Download The Mathematical Theory of Maxwell`s Equations

The Mathematical Theory of Maxwell’s Equations
Andreas Kirsch and Frank Hettlich
Department of Mathematics
Karlsruhe Institute of Technology (KIT)
Karlsruhe, Germany
c October 23, 2012
2
Contents
1 Introduction
7
1.1
Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
The Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Boundary and Radiation Conditions
. . . . . . . . . . . . . . . . . . . . . .
14
1.5
Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2 Expansion into Wave Functions
29
2.1
Separation in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . .
29
2.2
Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3
Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.4
The Boundary Value Problem for the Laplace Equation in a Ball . . . . . . .
54
2.5
Bessel Functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.6
The Boundary Value Problems for the Helmholtz Equation for a Ball . . . .
64
2.7
Spherical Vector Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.8
The Boundary Value Problems for Maxwell’s Equations for a Ball . . . . . .
73
Bibliography
75
3
4
CONTENTS
Preface
This book arose from a lecture on Maxwell’s equations given by the authors between ?? and
2009.
The emphasis is put on three topics which are clearly structured into Chapters 2, ??, and
??. In each of these chapters we study first the simpler scalar case where we replace the
Maxwell system by the scalar Helmholtz equation. Then we investigate the time harmonic
Maxwell’s equations.
In Chapter 1 we start from the (time dependent) Maxwell system in integral form and derive
...
5
6
CONTENTS
Chapter 1
Introduction
1.1
Maxwell’s Equations
Electromagnetic wave propagation is described by particular equations relating five vector
fields E, D, H, B, J and the scalar field ρ, where E and D denote the electric field
(in V /m) and electric displacement (in As/m2 ) respectively, while H and B denote the
magnetic field (in A/m) and magnetic flux density (in V s/m2 = T =Tesla). Likewise,
J and ρ denote the current density (in A/m2 ) and charge density (in As/m3 ) of the
medium. Here and throughout the lecture we use the rationalized MKS-system, i.e. V ,
A, m and s. All fields will be assumed to depend both on the space variable x ∈ R3 and on
the time variable t ∈ R.
The actual equations that govern the behavior of the electromagnetic field, first completely
formulated by Maxwell, may be expressed easily in integral form. Such a formulation has the
advantage of being closely connected to the physical situation. The more familiar differential
form of Maxwell’s equations can be derived very easily from the integral relations as we will
see below.
In order to write these integral relations, we begin by letting S be a connected smooth surface
with boundary ∂S in the interior of a region Ω0 where electromagnetic waves propagate. In
particular, we require that the unit normal vector ν(x) for x ∈ S be continuous and directed
always into “one side” of S, which we call the positive side of S. By τ (x) we denote the unit
vector tangent to the boundary of S at x ∈ ∂S. This vector, lying in the tangent plane of S
together with a vector n(x), x ∈ ∂S, normal to ∂S is oriented so as to form a mathematically
positive system (i.e. τ is directed counterclockwise when we sit on the positive side of S, and
n(x) is directed to the outside of S). Furthermore, let Ω ∈ R3 be an open set with boundary
∂Ω and outer unit normal vector ν(x) at x ∈ ∂Ω. Then Maxwell’s equations in integral form
7
8
CHAPTER 1. INTRODUCTION
state:
Z
d
H · τ dℓ =
dt
E · τ dℓ = −
d
dt
ZZ
D · ν ds =
J · ν ds (Ampère’s Law)
(1.1a)
S
Z
B · ν ds (Law of Induction)
(1.1b)
ρ dx
(Gauss’ Electric Law)
(1.1c)
Ω
∂Ω
Z
Z
S
∂S
Z
D · ν ds +
S
∂S
Z
Z
B · ν ds = 0 (Gauss’ Magnetic Law)
(1.1d)
∂Ω
To derive the Maxwell’s equations in differential form we consider a region Ω0 where µ and ε
are constant (homogeneous medium) or at least continuous. In regions where the vector
fields are smooth functions we can apply the Stokes and Gauss theorems for surfaces S and
solids Ω lying completely in Ω0 :
Z
curl F · ν ds =
S
Z
F · τ dℓ (Stokes),
(1.2)
F · ν ds (Gauss),
(1.3)
∂S
ZZ
div F dx =
Ω
Z
∂Ω
where F denotes one of the fields H, E, B or D. We recall the differential operators (in
cartesian coordinates):
div F(x) =
3
X
∂Fj
j=1




curl F(x) = 



∂xj
(x)
∂F3
(x) −
∂x2
∂F1
(x) −
∂x3
∂F2
(x) −
∂x1
(divergenz, “Divergenz”)
∂F2
(x)
∂x3
∂F3
(x)
∂x1
∂F1
(x)
∂x2








(curl, “Rotation”) .
