^{t})⟩ by divergence Theorem
= t⟨B, div(∇ψ) ∘ H^{t}⟩ by (9)
= t⟨B, (Δψ) ∘ H^{t}⟩ by definition of Δ

= t⟨B^{t}, Δ(ψ)⟩ = t⟨Δ(B^{t}), ψ⟩ by various definitions

from which the result follows.

We collect some further information about t-derivatives of some of the t-parametrized distributions considered. From Proposition 4.1 and the Fundamental Theorem of Calculus, we immediately derive

\frac{d}{dt}(B_{t}) = S_{t}.  (13)

In dimension 1, we have

\frac{d}{dt}(S_{t}) = Δ(B_{t});  (14)

for,

\frac{d}{dt}⟨S_{t}, ψ⟩ = \frac{d}{dt}⟨ψ(t) + ψ(−t)⟩ = ψ′(t) − ψ′(−t),

whereas

⟨ΔB_{t}, ψ⟩ = ⟨B_{t}, ψ′′⟩ = ∫_{−t}^{t} ψ′′(t) dt,

and the result follows from the Fundamental Theorem of Calculus.

The equation (14) implies the following equation if n = 1; we shall prove that it also holds if n ≥ 2:

t · \frac{d}{dt}(S_{t}) = (n − 1)S_{t} + t · Δ(B_{t}).  (15)

For, differentiate S_{t} = t^{n−1} · S^{t} to get

\frac{d}{dt}(S_{t}) = (n − 1)t^{n−2} · S^{t} + t^{n−1} \frac{d}{dt}(S^{t}),

which by Theorem 6.1 and the definition of B_{t} in terms of B^{t} is = (n−1)t^{n−2} ·S^{t} +Δ(B_{t}). Multiplying this equation by t and using the defining equation S_{t} = t^{n−1}S^{t} gives the result.

We we shall finally argue that

t · \frac{d}{dt}(B^{t}) = S^{t} − nB^{t}.  (16)

For, differentiating the defining equation B_{t} = t^{n}·B^{t} gives d/dtB_{t} = nt^{n−1}·B^{t}+t^{n}·d/dtB^{t}. Now the left hand side here is S_{t}, by (13), so we conclude that t^{n−1} · S^{t} = nt^{n−1} · B^{t} + t^{n} · d/dtB^{t}. If t were invertible, we would conclude by cancelling t^{n−1} in this equation. But since the equation holds for all t, we may cancel it in any case: a consequence of the integration axiom is the Lavendhomme Cancellation Principle, which says that if t · g(t) = 0 for all t, then g(t) = 0 for all t, see [8] Ch.1 Prop. 15. Applying this principle n − 1 times then yields (16).

7. Wave equation

Let Δ denote the Laplace operator ∑ ∂^{2}/∂x_{i}^{2} on R^{n}. We shall consider the wave equation (WE) in R^{n}, (for n = 1, 2, 3),

\frac{d^{2}}{dt^{2}} Q(t) = ΔQ(t)  (17)

as a second order ordinary differential equation on the Euclidean vector space D′_{c}(R^{n}) of distributions of compact support; in other words, we are looking for functions

Q : R → D′_{c}(R^{n})

so that for all t ∈ R, Q̈(t) = Δ(Q(t)) (viewing Δ as a map D′_{c}(R^{n}) → D′_{c}(R^{n}).) Consider a function f : R → V, where V is a Euclidean vector space (we are interested

in V = D′_{c}(R^{n})) . Then we call the pair of vectors in V consisting of f(0) and f^{·} (0) the initial state of f. We can now, for each of the cases n = 1, n = 3, and n = 2 describe fundamental solutions to the wave equations. (The case n = 2 is less explicit, and is derived “by projection” from the one in dimension 3.) By fundamental solutions, we mean solutions whose initial state is either a constant times (δ(\underline{0}, 0)), or a constant times (0, δ(\underline{0}).

