towards a differential vector calculus in three...

Aim and motivationIntroduction

Some elements of fractal geometryMetric vector calculus

Vector calculus in continuumBasic facts of quaternionic analysis

Fundamental identitiesMaxwell equations

Towards a differential vector calculus in threedimensional continuum with fractal metric

Juan Bory Reyes(joint work with Alexander Balankin and Michael Shapiro)

ESIME-ESFM. IPN. Mexico.

CIMPA Summer Research School in”Mathematical modeling in Biology and Medicine”

Santiago de Cuba, 10 June, 2016

ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric

Motivation

One way to deal with physical problems on nowhere differentiablefractals is the embedding of these problems into the correspondingproblems for continuum with a proper fractal metric.

On this way different definitions of the fractal metric weresuggested to account for the essential fractal features.

Motivation

One way to deal with physical problems on nowhere differentiablefractals is the embedding of these problems into the correspondingproblems for continuum with a proper fractal metric.

On this way different definitions of the fractal metric weresuggested to account for the essential fractal features.

In this work we develop the metric differential vector calculus in athree-dimensional continuum with a non-Euclidean metric

The metric differential forms and Laplacian are introduced, fundamentalidentities for metric differential operators are established and integraltheorems are proved by employing the metric version of the quaternionicanalysis for the Moisil-Teodoresco operator

It should be emphasized that the metric vector calculus developed in thiswork provides a comprehensive mathematical formalism for thecontinuum with any suitable definition of fractal metric. This offers anovel tool to study physics on fractals.

Introduction

The interplay between geometry and physics is fundamental forunderstanding physical phenomena. In this regard, the fractalgeometry allows better understanding of many real world systems.

The scaling properties of fractals can be characterized by a set offractional dimensionality. Accordingly, fractal approach allows us tostore the data related to all scales of observation of a complexsystem using a relatively small number of parameters.

Introduction

The interplay between geometry and physics is fundamental forunderstanding physical phenomena. In this regard, the fractalgeometry allows better understanding of many real world systems.

The scaling properties of fractals can be characterized by a set offractional dimensionality. Accordingly, fractal approach allows us tostore the data related to all scales of observation of a complexsystem using a relatively small number of parameters.

Functions defined on fractals are non-differentiable in theconventional sense and demand the development of novel tools todeal with problems on fractals within a continuum framework.

Vector calculus (or vector analysis) is a branch of mathematicswhich is concerned with differentiation and integration of vectorfields. A metric version of the quaternionic analysis for theMoisil-Teodoresco operator offers a basic tool for the modernvector calculus.

Functions defined on fractals are non-differentiable in theconventional sense and demand the development of novel tools todeal with problems on fractals within a continuum framework.

Vector calculus (or vector analysis) is a branch of mathematicswhich is concerned with differentiation and integration of vectorfields. A metric version of the quaternionic analysis for theMoisil-Teodoresco operator offers a basic tool for the modernvector calculus.

Bibliography

Balankin A S 2013 Stresses and strains in a deformable fractalmedium and in its fractal continuum model Phys. Lett. A 377 2535.

Balankin A S 2013 Mena B, Patino J, Morales D Electromagneticfields in fractal continua Phys. Lett. A 377 783.

Kravchenko V, Shapiro M 1996 Integral representations for spatialmodels of mathematical physics. Pitman Res. Notes in Math. Ser.351. Longman, Harlow.

Balankin A S, Bory Reyes J, Shapiro M 2015 Towards a physics onfractals: differential vector calculus in three-dimensional continuumwith fractal metric. Physica A. Statistical Mechanics and itsApplications, 444, 345–359, 2016.

Some elements of fractal geometry

Hausdorff measure

Let E ⊂ R3. Then for any s ≥ 0 the s-dimensional Hausdorffmeasure Hs(E ) may be defined by:

Hs(E ) = lımδ→0

∞∑k=1

(diam Bk)s : E ⊂∞⋃k=1

Bk , diam Bk < δ

the infimum being taken over all countable δ-coverings Bk of Ewith open or closed balls. For s = 3, H3 coincides, up to a positivemultiplicative constant, with the Lebesgue measure L3 en R3.

Hausdorff dimension

Let E ⊂ R3 be compact. dimH(E ) = inf s ≥ 0 : Hs(E ) < +∞ .

