FLUX ACROSS NONSMOOTH BOUNDARIES AND THE FRACTAL DIVERGENCE THEOREM
JENNY HARRISON DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY Abstract. Replacing parametrization of a domain with polyhedral approximations we give an optimal extension of the Divergence Theorem. Permitted domains of integration, called chainlets, range from smooth submanifolds to structures that may not be locally Euclidean and have no tangent vectors defined anywhere. One may still calculate divergence over the domain, and flux through its boundary which itself may have no normal vectors defined anywhere.

Introduction The real numbers R are a completion of the rationals via the Euclidean metric. Continuity properties of real numbers, relative to the Euclidean metric, are at the heart of real analysis. Similarly, one may consider the vector space of p-dimensional simplicial chains k ai σi i=1 in Rn and their completion w.r.t. a norm∗ . The Banach space obtained on completion has limit points that can be written as conditionally convergent series of simplicial chains,
∞

A=
i=1

ai σi ,

called chainlets. In [H1] the author defined a family of norms giving geometric meaning to these infinite series of weighted simplexes and thus to chainlets. (See §1, below.)
Date: November 14, 1998. 1991 Mathematics Subject Classification. Primary: 06B10; Secondary: 06B40. Key words and phrases. fractal, differential form, chainlet, Hodge, Laplace, Stokes’ theorem, flux, curl, divergence. ∗ One may also work with ambient spaces of Riemannian manifolds. (See [H3].)
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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY
∞ i=1

The integral of a smooth form ω over a chainlet using term by term integration
∞
∞ i=1

ai σi is defined

ω=
a i σi i=1 a i σi

ω.

Examples of chainlets include smooth submanifolds, fractals, vector fields, Dirac delta masses, Cantor sets, and stable manifolds and the theory shows how they all fit together continuously into Banach spaces. Some examples are further described in §2 and §3. The author [H4] has shown that every current and distribution corresponds to a chainlet. This not only provides a large source of examples, but it can be used to show a number of generalizations of classical results are optimal. While distributions and currents are defined abstractly as linear functionals on functions and differential forms, resp., we emphasize that chainlets have concrete geometric definition. The classical theory of differential manifolds relies heavily on results of linear algebra of tangent spaces. Much of the work involves taking partitions of unity or checking coherence in the overlap maps. These techniques are not necessary or even valid for the theory of chainlets which are not assumed to be locally Euclidean and thus may have no tangent spaces. Instead, one replaces linear algebra on tangent spaces with analysis on simplexes. Parametrization is replaced with simplicial approximation. Partitions of unity are replaced with algebraic sums of chains. The unit normal bundle of a smooth submanifold B is replaced with ∗B, the chainlet that is the geometric Hodge * of B. (See §3.) Because of continuity of fundamental operators, results such as Stokes’ theorem for simplexes carry immediately over to chainlets. Other important results of calculus, algebraic topology, differential topology, and measure theory extend to chainlets giving a common language for these theories. 1. Dipoles and norms An oriented p-simplex in Rn is the oriented convex hull of p + 1 points in Rn . We assume all simplexes are oriented henceforth. A simplicial chain in Rn is a formal sum of simplexes in Rn with real coefficients. We may assume that integration of smooth forms is defined over simplexes and thus over simplicial chains, and that Stokes’ theorem is valid for simplicial chains ∂S ω = S dω. The mass of a simplicial chain i ai σi is simply the weighted sum of masses of the simplexes |S|0 = i |ai |m(σi ) where m denotes p-dimensional Lebesgue measure. If v ∈ Rn is a vector let |v| denotes its length. If σ is a p-simplex in

DIVERGENCE THEOREM

3

Rn and v ∈ Rn , define Tv σ as the translate of σ by v. Its orientation is

naturally induced from the orientation given on σ. Mass of simplicial chains does not naturally measure geometric continuity. For example, the simplicial chain σ − Ttv σ has mass that is twice the mass of σ unless t = 0. This problem is partially circumvented with polyhedral chains. Polyhedral chains are equivalence classes of simplicial chains satisfying S ∼ T ⇐⇒
S