With these formulas we can eliminate the boundary integrals in (1.1a-1.1d). We then use
the fact that we can vary the surface S and the solid Ω in D arbitrarily. By equating the
integrands we are led to Maxwell’s equations in differential form so that Ampère’s Law,
the Law of Induction and Gauss’ Electric and Magnetic Laws, respectively, become:
9
1.1. MAXWELL’S EQUATIONS
∂B
+ curlx E = 0
(Faraday’s Law of Induction, “Induktionsgesetz”)
∂t
∂D
− curlx H = −J (Ampere’s Law, “Durchflutungsgesetz”)
∂t
divx D = ρ
(Gauss’ Electric Law, “Coulombsches Gesetz”)
divx B = 0
(Gauss’ Magnetic Law)
We note that the differential operators are always taken w.r.t. the spacial variable x (not
w.r.t. time t!). Therefore, in the following we often drop the index x.
Physical remarks:
• The law of induction describes how a time-varying magnetic field effects the electric
field.
• Ampere’s Law describes the effect of the current (external and induced) on the magnetic field.
• Gauss’ Electric Law describes the sources of the electric displacement.
• The forth law states that there are no magnetic currents.
• Maxwell’s equations imply the existence of electromagnetic waves (as ligh, X-rays, etc)
in vacuum and explain many electromagnetic phenomena.
• Literature wrt physics: J.D. Jackson, Klassische Elektrodynamik, de Gruyter Verlag
Historical Remark:
• Dates: André Marie Ampère (1775–1836), Charles Augustin de Coulomb (1736–1806),
Michael Faraday (1791–1867), James Clerk Maxwell (1831–1879)
• It was the ingeneous idea of Maxwell to modify Ampere’s Law which was known up to
that time in the form curl H = J for stationary currents. Furthermore, he collected
the four equations as a consistent theory to describe the electromagnetic fields. (James
Clerk Maxwell, Treatise on Electricity and Magnetism, 1873).
Conclusion 1.1 Gauss’ Electric Law and Ampere’s Law imply the equation of continuity
∂ρ
∂D
= div
= div curl H − J = − div J
∂t
∂t
since div curl = 0.
10
1.2
CHAPTER 1. INTRODUCTION
The Constitutive Equations
In this general setting the equation are not yet consistent (more unknown than equations).
The Constitutive Equations couple them:
D = D(E, H) and B = B(E, H)
The electric properties of material are complicated. In general, they not only depend on the
molecular character but also on macroscopic quantities as density and temperature of the
material. Also, there are time-dependent dependencies as, e.g., the hysteresis effect, i.e. the
fields at time t depend also on the past.
As a first approximation one starts with representations of the form
D = E + 4πP
and B = H − 4πM
where P denotes the electric polarisation vector and M the magnetization of the material.
These can be interpreted as mean values of microscopic effects in the material. Analogously,
ρ and J are macroskopic mean values of the free charge and current densities in the medium.
If we ignore ferro-electric and ferro-magnetic media and if the fields are small one can model
the dependencies by linear equations of the form
D = εE
and B = µH
with matrix-valued functions ε : R3 → R3×3 (dielectric tensor), and µ : R3 → R3×3
(permeability tensor). In this case we call the media inhomogenous and anisotropic.
The special case of an isotropic medium means that polarization and magnetisation do
not depend on the directions. In this case they are just real valued functions, and we have
D = εE
and B = µH
with functions ε, µ : R3 → R.
In the simplest case these functions ε and µ are constant. This is the case, e.g., in vacuum.
We indicated already that also ρ and J can depend on the material and the fields. Therefore,
we need a further equation. In conducting media the electric field induces a current. In a
linear approximation this is described by Ohm’s Law:
J = σE + Je
where Je is the external current density. For isotropic media the function σ : R3 → R is
called the conductivity.