Fractal

E ⊂ R3 bounded is called fractal (in the sense of Mandelbrot) if dimH(E )strictly exceeds the fractal’s topological dimension of E .

Box dimension

dim(E ) = lımε→0 sup log NE (ε)− log ε , where NE (ε) = stands for the

minimal number of ε-balls needed to cover E .

dimH(E ) ≤ dim(E )

Hausdorff dimension

Fractal

Box dimension

Hausdorff dimension

Fractal

Box dimension

Hausdorff dimension

Fractal

Box dimension

Metric vector calculus

Metric spaces

(R3, d(x , y)), (x , y) ∈ R3 × R3, (x = (x1, x2, x3); y = (y1, y2, y3); etc),d(x , y) ≥ 0, d(x , y) = 0⇔ x = y , d(x , z) ≤ d(x , y) + d(y , z).

Case of study

d(x , y) := (∑3

s=1 d2s (x , y))

12 , ds(x , y) := |ϑs(x)− ϑs(y)|, ϑs(•) = φ(•s),

with φ : R→ R such that φ(ξ), φ(ξ)/ξ y φ(0) = 0.

Examples

φ(xs) := signxs|xs |ζs , φ(xs) := |1 +xsl0|ζs, xs ≥ 0, where ζs denotes a

scaling exponent and l0 is the lower cutoff of fractal behavior along xs .

Metric spaces

Case of study

d(x , y) := (∑3

s=1 d2s (x , y))

12 , ds(x , y) := |ϑs(x)− ϑs(y)|, ϑs(•) = φ(•s),

Examples

Metric spaces

Case of study

d(x , y) := (∑3

s=1 d2s (x , y))

12 , ds(x , y) := |ϑs(x)− ϑs(y)|, ϑs(•) = φ(•s),

Examples

Bibliography

Balankin A S, Mena B, Patino J and Morales D 2013Electromagnetic fields in fractal continua Phys. Lett. A 377 783

Balankin A S 2013 Stresses and strains in a deformable fractalmedium and in its fractal continuum model Phys. Lett. A 377 2535

Metric vector calculus (Cont.)

Metric partial derivatives

Let f : (R3, d(x , y))→ R :

∇d1 [f ](x0) := lımd1(x,x0)→0

f (x1,x02 ,x

03 )−f (x0

1 ,x02 ,x

d1(x,x0)

∇d1 [f ](x0) exists⇐⇒ ∇1[f ](x0) exists, ∇1 :=

∇d1 [f ](x0) = 1

c1(x01 ).∇1[f ](x0),

c1(x01 ) := lımh→0

φ(x01 +h)−φ(x0

1 )h = φ

′(x0

Let f : (R3, d(x , y))→ R :

∇d1 [f ](x0) := lımd1(x,x0)→0

f (x1,x02 ,x

03 )−f (x0

1 ,x02 ,x

d1(x,x0)

∇d1 [f ](x0) = 1

c1(x01 ).∇1[f ](x0),

c1(x01 ) := lımh→0

φ(x01 +h)−φ(x0

1 )h = φ

′(x0

Let f : (R3, d(x , y))→ R :

∇d1 [f ](x0) := lımd1(x,x0)→0

f (x1,x02 ,x

03 )−f (x0

1 ,x02 ,x

d1(x,x0)

∇d1 [f ](x0) = 1

c1(x01 ).∇1[f ](x0),

c1(x01 ) := lımh→0

φ(x01 +h)−φ(x0

1 )h = φ

′(x0

Vector calculus in continuum

Ω ⊂ R3x Ξ ⊂ R3

ξ , ξ = (ξ1, ξ2, ξ3).

Change of variables

ϕ : (x1, x2, x3) ∈ Ω→ (ξ1 = ϕ1(x1, x2, x3), ξ2 = ϕ2(x1, x2, x3), ξ3 =ϕ3(x1, x2, x3)) = ϕ(x1, x2, x3) ∈ Ξ tal que ϕ ∈ C 2(Ω)

ϕ−1 : (ξ1, ξ2, ξ3) ∈ Ξ→ (x1, x2, x3) ∈ Ω

Operators of change of variables

Wϕ : f ∈ C 2(Ξ)→ f ϕ =: f ∈ C 2(Ω),

Wϕ−1 : f ∈ C 2(Ω)→ f ϕ−1 =: f ∈ C 2(Ξ).