7.1. Theorem. In dimension 1: The function R → D′_{c}(R) given by

t ↦ S^{t}(= S_{t})

is a solution of the WE; its initial state is 2(δ(\underline{0}), 0).

The function R → D′_{c}(R) given by

t ↦ B_{t}

is a solution of the WE with initial state 2(0, δ(\underline{0})).

Proof. We have d/dt(B_{t}) = S_{t} by (13), and d/dt(S_{t}) = Δ(B_{t}), by (14). This establishes the WE for B_{t}. Since Δ and d/dt commute, it therefore follows that WE also holds for d/dt(B_{t}) = S_{t}. The initial position of the solution S_{t} is S_{0} = S^{0} = 2δ(\underline{0}), by (7), and the initial velocity Δ(B_{0}) by (14), which is 0 since B_{0} = 0. The initial state of the solution B_{t} is B_{0} = 0, and the initial velocity is S_{0}, as we already calculated, so is = 2δ(\underline{0}).

7.2. Theorem. In dimension 3: The function R → D′_{c}(R^{3}) given by

t ↦ S^{t} + t^{2}Δ(B^{t})

is a solution of the WE with initial state 4πδ(\underline{0}), 0). The function R → D′_{c}(R^{3}) given by

t ↦ t · S^{t}

is a solution of the WE with initial state 4π(0, δ(\underline{0})).

Proof. We calculate first d/dt of t · S^{t}, using Theorem 6.1:

\frac{d}{dt}(t · S^{t}) = S^{t} + t^{2} · Δ(B^{t}),  (18)

and so by Theorem 6.1 again,

\frac{d^{2}}{dt^{2}} (t · S^{t}) = t · Δ(B^{t}) + 2 · t · Δ(B^{t}) + t^{2} · Δ(\frac{d}{dt}B^{t})

= 3 · t · Δ(B^{t}) + t · Δ(t · \frac{d}{dt}B^{t})

= 3 · t · Δ(B^{t}) + t · Δ(S^{t} − 3B^{t}),

using (16), and now by linearity of Δ, the terms involving Δ(B^{t}) cancel, so we are left with the equation

\frac{d^{2}}{dt^{2}} (t · S^{t}) = Δ(t · S^{t}),  (19)

which establishes WE for t · S^{t}.

Since d/dt and Δ commute, and since t · S^{t} is a solution, then so is its t-derivative (calculated in (18) above), i.e. S^{t} + t^{2} · Δ(B^{t}) is a solution. The assertions about initial position and velocity follow from (7), (using Theorem 6.1 to calculate the initial velocity of the solution S^{t} + t^{2}Δ(B^{t})).

Recall that we considered the orthogonal projection p : R^{3} → R^{2}. Applying covariant functorality, we get for any distribution Q on R^{3} of compact support a distribution p_{∗}(Q) on R^{2}, also of compact support.

7.3. Theorem. In dimension 2: The function R → D′_{c}(R^{2}) given by

t ↦ p_{∗}(S^{t} + t^{2}Δ(B^{t}))

is a fundamental solution of the WE in dimension 2; its initial state is 4π(δ(\underline{0}), 0).The function R → D′_{c}(R^{2}) given by

t ↦ p_{∗}(t · S^{t})

is a fundamental solution of the WE in dimension 2; its initial state is 4π(0, δ(\underline{0})).

(Note: The S^{t} and B^{t} in the statement of the theorem are those of R^{3}.)

Proof. The fact that the distributions in question are solutions of the WE is immediate from the Proposition 2.2 (“p_{∗} commutes with Δ”) and from the fact that p_{∗} : D′_{c}(R^{3}) → D′_{c}(R^{2}) is linear, and hence commutes with formation of d/dt; also, D′_{c}(p) sends Dirac distribution at \underline{0} ∈ R^{3} to Dirac distribution at \underline{0} ∈ R^{2}, so the initial values and velocities are as claimed.

An explicit integral expression for the two fundamental solutions here, obtained by projection, requires more assumptions, in particular, a square root formation, as is known from classical descriptions of the solutions in terms of “Poisson’s kernel”. We may express this by saying that the distributional solutions presented exist under our weak assumptions, but that they are not presented by functions (densities).