ω=
T

ω

for all smooth ω. Write A = [S] and define A ω = S ω. For example, −σ is identified with the same simplex as σ but with the opposite orientation. This definition takes into account overlapping simplexes with the opposite sign. The region of overlap is canceled. Polyhedral chains have naturally defined mass |A|0 = inf{|S|0 : A = [S]}. The mass of an n-dimensional polyhedral chain A = [σ − Ttv σ] in Rn is a continuous function of t. Because of cancellation of overlapping, oppositely oriented simplexes, the mass tends to zero as t → 0. Unfortunately, mass alone is not enough to conclude that a chain and its nearby translate are close to each other. Consider a 2-simplex σ in R3 and v a vector not in the plane of σ. Then σ and Ttv σ are disjoint if t = 0. The mass of [σ − Ttv σ] is again exactly twice the mass of σ, until t = 0, at which point the mass becomes 0. We need finer norms than mass to be sensitive to geometric continuity. Dipoles. A simple p-dimensional 0-dipole in Rn is defined to be a psimplex σ 0 with diameter ≤ 1. A simple p-dimensional 1-dipole is a p-chain of the form σ 1 = σ 0 − Tv1 σ 0 where |v1 | ≤ 1 and σ 0 is disjoint from Tv1 σ 0 . We inductively define simple j-dipoles. Given a vector vj with |vj | ≤ 1 and a simple (j − 1)dipole σ j−1 disjoint from Tvj σ j−1 , define the simple p-dimensional jdipole σ j as the simplicial p-chain σ j = σ j−1 − Tvj σ j−1 . Thus σ j is generated by vectors v1 , . . . , vj , each with norm ≤ 1, and a simplex σ 0 , where all translations of σ 0 through the vectors vi are disjoint.

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

Figure 1. A simple 2-dipole, or quadrupole For j ≥ 0, a j-dipole in Rn , is a polyhedral chain represented by a k j formal sum of simple j-dipoles, Dj = i=1 ai σi with real coefficients ai . Dipole mass. Given a simple j-dipole σ j , generated by a simplex σ 0 and constant vector fields v1 , . . . , vj , with |vi | ≤ |vj | ≤ 1, 1 ≤ i ≤ j, define its j-dipole mass σj
j

= |σ 0 |0 |v1 | · · · |vj |.

For example, suppose σ 1 is a 1-dimensional 1-dipole, forming the oppositely oriented sides of a parallelogram. If each of these sides has length ε and the other sides each has length δ then the dipole mass σ 1 1 = εδ, regardless of the angle formed by the parallelogram. Even if the parallelogram is degenerate, the dipole mass is the same. For j-dipoles Dj , j ≥ 0, define j-dipole mass as
k

D

j

j

= inf
i=1

j |ai | σi

j

,
k i=1 j ai σi .

where the infimum is taken over all representatives Dj = In particular, D0
0

= |D0 |0 .

r-norms. Let A be a polyhedral chain in Rn and r ∈ Z, r ≥ 1. Define
r

(1)

|A|r = inf
s=0

Ds

s

+ |C|r−1

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5

where the infimum is taken over all dipole decompositions
r

A=
s=0

Ds + ∂C.

These norms in this form were introduced in [H5], although earlier versions appeared in [H0]. We denote the Banach space of pdimensional polyhedral chains completed with the r-norm by Ar . p Lemma 1.1. If A is polyhedral, |A|s ≤ |A|r for all r ≤ s. Proof. This follows directly from the definitions of the norms. In [H6] the author shows that the s-norm of a chainlet in Ar is wellp defined by taking suitable limits of polyhedral chains and lower semicontinuous. This implies that the Banach spaces Ar are nested and get p larger and larger, including more and more strange and pathological limit points as r increases. For example, we see in §2 that the Dirac delta function is represented by a chainlet in A1 , its rth derivative (in 1 the sense of distributions) by a chainlet in Ar+1 . 1 r,Lip For r ∈ Z+ , let Bp denote the real linear space of p-forms in Rn with bounded norm ω C r,Lip . That is, the r derivatives of each component function of ω exist, have uniformly bounded sup norm and satisfy a uniform Lipschitz condition.

Integration over chainlets.
r−1,Lip Theorem 1.2. For A a polyhedral p-chain in Rn and ω ∈ Bp then

ω ≤ (p + 1) ω
A

C r−1,Lip |A|r .