Remark: If σ = 0 the the material is called dielectric. In vacuum we have σ = 0,
ε = ε0 ≈ 8.854 · 10−12 AS/V m, µ = µ0 = 4π · 10−7 V s/Am. In anisotropic media, also the
function σ is matrix valued.
11
1.3. SPECIAL CASES
1.3
Special Cases
Vacuum
In regions of vacuum with no charge distributions and (external) currents (i.e. (ρ = 0, Je =
0) the law of induction takes the form
µ0
∂H
+ curl E = 0 .
∂t
Differentiation wrt time t and use of Ampere’s Law yields
µ0
∂2H
1
curl curl H = 0 ,
+
2
∂t
ε0
i.e.
ε 0 µ0
∂2H
+ curl curl H = 0 .
∂t2
√
√
1/ ε0 µ0 has the dimension of velocity and is called the speed of light: c0 = ε0 µ0 .
From curl curl = ∇ div −∆ it follows that the components of H are solutions of the linear
wave equation
∂2H
c20 2 − ∆H = 0 .
∂t
Analogously, one derives the same equation for the electric field:
c20
∂2E
− ∆E = 0 .
∂t2
Remark: Heinrich Rudolf Hertz (1857–1894) showed also experimentally the existence of
electromagenetic waves about 20 years after Maxwell’s paper (in Karlsruhe!).
Electrostatics
If E is in some region Ω constant wrt time t (i.e. in the static case) the law of induction
reduces to
curl E = 0 in Ω .
Therefore, if Ω is simply connected there exists a potential u : Ω → R with E = −∇u in Ω.
Gauss’ Electric Law yields in homogeneous media the Poisson equation
ρ = div D = − div (ε0 E) = −ε0 ∆u
for the potential u. The electrostatics is described by this basic elliptic partial differential
equation ∆u = −ρ/ε0 . Mathematically, this is the subject of potential theory.
12
CHAPTER 1. INTRODUCTION
Magnetostatics
The same technique does not work in magnetostatics since, in general, curl H 6= 0. However,
since
div B = 0
we conclude the existence of a vector potential A : R3 → R3 with B = − curl A in D.
Substituting this into Ampere’s Law yields (for homogeneous media Ω) after multiplication
with µ0
−µ0 J = curl curl A = ∇ div A − ∆A .
Since curl ∇ = 0 we can add gradients ∇u to A without changing B. We will see later that
we can choose u such that the resulting potential A satisfies div A = 0. This choice of
normalization is called Coulomb gauge.
With this normalization we also get in the magnetostatic case the Poisson equation
∆A = −µ0 J .
We note that in this case the Laplacian has to be taken component wise.
Time Harmonic Fields
Under the assumptions that the fields allow a Fourier transformation w.r.t. time we set
E(x; ω) = (Ft E)(x; ω) =
Z
E(x, t) eiωt dt ,
H(x; ω) = (Ft H)(x; ω) =
Z
H(x, t) eiωt dt ,
R
R
etc. We note that the fields E, H etc are now complex valued, i.e, E(·; ω), H(·; ω) : R3 → C3
(and also the other fields). Although they are vector fields we denote them by capital
Latin letters only. Maxwell’s equations transform into (since Ft (u′ ) = −iωFt u) the time
harmonic Maxwell’s equations
−iωB + curl E = 0 ,
iωD + curl H = σE + Je ,
div D = ρ ,
div B = 0 .
Remark: The time harmonic Maxwell’s equation can also be derived from the assumption
that all fields behave periodically w.r.t. time with the same frequency ω. Then the forms
E(x, t) = e−iωt E(x), H(x, t) = e−iωt H(x), etc satisfy the time harmonic Maxwell’s equations.
13
1.3. SPECIAL CASES
With the constitutive equations D = εE and B = µH we arrive at
−iωµH + curl E = 0 ,
(1.4a)
iωεE + curl H = σE + Je ,
(1.4b)
div (εE) = ρ ,
(1.4c)
div (µH) = 0 .
(1.4d)
Eliminating H or E, respectively, from (1.4a) and (1.4b) yields
curl
and
curl
1
curl E
iωµ
1
curl H
iωε − σ
+ (iωε − σ) E = Je .