We haven’t touched the notion of support, but when defined (in a context where it makes sense), the two fundamental solutions , S^{t} and t · S^{t} in dimension 1 and 3 will have support only on the geometric sphere of radius t (which is of “codimension” 1), whereas the solution p_{∗}(t · S^{t}) will have support in the direct image in R^{2} of S^{t} ⊆ R^{3}, and be of codimension 0. This accounts for the Huygens Principle that in a 2-dimensional world, sounds cannot be sharp signals, cf. e.g. [15] p. 227.

One might of course also derive one-dimensional fundamental solutions by orthogonal projection along q : R^{3} → R. Since fundamental solutions are unique modulo constants, we conclude that q_{∗}(S^{t} +t^{2}Δ(B^{t})) is proportional to the 1-dimensional S^{t} (whose support is a 2-point set).

Combining Theorem 7.2 with Proposition 3.2, we can obtain information about S^{t}, and other spheres and balls, for nilpotent t. As examples, we shall prove

7.4. Proposition. If t^{5} = 0, then in dimension 1,

B_{t} = 2[t · δ(\underline{0}) + \frac{t^{3}}{3!} · δ(\underline{0})′′],

and in dimension 3,

S^{t} = 4π[t · δ(\underline{0}) + \frac{t^{3}}{3!} · Δ(δ(\underline{0}))].

Proof. We prove the second assertion only. (The proof of the first one is similar, using Theorem 7.1.) We already observed in (7) that, in dimension 3,

S^{0} = 4π · δ(\underline{0}).

Now the two expressions above are both solutions to WE with initial state (0, 4πδ(\underline{0}))—the left hand side by Theorem 7.2, and the right hand side by Proposition 3.2, with Γ = Δ, v = 0, w = 4πδ(\underline{0}).

References

[1] Bitsadze, A.V., Equations of Mathematical Physics. MIR Publishers Moscow 1980.

[2] Froelicher, A. and A. Kriegl, Linear spaces and differentiation theory, John Wiley and Sons 1988.

[3] Kock, A., Strong functors and monoidal monads. Archiv der Math. 23 (1972), 113-120.

[4] Kock, A., Synthetic Differential Geometry. Cambridge University Press 1981.

[5] Kock, A., Some problems and results in synthetic functional analysis, in Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Various Publication Series 35 (1983), 168-191.

[6] Kock, A. and G.E. Reyes, Some differential equations in SDG. arXiv:math. CT/0104164.

[7] Lang, S., Analysis II. Addison-Wesley 1969.

[8] Lavendhomme, R., Basic Concepts Of Synthetic Differential Geometry. Kluwer Academic Publishers 1996.

[9] Lawvere, F.W., Categories of Space and of Quantity. The Space of Mathematics: Philosophical, Epistemological and Historical Explorations. (International Symposium on Structures in Mathematical Theories. San Sebastian, Spain 1990), De-Gruyter, Berlin 1992, 14-30.

[10] Lawvere, F.W., Foundations and Applications, Axiomatization and Education. Bulletin of Symbolic Logic 9 (2003), 213-234.

[11] Moerdijk, I. and G.E. Reyes, Models for Smooth Infinitesimal Analysis. Springer-Verlag 1991.

[12] Quê, Ngo Van and G.E. Reyes, Smooth functors and synthetic calculus. The L.E.J. Brouwer Centenary Symposium, North Holland 1982, 377-395.

[13] de Rham, G., Variétés différentiables. Hermann Paris 1973.

[14] Schwartz, L., Méthodes mathématiques pour les sciences physiques. Hermann 1998.

[15] Strauss, W.A., Partial Differential Equations. John Wiley and Sons 1992.

Dept. of Mathematical Sciences, University of Aarhus, DK 8000 Aarhus C, Denmark

Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, H3C 3J7

Email: kock@imf.au.dk
      reyes@dms.umontreal.ca

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