This is proved in [H5]. r−1,Lip over a chainlet in Ar . A in Rn The integral of a form ω ∈ Bp p is defined by taking limits. If Ak → A are polyhedral chains in Rn converging to A in the r-norm, define ω = lim
A

k→∞

ω.
Ak

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

This is well-defined because of Theorem 1.2. This is equivalent† to the alternate definition given in the introduction. If A = ∞ ai σi then i=1
∞

ω=
A i=1 a i σi

ω.

Another consequence of Theorem 1.2 is that | |r is a norm. Suppose A = 0 is a polyhedral chain. Then there exists an ∞-smooth form ω such that A ω = 0. Thus 0<
A

ω ≤ (p + 1) ω

C r−1,Lip |A|r .

Hence |A|r = 0. 2. Examples We have chosen four far-ranging examples to illustrate chainlets. Support of a chainlet The support of a polyhedral chain A is a closed set spt(A) defined as follows: x ∈ Rn \spt(A) iff there exists a neighborhood U of x in Rn such that if ω is any smooth form supported in U then A ω = 0. The support spt(A) of a chainlet AinAr is the set p of points q ∈ Rn such that for every ε > 0 there exists a differential form ω ∈ B r such that A ω = 0 and ω(p) = 0 outside Bε (q), the ball of radius ε about q. It is important to keep in mind that there is much more to a chainlet than the subset of Rn that forms its support. We will see that there may be many chainlets supported in a given set. For a trivial example, consider a positively oriented two simplex σ in R2 . The chains λσ, λ ∈ R, are distinct chains, with the same support. A more interesting example is the solenoid, seen below, which naturally supports quite different chainlets. 1. Van Koch Snowflake One may write the snowflake arc S as a sum of simplicial chains ∞ Sk where for k ≥ 1, Sk is the k=0 sum of 4k boundaries of triangles σk each of side length 3−k . We show this series converges w.r.t. to the 1-norm: The partial sums satisfy Sk + · · · + Sj = ∂(σk + · · · + σj ). Thus |Sk + · · · + Sj |1 ≤ |σk |0 + · · · + |σj |0 < 4k /32k .
It is worth noting that if an infinite series is conditionally convergent w.r.t. a given norm then the sequence of partial sums converges w.r.t. the norm. Conversely, if xn → x is a sequence converging w.r.t. a norm, then the infinite series x0 + ∞ k=1 xk − xk−1 conditionally converges w.r.t. the norm.
†

DIVERGENCE THEOREM

7

S0 0

S1

S2

S 0 + S 1+ S 2

Figure 2. The snowflake as a sum of simplexes Since the r.h.s. tends to 0 as k, j → ∞, we know the infinite sum S is a well defined chainlet. We conclude that the snowflake is a current and we may integrate Lipschitz differential forms over it. 2. Dirac delta function and its derivatives We work in dimension one for simplicity of notation, but the construction can be extended to any dimension. Fix p ∈ R1 . For each k ≥ 0, let Qk be a positively oriented interval with length 2−k and centered at p. We claim that the sequence of polyhedral chains Dk = 2k Qk converges w.r.t. the 1-norm. Notice that the mass of each chain is one. It suffices to estimate |Dk − Dk+1 |1 We show the difference Dk − Dk+1 is a 1-dimensional 1-dipole, a sum of 4 weighted simple dipoles. Divide Dk into 2 intervals Qk with disjoint interiors, of length 2−(k+1) and weighted by 2k . Now Dk+1 can also be written

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

as the sum of 2 intervals Pk of length 2−(k+1) and weight 2k , but the line segments are identical to each other. Since the distance between the line segments of Pk and those of Qk is less than 2−k we deduce |Dk − Dk+1 |1 ≤ 22−k 2k 2−(k+1) = 2−k . We conclude that the sequence Dk is Cauchy in the 1-norm and its limit D has support p. The limit is canonically associated to the Dirac delta function. Since D ∈ A1 , we may integrate smooth 1 1-forms φdx over it. Hence φdx = lim
D k→∞

