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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
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Aim
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Aim
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Aim
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
inf
∞∑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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 )
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 )
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 )
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 )
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 .
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 .
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 .
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
03 )
d1(x,x0)
∇d1 [f ](x0) exists⇐⇒ ∇1[f ](x0) exists, ∇1 :=
∂
∂x1
∇d1 [f ](x0) = 1
c1(x01 ).∇1[f ](x0),
c1(x01 ) := lımh→0
φ(x01 +h)−φ(x0
1 )h = φ
′(x0
1 ).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
03 )
d1(x,x0)
∇d1 [f ](x0) exists⇐⇒ ∇1[f ](x0) exists, ∇1 :=
∂
∂x1
∇d1 [f ](x0) = 1
c1(x01 ).∇1[f ](x0),
c1(x01 ) := lımh→0
φ(x01 +h)−φ(x0
1 )h = φ
′(x0
1 ).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
03 )
d1(x,x0)
∇d1 [f ](x0) exists⇐⇒ ∇1[f ](x0) exists, ∇1 :=
∂
∂x1
∇d1 [f ](x0) = 1
c1(x01 ).∇1[f ](x0),
c1(x01 ) := lımh→0
φ(x01 +h)−φ(x0
1 )h = φ
′(x0
1 ).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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)).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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)).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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)).
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
Canonical Nablas
∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.
Metric Nablas
∇dx = ∇d
1e1 +∇d2e2 +∇d
3e3,
∇ξ = ∇1ξe1 + ∇2
ξe2 + ∇3ξe3 := c1(ξ1)
∂
∂ξ1e1 + c2(ξ2)
∂
∂ξ2e2 + c3(ξ3)
∂
∂ξ3e3.
∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
Canonical Nablas
∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.
Metric Nablas
∇dx = ∇d
1e1 +∇d2e2 +∇d
3e3,
∇ξ = ∇1ξe1 + ∇2
ξe2 + ∇3ξe3 := c1(ξ1)
∂
∂ξ1e1 + c2(ξ2)
∂
∂ξ2e2 + c3(ξ3)
∂
∂ξ3e3.
∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
Canonical Nablas
∇x = ∇1e1 +∇2e2 +∇3e3, ∇ξ = ∇ξ,1e1 +∇ξ,2e2 +∇ξ,3e3.
Metric Nablas
∇dx = ∇d
1e1 +∇d2e2 +∇d
3e3,
∇ξ = ∇1ξe1 + ∇2
ξe2 + ∇3ξe3 := c1(ξ1)
∂
∂ξ1e1 + c2(ξ2)
∂
∂ξ2e2 + c3(ξ3)
∂
∂ξ3e3.
∇ξ = Wϕ−1 ∇dx Wϕ, ∇ξ = Wϕ−1 ∇x Wϕ
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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].
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Vector calculus in continuum (Cont.)
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].
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
].
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
].
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
].
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Moisil-Teodoresco systems
f = f0 + ~f : R3x → H.
Canonical Moisil-Teodoresco systemsdivx ~f = 0,
gradx f0 + rotx ~f = 0,
divξ
~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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Moisil-Teodoresco systems
f = f0 + ~f : R3x → H.
Canonical Moisil-Teodoresco systemsdivx ~f = 0,
gradx f0 + rotx ~f = 0,
divξ
~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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Moisil-Teodoresco systems
f = f0 + ~f : R3x → H.
Canonical Moisil-Teodoresco systemsdivx ~f = 0,
gradx f0 + rotx ~f = 0,
divξ
~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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Operators of Moisil-Teodoresco
Operadores de Moisil-Teodoresco canonicos
Dx =3∑
k=1
∂
∂xkik , Dξ =
3∑k=1
∂
∂ξkik .
Metric operators of Moisil-Teodoresco
Ddx =
3∑k=1
1
ck(xk)
∂
∂xkik , Dξ =
3∑k=1
ck(ξk)∂
∂ξkik .
D2x = −∆x , [Dd ]2 = −∆d
x , D2ξ = −∆ξ, [Dξ]
2 = −∆ξ.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Operators of Moisil-Teodoresco
Operadores de Moisil-Teodoresco canonicos
Dx =3∑
k=1
∂
∂xkik , Dξ =
3∑k=1
∂
∂ξkik .
Metric operators of Moisil-Teodoresco
Ddx =
3∑k=1
1
ck(xk)
∂
∂xkik , Dξ =
3∑k=1
ck(ξk)∂
∂ξkik .
D2x = −∆x , [Dd ]2 = −∆d
x , D2ξ = −∆ξ, [Dξ]
2 = −∆ξ.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Operators of Moisil-Teodoresco
Operadores de Moisil-Teodoresco canonicos
Dx =3∑
k=1
∂
∂xkik , Dξ =
3∑k=1
∂
∂ξkik .
Metric operators of Moisil-Teodoresco
Ddx =
3∑k=1
1
ck(xk)
∂
∂xkik , Dξ =
3∑k=1
ck(ξk)∂
∂ξkik .
D2x = −∆x , [Dd ]2 = −∆d
x , D2ξ = −∆ξ, [Dξ]
2 = −∆ξ.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Quaternionic hyperholomorfy
Quaternionic hyperholomorfy
Dx [f ] =3∑
j=0
3∑k=1
∂fj∂xk
ik ij = −divx ~f + gradx f0 + rotx ~f .
[f ]Dx =3∑
j=0
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Quaternionic hyperholomorfy
Quaternionic hyperholomorfy
Dx [f ] =3∑
j=0
3∑k=1
∂fj∂xk
ik ij = −divx ~f + gradx f0 + rotx ~f .
[f ]Dx =3∑
j=0
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 ],
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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 ],
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
]ds =
∫Ωx
rotx ~f dVx .
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
]ds =
∫Ωx
rotx ~f dVx .
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Time-harmonic Maxwell system
Let ~E : R3x → R3
x , ~H : R3x → R3
x , ~E : R3ξ → R3
ξ , ~H : R3ξ → R3
ξ
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,
rotξ
~H = −iωε~E + ~j ,
rotξ~E = iωµ~H,
divξ~E = ρ
ε ,
divξ~H = 0.
and the continuity equation as:
divx ~j = iωρ divξ ~j = iωρ,
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
Time-harmonic Maxwell system
Let ~E : R3x → R3
x , ~H : R3x → R3
x , ~E : R3ξ → R3
ξ , ~H : R3ξ → R3
ξ
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,
rotξ
~H = −iωε~E + ~j ,
rotξ~E = iωµ~H,
divξ~E = ρ
ε ,
divξ~H = 0.
and the continuity equation as:
divx ~j = iωρ divξ ~j = iωρ,
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
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
ε ,
divξ~H = 0,
and the continuity equation as:
divd ~j = iωρ, divξ~j = iωρ.
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
Aim and motivationIntroduction
Some elements of fractal geometryMetric vector calculus
Vector calculus in continuumBasic facts of quaternionic analysis
Fundamental identitiesMaxwell equations
THANK YOU
ESIME-Zacatenco-IPN A differential vector calculus in continuum with fractal metric
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