Ω ⊂ R3x Ξ ⊂ R3

ξ , ξ = (ξ1, ξ2, ξ3).

Change of variables

ϕ−1 : (ξ1, ξ2, ξ3) ∈ Ξ→ (x1, x2, x3) ∈ Ω

Wϕ : f ∈ C 2(Ξ)→ f ϕ =: f ∈ C 2(Ω),

Wϕ−1 : f ∈ C 2(Ω)→ f ϕ−1 =: f ∈ C 2(Ξ).

Ω ⊂ R3x Ξ ⊂ R3

ξ , ξ = (ξ1, ξ2, ξ3).

Change of variables

ϕ−1 : (ξ1, ξ2, ξ3) ∈ Ξ→ (x1, x2, x3) ∈ Ω

Wϕ : f ∈ C 2(Ξ)→ f ϕ =: f ∈ C 2(Ω),

Wϕ−1 : f ∈ C 2(Ω)→ f ϕ−1 =: f ∈ C 2(Ξ).

Ω ⊂ R3x Ξ ⊂ R3

ξ , ξ = (ξ1, ξ2, ξ3).

Change of variables

ϕ−1 : (ξ1, ξ2, ξ3) ∈ Ξ→ (x1, x2, x3) ∈ Ω

Wϕ : f ∈ C 2(Ξ)→ f ϕ =: f ∈ C 2(Ω),

Wϕ−1 : f ∈ C 2(Ω)→ f ϕ−1 =: f ∈ C 2(Ξ).

Vector calculus in continuum (Cont.)

A := Wϕ−1 A Wϕ

B := Wϕ B Wϕ−1 .

A ∈ L(C 2(Ω))→Wϕ−1 A Wϕ = A ∈ L(C 2(Ξ)),

B ∈ L(C 2(Ξ))→Wϕ B Wϕ−1 = B ∈ L(C 2(Ω)).

Change of variables related to metric derivative

ξ ↔ ϕ(x) = (ϑ1(x1, x2, x3), ϑ2(x1, x2, x3), ϑ3(x1, x2, x3)),

x ↔ ϕ−1(ξ) = (ϑ−11 (ξ1, ξ2, ξ3), ϑ−1

2 (ξ1, ξ2, ξ3), ϑ−13 (ξ1, ξ2, ξ3)).

A := Wϕ−1 A Wϕ

B := Wϕ B Wϕ−1 .

A ∈ L(C 2(Ω))→Wϕ−1 A Wϕ = A ∈ L(C 2(Ξ)),

B ∈ L(C 2(Ξ))→Wϕ B Wϕ−1 = B ∈ L(C 2(Ω)).

x ↔ ϕ−1(ξ) = (ϑ−11 (ξ1, ξ2, ξ3), ϑ−1

2 (ξ1, ξ2, ξ3), ϑ−13 (ξ1, ξ2, ξ3)).

A := Wϕ−1 A Wϕ

B := Wϕ B Wϕ−1 .

A ∈ L(C 2(Ω))→Wϕ−1 A Wϕ = A ∈ L(C 2(Ξ)),

B ∈ L(C 2(Ξ))→Wϕ B Wϕ−1 = B ∈ L(C 2(Ω)).

x ↔ ϕ−1(ξ) = (ϑ−11 (ξ1, ξ2, ξ3), ϑ−1

2 (ξ1, ξ2, ξ3), ϑ−13 (ξ1, ξ2, ξ3)).

Canonical Nablas

∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.

Metric Nablas

∇dx = ∇d

1e1 +∇d2e2 +∇d

∇ξ = ∇1ξe1 + ∇2

ξe2 + ∇3ξe3 := c1(ξ1)

∂ξ1e1 + c2(ξ2)

∂ξ2e2 + c3(ξ3)

∂ξ3e3.

∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ

Canonical Nablas

∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.

Metric Nablas

∇dx = ∇d

1e1 +∇d2e2 +∇d

∇ξ = ∇1ξe1 + ∇2

ξe2 + ∇3ξe3 := c1(ξ1)

∂ξ1e1 + c2(ξ2)

∂ξ2e2 + c3(ξ3)

∂ξ3e3.

∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ

Canonical Nablas

∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.