φdx = φ(p) = δ(φ).
Dk

The derivative of the Dirac delta function can also be realized geometrically, but as a chainlet B ∈ A2 . One considers the quadrupoles 1 (or 2-dipoles) formed by small oppositely oriented intervals centered at the endpoints of the Dk . It is left as an exercise to show that B φdx = δ (φ). 3. Toral solenoid Let T be the 2-torus in R3 and f : T → T a smooth hyperbolic mapping that contracts the torus in one direction, expands it in the other and then wraps the torus around inside itself twice. The solenoid is defined as the intersection ∞ n n=1 f T. It is a set of points that supports many chainlets. For example, let Q be the solid torus positively oriented and A0 = Q/|Q|0 . For k ≥ 0, let Ak+1 = f (Ak )/|f (Ak )|0 . Since the mass stays constant, the analysis here is similar to that for the Dirac delta function and one can use dipoles to show that Ak converges to a nonzero chainlet in A1 with support the solenoid. One 3 can also find chainlets in A1 with support the solenoid as follows. 1 Let B0 be the oriented core circle in the torus which is not null homotopic.. For k ≥ 0, let Bk+1 = f (Bk )/|f (Bk )|0 . Then Bk forms a Cauchy sequence in A1 and thus converges to a chainlet B ∈ A1 . 1 p It is also possible to find chainlets in A1 with support the solenoid 0 by choosing a countable dense subset and forming a Dirac mass at each of these points so that their total mass is finite. In the next section we find a chainlet in A1 with support the solenoid. 2 4. Graphs of L1 functions The graph Γ of a nonnegative L1 function over an interval [a, b] supports a chainlet. One merely approximates Γ with graphs of monotone increasing step functions Γn . These Γn are naturally oriented to form simplicial chains and these form a Cauchy sequence in the 1-norm. The difference Γn −Γn+1 is a dipole and so its 1-norm is bounded above by the area between

DIVERGENCE THEOREM

9

Figure 3. Toral solenoid the two graphs Γn and Γm . Thus {Γn } forms a Cauchy sequence in the 1-norm, converging to a chainlet Γ whose support is in the graph of f . One can think of Γ as the x-component of the graph of f . 3. Div, grad, and curl for fractals Banach spaces of chainlets have standard operators defined on them. In this paper we consider the boundary, pushforward, and geometric Hodge * operators. Each is first defined for simplexes and extended to simplicial chains by linearity. Differential forms are used to prove the operators are well-defined on polyhedral chains. Finally, the operators are proved to be continuous w.r.t. the norms, showing they are defined on chainlets. In practice, most of the work comes in establishing the first and last steps . For each operator there is a duality theorem relating chainlets to differential forms. We demonstrate this method of proof for the boundary operator. 3.1. Boundary operator. The boundary of a simplicial chain is defined in the standard way. If S ∼ T are simplicial chains we apply Stokes’ theorem for simplicial chains to deduce ω=
∂S S

dω =
T

dω =
∂T

ω.

Hence ∂S ∼ ∂T, implying that the boundary operator is well-defined on polyhedral chains. The boundary operator on polyhedral chains is bounded w.r.t. the r-norms. That is, Lemma 3.1. |∂A|r+1 ≤ |A|r .

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

Proof. This follows immediately from the definition of the r-norms. We may thus define the boundary ∂A of an chainlet A ∈ Ar . In p particular, the boundary operator ∂ : Ar −→ Ar+1 p p−1 is defined for r ≥ 0. It restricts to the usual boundary operator on polyhedral chains and satisfies Stokes’ theorem. The boundary operator is dual to the exterior derivative of forms, leading to Stoke’s theorem for chainlets. Theorem 3.2. Generalized Stokes’ theorem Let r ≥ 0. If A is an r-generalized p-chain and ω is a differential (p − 1)-form of class B r+1 then ∂A is a (r + 1)-generalized (p − 1)-chain, dω is differential p-form of class B r and dω =
A ∂A

ω.

Proof. By Stokes’ theorem for simplexes, continuity of the boundary operator and of the integral,
A

dω =

∞ i=1

a i σi

dω =

∞ i=1

ai

σi

dω = = = =

∞ i=1
∞ i=1

ai

∂σi

ω

ai ∂σi

∂ ∂A

ω.