+ iωµ H = curl
1
Je ,
iωε − σ
(1.5)
(1.6)
respectively. Usually, one writes these equations in a slightly different way by introducing the
constant values ε0 > 0 and µ0 > 0 in vacuum and relative values (dimensionless!) µr , εr ∈ R
and εc ∈ C, defined by
µr =
µ
,
µ0
εr =
ε
,
ε0
εc = ε r + i
σ
.
ωε0
Then equations (1.5) and (1.6) take the form
1
curl E − k 2 εc E = iωµ0 Je ,
curl
µr
1
1
2
curl H − k µr H = curl
Je ,
curl
εc
εc
(1.7)
(1.8)
√
with the wave number k = ω ε0 µ0 . In vacuum we have εc = 1, µr = 1 and thus
curl curl E − k 2 E = iωµ0 Je ,
(1.9)
curl curl H − k 2 H = curl Je .
(1.10)
Example 1.2 In the case Je = 0 and in vacuum the fields
E(x) = p eik d·x
and H(x) = (p × d) eik d·x
are solutions of the time harmonic Maxwell’s equations (1.9), (1.10) provided d is a unit
vector in R3 and p ∈ C3 with p · d = 01 . Such fields are called plane time harmonic fields
with polarization vector p ∈ C3 and direction d.
1
We set p · d =
P3
j=1
pj dj even for p ∈ C3
14
CHAPTER 1. INTRODUCTION
We make the assumption εc = 1, µr = 1 for the rest of Section 1.3. Taking the divergence of
these equations yield div H = 0 and k 2 div E = −iωµ0 div Je , i.e. div E = −(i/ωε0 ) div Je .
Comparing this to (1.4c) yields the time harmonic version of the equation of continuity
div Je = iωρ .
With the vector identity curl curl = −∆ + div ∇ equations (1.10) and (1.9) can be written
as
1
∇ρ ,
ε0
(1.11)
∆H + k 2 H = − curl Je .
(1.12)
∆E + k 2 E = −iωµ0 Je +
1.4
Boundary and Radiation Conditions
Maxwell’s equations hold only in regions with smooth parameter functions εr , µr and σ. If we
consider a situation in which a surface S separates two homogeneous media from each other,
the constitutive parameters ε, µ and σ are no longer continuous but piecewise continuous
with finite jumps on S. While on both sides of S Maxwell’s equations (1.4a)–(1.4d) hold,
the presence of these jumps implies that the fields satisfy certain conditions on the surface.
To derive the mathematical form of this behaviour (the boundary conditions) we apply the
law of induction (1.1b) to a narrow rectangle-like surface R, containing the normal n to the
surface S and whose long sides C+ and C− are parallel to S and are on the opposite sides of
it, see the following figure.
15
1.4. BOUNDARY AND RADIATION CONDITIONS
When we let the height of the narrow sides, AA′R and BB ′ , approach zero then C+ and C−
∂
approach a curve C on S, the surface integral ∂t
B · ν ds will vanish in the limit since the
R
field remains finite (note, that the
R normal to R lying in the tangential plane
R normal ν is the
of S). Hence, the line integrals C E + · τ dℓ and C E − · τ dℓ must be equal. Since the curve
C is arbitrary the integrands E + · τ and E − · τ coincide on every arc C, i.e.
n × E + − n × E − = 0 on S .
(1.13)
A similar argument holds for the magnetic field in (1.1a) if the current distribution J =
σE + Je remains finite. In this case, the same arguments lead to the boundary condition
n × H+ − n × H− = 0 on S .
(1.14)
If, however, the external current distribution is a surface current, i.e. if Je is of the form
Je (x + τ n(x)) = Js (x)δ(τ ) for small
R τ and x ∈ S and withRtangential surface field Js and σ
is finite, then the surface integral R Je · ν ds will tend to C Js · ν dℓ, and so the boundary
condition is
n × H+ − n × H− = Js
on S .
(1.15)
We will call (1.13) and (1.14) or (1.15) the transmission boundary conditions.
A special and very important case is that of a perfectly conducting medium with boundary S. Such a medium is characterized by the fact that the electric field vanishes inside this
medium, and (1.13) reduces to
n × E = 0 on S
(1.16)
Another important case is the impedance- or Leontovich boundary condition
n × H = λ n × (E × n) on S
(1.17)
which, under appropriate conditions, may be used as an approximation of the transmission
conditions.