Metric Nablas

∇dx = ∇d

1e1 +∇d2e2 +∇d

∇ξ = ∇1ξe1 + ∇2

ξe2 + ∇3ξe3 := c1(ξ1)

∂ξ1e1 + c2(ξ2)

∂ξ2e2 + c3(ξ3)

∂ξ3e3.

∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ

Canonical Laplacianos

∆x := ∇x ∇x , ∆ξ := ∇ξ ∇ξ.

Metric Laplacianos

∆dx := ∇d

x ∇dx , ∆ξ := ∇ξ ∇ξ.

∆ξ = Wϕ−1 ∆dx Wϕ, ∆ξ = Wϕ−1 ∆x Wϕ

Scalar and Vector fields

f0 : R3x → R, ~f := (f1, f2, f3) : R3

x → R3x ,

f0 : R3ξ → R, ~f := (f1, f2, f3) : R3

ξ → R3ξ.

∆x := ∇x ∇x , ∆ξ := ∇ξ ∇ξ.

Metric Laplacianos

∆dx := ∇d

x ∇dx , ∆ξ := ∇ξ ∇ξ.

∆ξ = Wϕ−1 ∆dx Wϕ, ∆ξ = Wϕ−1 ∆x Wϕ

f0 : R3x → R, ~f := (f1, f2, f3) : R3

x → R3x ,

f0 : R3ξ → R, ~f := (f1, f2, f3) : R3

ξ → R3ξ.

∆x := ∇x ∇x , ∆ξ := ∇ξ ∇ξ.

Metric Laplacianos

∆dx := ∇d

x ∇dx , ∆ξ := ∇ξ ∇ξ.

∆ξ = Wϕ−1 ∆dx Wϕ, ∆ξ = Wϕ−1 ∆x Wϕ

f0 : R3x → R, ~f := (f1, f2, f3) : R3

x → R3x ,

f0 : R3ξ → R, ~f := (f1, f2, f3) : R3

ξ → R3ξ.

∆x := ∇x ∇x , ∆ξ := ∇ξ ∇ξ.

Metric Laplacianos

∆dx := ∇d

x ∇dx , ∆ξ := ∇ξ ∇ξ.

∆ξ = Wϕ−1 ∆dx Wϕ, ∆ξ = Wϕ−1 ∆x Wϕ

f0 : R3x → R, ~f := (f1, f2, f3) : R3

x → R3x ,

f0 : R3ξ → R, ~f := (f1, f2, f3) : R3

ξ → R3ξ.

Canonical grad, div, rot

gradx [f0] := ∇x f0 = ∇1[f0]e1 +∇2[f0]e2 +∇3[f0]e3,

rotx [~f ] :=[∇x , ~f

(∇2[f3]−∇3[f2])e1 + (∇3[f1]−∇1[f3])e2 + (∇1[f2]−∇2[f1])e3,

divx [~f ] :=⟨∇x , ~f

⟩= ∇1[f1] +∇2[f2] +∇3[f3].

gradξ[f0] := ∇ξ f0 = ∇1[f0]e1 +∇2[f0]e2 +∇3[f0]e3,

rotξ[~f ] :=[∇ξ, ~f

(∇2[f3]−∇3[f2])e1 + (∇3[f1]−∇1[f3])e2 + (∇1[f2]−∇2[f1])e3,

divξ[~f ] :=⟨∇ξ, ~f

⟩= ∇1[f1] +∇2[f2] +∇3[f3].

Metric grad, div, rot

gradd [f0] := ∇d f0 = ∇d1 [f0]e1 +∇d

2 [f0]e2 +∇d3 [f0]e3,

rotd [~f ] :=[∇d , ~f

(∇d2 [f3]−∇d

3 [f2])e1 + (∇d3 [f1]−∇d

1 [f3])e2 + (∇d1 [f2]−∇d

2 [f1])e3,

divd [~f ] :=⟨∇d , ~f

⟩= ∇d

1 [f1] +∇d2 [f2] +∇d

3 [f3].

ˆgradξ[f0] := ∇ξ f0 = ∇1ξ[f0]e1 + ∇1

ξ[f0]e2 + ∇1ξ[f0]e3,

ˆrotξ[~f ] :=[∇ξ, ~f

(∇2ξ[f3]− ∇3

ξ[f2])e1 + (∇3ξ[f1]− ∇1

ξ[f3])e2 + (∇1ξ[f2]− ∇2

ξ[f1])e3,

divξ[~f ] :=⟨∇ξ, ~f

⟩= ∇1

ξ[f1] + ∇2ξ[f2] + ∇3

ξ[f3].