∞ i=1

ω ω a i σi

In [H4] it is shown that this generalization of Stokes’ theorem is optimal for integrands of smooth forms. That is, all possible domains of integration arise as chainlets. For 1-dimensional chains A Stokes’ theorem implies that the Fundamental Theorem of Calculus is valid for all r-generalized 1-chains, e.g., fractal arcs. Here, ω is taken to be a function f and df is its gradient. For arcs A with endpoints p and q, q this is usually written p f (x)dx = A df. 3.2. Pushforward operator. If f is a mapping of class B r+1 , r ≥ 0, one can define the pushforward operator or change of variables operator f : Ar → Ar . The pushforward operator on chainlets is dual to the p p pullback operator on forms leading to a change of variables theorem for chainlets. Theorem 3.3. Let r ≥ 0. If A is an r-generalized p-chain, and ω ∈ Br and f ∈ Br+1 then ω=
fA A

f ω.

DIVERGENCE THEOREM

11

A

Figure 4. A polyhedral approximation to ∗R (See [H1] for more details.) Several new operators are defined on chainlets. The main one we discuss here is the geometric Hodge * operator ∗. This, along with the boundary operator, leads to optimal Green and Gauss theorems for chainlets. Combinations and modifications of these operators leads to geometric Laplace operators, Lie derivative, Dirac operators, and coboundary operators on chainlets. 3.3. Geometric Hodge star operator. ∗ : Ar −→ Ar is defined p n−p in ([H3]) for r > 0. To give the idea, Figure 3.2 illustrates a polyhedral approximation to ∗R where R is the oriented rectangle depicted in R3 . ∗R is found by taking a limit in the 1-norm of similar sums of tiny equally spaced 1-simplexes, orthogonal to R and whose total length for each sum is the same as the area of R. Even in this simplest example of a rectangle ∗R is not locally Euclidean, showing that fractal-like structures are naturally associated to smooth ones. (See Figure 3.2.) Theorem 3.4. Generalized Hodge star theorem If A is an rgeneralized p-chain, and ω ∈ Br then ∗A is a chainlet in Ar , ∗ω ∈ Br p and ∗ω =
A

ω.
∗A

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

For the proof, see [H3]. Using a combinatorial definition, others have defined a local dual to the Hodge star operator, but the integral equation of Theorem 3.4 does not hold. The boundary and geometric Hodge star operators lead to generalized Green and Gauss theorems. Let S be a smooth, oriented surface with boundary in R3 . The usual way to integrate the curl of a vector field X over S is to integrate the dot product of curlX with the unit normal vector field to S. According to Green’s Theorem, this quantity equals the integral of X over ∂S. We have already seen we do not require the existence of tangents to ∂S to calculate the integral of X over it. By working with the differential 1-form ω associated to X via Euclidean coordinates and the geometric Hodge * operator ∗, we no longer require the existence of any unit normals to S to integrate curlX over A. Instead we use the geometric Hodge star operator applied to A, which is always defined if A is a chainlet. If ω is the differential 1-form representing the vector field X it is well known that ∗dω represents curlX. We see now that
∗S

∗dω represents the integral of the curlX over S.

Theorem 3.5. Fractal Green’s theorem If S is an chainlet in Ar p and ω ∈ Br+1 then ∂S is a (r + 1)-generalized (p − 1)-chain, ∗S is r-generalized, ∗dω ∈ Br and ω=
∂S ∗S ∂S

∗dω.
S

Proof. By Theorems 3.2 and 3.4

ω=

dω =

∗S

∗dω.

For smooth surfaces with smooth boundary, we have ∂S X · nds = ω. Since ∗dω corresponds to curlX. It follows that ∗S ∗dω = ∂S curlX · ndA. Therefore the preceding theorem generalizes and simS plifies Green’s theorem: S curlX · ndA = ∂S X · nds. If A is a chainlet then ∗∂A plays the role of a normal vector field on its boundary, even though the boundary of A may have no normal vectors defined. Thus one may calculate flux across fractal boundaries and obtain a fractal divergence theorem. The usual way to calculate flux of a vector field X across a boundary of a smooth solid region D in space is to integrate the dot product of X with the unit normal vector field to ∂D over the domain ∂D. According to Gauss’ Divergence Theorem, this quantity equals the integral of the divergence of X over D. By working with the differential form ω associated to X via Euclidean coordinates and the operator ∗, we no

DIVERGENCE THEOREM

13

longer require the existence of any unit normals to ∂D to calculate the flux of X across ∂D. We see that ω represents the flux of X across ∂D.
∗∂D

Theorem 3.6. Fractal divergence theorem If D is an r-generalized p-chain and ω is of class B r+1 then ω=
∗∂D D

d ∗ ω.