Basic facts of quaternionic analysis

H the set of real quaternions. a =∑3

k=0 ak ik := a0 + ~a whereak ⊂ R, i0 is the multiplicative unit and ik | k = 1, 2, 3 are thequaternionic imaginary units.

i0ij = ij i0, j = 1, 2, 3, i2j = −i0, j = 1, 2, 3,i1i2 = i3 = −i2i1, i2i3 = i1 = −i3i2, i3i1 = i2 = −i1i3.

Quaternionic product

ab = a0b0 −⟨~a, ~b

⟩+ a0

~b + b0~a +[~a, ~b

]~a~b = −

⟨~a, ~b

⟩+[~a, ~b

ab = a0b0 −⟨~a, ~b

⟩+ a0

~b + b0~a +[~a, ~b

~a~b = −⟨~a, ~b

⟩+[~a, ~b

ab = a0b0 −⟨~a, ~b

⟩+ a0

~b + b0~a +[~a, ~b

]~a~b = −

⟨~a, ~b

⟩+[~a, ~b

Moisil-Teodoresco systems

f = f0 + ~f : R3x → H.

Canonical Moisil-Teodoresco systemsdivx ~f = 0,

gradx f0 + rotx ~f = 0,

~f = 0,

gradξ f0 + rotξ~f = 0.

Metric Moisil-Teodoresco systemsdivd [~f ] = 0,

gradd [f0] + rotd [~f ] = 0,

divξ[

~f ] = 0,

ˆgradξ[f0] + ˆrotξ[~f ] = 0.

f = f0 + ~f : R3x → H.

~f = 0,

gradd [f0] + rotd [~f ] = 0,

divξ[

~f ] = 0,

f = f0 + ~f : R3x → H.

~f = 0,

gradd [f0] + rotd [~f ] = 0,

divξ[

~f ] = 0,

Operators of Moisil-Teodoresco

Operadores de Moisil-Teodoresco canonicos

Dx =3∑

∂xkik , Dξ =

3∑k=1

∂ξkik .

Metric operators of Moisil-Teodoresco

3∑k=1

ck(xk)

∂xkik , Dξ =

3∑k=1

ck(ξk)∂

∂ξkik .

D2x = −∆x , [Dd ]2 = −∆d

x , D2ξ = −∆ξ, [Dξ]

2 = −∆ξ.

Dx =3∑

∂xkik , Dξ =

3∑k=1

∂ξkik .

3∑k=1

ck(xk)

∂xkik , Dξ =

3∑k=1

ck(ξk)∂

∂ξkik .

D2x = −∆x , [Dd ]2 = −∆d

x , D2ξ = −∆ξ, [Dξ]

2 = −∆ξ.

Dx =3∑

∂xkik , Dξ =

3∑k=1

∂ξkik .

3∑k=1

ck(xk)

∂xkik , Dξ =

3∑k=1

ck(ξk)∂

∂ξkik .

D2x = −∆x , [Dd ]2 = −∆d

x , D2ξ = −∆ξ, [Dξ]

2 = −∆ξ.

Quaternionic hyperholomorfy

Dx [f ] =3∑

3∑k=1

∂fj∂xk

ik ij = −divx ~f + gradx f0 + rotx ~f .

[f ]Dx =3∑

3∑k=1

∂fj∂xk

ij ik = −divx ~f + gradx f0 − rotx ~f .

A function f ∈ C 1(Ωx ,H) will be called left hyperholomorphic ifDx [f ](x) = 0 in Ωx . Similarly, f is right hyperholomorphic if[f ]Dx(x) = 0.

Quaternionic hyperholomorfy

Dx [f ] =3∑

3∑k=1

∂fj∂xk

ik ij = −divx ~f + gradx f0 + rotx ~f .

[f ]Dx =3∑

3∑k=1

∂fj∂xk

ij ik = −divx ~f + gradx f0 − rotx ~f .

A function f ∈ C 1(Ωx ,H) will be called left hyperholomorphic ifDx [f ](x) = 0 in Ωx . Similarly, f is right hyperholomorphic if[f ]Dx(x) = 0.