Proof.

∗∂D

ω=

∂D

∗ω =

D

d ∗ ω.

For smooth surfaces, we have S X · ndA = ∗S ω. It is well known that d∗ω corresponds to divX. Therefore the preceding theorem generalizes and simplifies the Divergence Theorem of Gauss: ∂D X · ndA = divXdV. D Some authors have defined the integral over fractal boundaries using the integral of the derived form over the interior, i.e., using the generalized Stokes’ theorem, as the definition. Instead, the integrals in the preceding five theorems are defined independently and are shown to satisfy the generalized Stoke’s theorem. 3.4. Examples revisited. 1. Van Koch snowflake One may calculate flux of a vector field F across the snowflake S as ∗S ω where ω is the 1-form determined by F using the Euclidean inner product. 2. Dirac delta function and its derivatives Distributions and their derivatives can be realized more systematically using the operator ∗ defined in §3 below. We say a distribution c is associated to a chainlet A if c(φ) = A φdx for all test functions φ. In [H3] it is shown that if c is a distribution associated to the chainlet A then c is associated to ∗∂A. 3. Toral solenoid Recall the chainlet B in A1 found by iterating 1 the core circle via the mapping f . supported in the solenoid. A 2-chainlet in A1 can be found by applying the ∗ operator to B. 2 In some sense, this ∗B acts as a normal bundle to B. 4. Graph of an L1 function One may calculate flux of a vector field F across the x-component of the graph Γ of a nonnegative L1 function as ∗Γ ω where, again, ω is the 1-form determined by F . We give an important example. Let F = ye2 where {e1 , e2 } is the Euclidean basis of R2 . Then ω = ydy corresponds to F . Applying

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J. HARRISON DEPARTMENT OF MATHEMATICS U.C. BERKELEY

A

*A

∂*A

* ∂*A

Figure 5. Polyhedral approximation to the coboundary of a line segment Stokes’ theorem we calculate the flux of F across Γ to be ydy = −
Γ 1

ydx =
S

dxdy =
0

f (x)dx.

∗Γ

Here, S denotes the subgraph of f . This links Lebesgue theory to chainlets. Geometric Laplace operator The geometric Hodge * operator ∗ leads immediately to definitions of the geometric coboundary operator δ on chainlets, defined as δ = (−1)n(p+1)+1 ∗ ∂∗ (see figure 3), and the geometric Laplace operator ∆ = δ∂ + ∂δ. If A is an r-generalized pchain, then δA is an (r + 1)-generalized (p + 1)-dimensional chain and ∆A is an (r + 2)-generalized p-dimensional chain. It follows readily from Theorems 3.4 and 3.2 that for ω ∈ B r+1 and A an r-generalized p-chain δω =
A δA

ω.

and for ω ∈ Br+2 then ∆ω =
A ∆A

ω.

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15

References
Harrison, Jenny, Continuity of the integral as a function of the domain, to appear in The Journal of Geometric Analysis. [H-N1] Harrison, Jenny and Norton, Alec, Geometric integration on fractal curves in the plane, Indiana Journal, 40 (1991), 567–594. [W] Whitney, Hassler, Geometric Integration Theory Princeton University press, 1950. [H0] Harrison, Jenny, Stokes’ theorem on nonsmooth chains, Bulletin AMS, October 1993. [H2] Harrison, Jenny, rth order conditional convergence of infinite series of fractal domains, AMS, Contemporary Mathematics [H3] Harrison, Jenny, Geometric dual to the Hodge * operator with applications to Green’s theorem, preprint in preparation. [H4] Harrison, Jenny, Geometric realizations of distributions and currents, submitted. [H5] Harrison, Jenny, Isomorphisms of Differential forms and cochains, submitted. [H6] Harrison, Jenny, Fractal dimension and mass of nonsmooth domains, preprint in preparation E-mail address: harrison@math.berkeley.edu [H1]



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