Equivalent theories

Dx [f ] = Dx [f0 + ~f ] = 0⇔

divx ~f = 0,

gradx f0 + rotx ~f = 0.

Div-rot system

Dx [~f ] = 0⇔

divx ~f = 0,

rotx ~f = 0.

Equivalent theories

Dx [f ] = Dx [f0 + ~f ] = 0⇔

divx ~f = 0,

gradx f0 + rotx ~f = 0.

Div-rot system

Dx [~f ] = 0⇔

divx ~f = 0,

rotx ~f = 0.

Fundamental identities

D2x [f ] = −∆x [f ]⇔ Dx [−divx ~f + gradx f0 + rotx ~f ] = −∆x [f0 + ~f ]

Vectorial identities∆x [f0] = divx [gradx f0],

rotx [gradx f0] = 0,

divx [rotx ~f ] = 0

rotx [rotx ~f ] = ∇x(∇x [~f ])−∇x∇x [~f ],

Fundamental identities

D2x [f ] = −∆x [f ]⇔ Dx [−divx ~f + gradx f0 + rotx ~f ] = −∆x [f0 + ~f ]

Vectorial identities∆x [f0] = divx [gradx f0],

rotx [gradx f0] = 0,

divx [rotx ~f ] = 0

rotx [rotx ~f ] = ∇x(∇x [~f ])−∇x∇x [~f ],

Fundamental identities (Cont.)

Sean f , g ∈ C 1(Ωx ,H)∫Γxg(x) ~n(x) f (x) ds =

∫Ωx

([g ]Dx(x) f (x) + g(x)Dx [f ](x)) dVx .

Integral identities∫Γx

(g0∂f0∂~n − f0

∂g0∂~n )ds =

∫Ωx

g0∆x f0 − f0∆xg0 dVx ,∫Γx

⟨~n, ~f

⟩ds =

∫Ωx

divx ~f dVx ,

∫Γx

[~n, ~f

∫Ωx

rotx ~f dVx .

Fundamental identities (Cont.)

Sean f , g ∈ C 1(Ωx ,H)∫Γxg(x) ~n(x) f (x) ds =

∫Ωx

([g ]Dx(x) f (x) + g(x)Dx [f ](x)) dVx .

Integral identities∫Γx

(g0∂f0∂~n − f0

∂g0∂~n )ds =

∫Ωx

g0∆x f0 − f0∆xg0 dVx ,∫Γx

⟨~n, ~f

⟩ds =

∫Ωx

divx ~f dVx ,

∫Γx

[~n, ~f

∫Ωx

rotx ~f dVx .

Time-harmonic Maxwell system

Let ~E : R3x → R3

x , ~H : R3x → R3

x , ~E : R3ξ → R3

ξ , ~H : R3ξ → R3

vector fields en R3x y R3

ξ respectively.

In phasor form, time-harmonic Maxwell equations are given byrotx ~H = −iωε~E + ~j ,

rotx ~E = iωµ ~H,

divx ~E = ρε ,

divx ~H = 0,

~H = −iωε~E + ~j ,

rotξ~E = iωµ~H,

divξ~E = ρ

divξ~H = 0.

and the continuity equation as:

divx ~j = iωρ divξ ~j = iωρ,

Time-harmonic Maxwell system

Let ~E : R3x → R3

x , ~H : R3x → R3

x , ~E : R3ξ → R3

ξ , ~H : R3ξ → R3

vector fields en R3x y R3

ξ respectively.In phasor form, time-harmonic Maxwell equations are given by

rotx ~H = −iωε~E + ~j ,

rotx ~E = iωµ ~H,

divx ~E = ρε ,

divx ~H = 0,

~H = −iωε~E + ~j ,

rotξ~E = iωµ~H,

divξ~E = ρ

divξ~H = 0.

divx ~j = iωρ divξ ~j = iωρ,

Metric time-harmonic Maxwell system

rotd ~H = −iωε~E + ~j ,

rotd ~E = iωµ ~H,

divd ~E = ρε ,

divd ~H = 0,

ˆrotξ

~H = −iωε~E +~j ,

ˆrotξ~E = iωµ~E ,

divξ~E = ρ

divξ~H = 0,

divd ~j = iωρ, divξ~j = iωρ.

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towards a differential vector calculus in three...

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