The same kind of boundary occur also in the time harmonic case (where we denote the fields
by capital Latin letters).
Finally, we specify the boundary conditions to the E- and H-modes derived above. We
assume that the surface S is an infinite cylinder in x3 −direction with constant cross section.
Furthermore, we assume that the volume current density J vanishes near the boundary S and
that the surface current densities take the form Js = js ẑ for the E-mode and Js = js ν × ẑ
for the H-mode. We use the notation [v] := v|+ − v|− for the jump of the function v at the
boundary. Also, we abbreviate (only for this table) σ ′ = σ − iωε. We list the boundary
conditions in the following table.
16
CHAPTER 1. INTRODUCTION
Boundary condition
E-mode
transmission
impedance
H-mode
∂u µ ∂ν = 0 on S ,
[u] = 0 on S ,
′ ∂u σ ∂ν = −js on S ,
[u] = js on S ,
= −js on S , k 2 u − λ iωµ ∂u
= js on S ,
λ k 2 u + σ ′ ∂u
∂ν
∂ν
perfect conductor
∂u
∂ν
u = 0 on S ,
= 0 on S .
The situation is different for the normal components. We consider Gauss’ Electric and
Magnetic Laws and choose Ω to be a box which is separated by S into two parts Ω1 and Ω2 .
We apply (1.1c) first to all of Ω and then to Ω1 and Ω2 separately. The addition of the last
two formulas and the comparison with the first yields that the normal component D · n has
to be continuous as well as (application of (1.1d)) B · n. With the constitutive equations one
gets
n · (εr,1 E1 − εr,2 E2 ) = 0 on S
and n · (µr,1 H1 − µr,2 H2 ) = 0 on S .
Conclusion 1.3 The normal components of E and/or H are not continuous at interfaces
where εc and/or µr have jumps.
The Silver-Müller radiation condition
Reference Problems
During our course we will consider two classical boundary value problems.
• (Cavity with an ideal conductor as boundary) Let D ⊆ R3 be a bounded domain
with sufficiently smooth boundary ∂D and exterior unit normal vector ν(x) at x ∈ ∂D.
Let Je : D → C3 be a vector field. Determine a solution (E, H) of the time harmonic
Maxwell system
curl E − iωµH = 0 in D ,
curl H + (iωε − σ)E = Je
in D ,
ν × E = 0 on ∂D .
(1.18a)
(1.18b)
(1.18c)
17
1.4. BOUNDARY AND RADIATION CONDITIONS
ε c , µr
D
ν×E =0
• (Scattering by an ideal conductor) Given a bounded region D and some solution
E i and H i of the “unperturbed” time harmonic Maxwell system
curl E i − iµ0 H i = 0 in R3 ,
curl H i + iε0 E i = 0 in R3 ,
determine E, H of the Maxwell system
curl E − iµ0 H = 0 in R3 \ D ,
curl H + iε0 E = 0 in R3 \ D ,
such that E satisfies the boundary condition ν × E = 0 on ∂D, and E and H have the
decompositions into E = E s + E i and H = H s + H i in R3 \ D with some scattered
field E s , H s which satisfy the Silver-Müller radiation condition
r
ε0 s
x
lim |x| H (x) ×
−
E (x)
= 0
|x|→∞
|x|
µ0
r
µ0 s
x
s
+
H (x)
= 0
lim |x| E (x) ×
|x|→∞
|x|
ε0
s
uniformly with respect to all directions x/|x|.
Remark: For general µr , εc ∈ L∞ (R3 ) we have to give a correct interpretation of the
differential equations (“variational or weak formulation”) and transmission conditions
(“trace theorems”).
18
CHAPTER 1. INTRODUCTION
A
A
A
A
A
A E i , H i
A
A
A
A
A
A
D
ν×E =0
H
HH
1.5
H
HH
j
E s, H s
Vector Calculus
In this subsection we collect the most important formulas from vector calculus.
Table of Differential Operators and Their Properties
We assume that all functions are sufficiently smooth. In cartesian coordinates:
Operator
Application to function
∇
, ∂u , ∂u )⊤
∇u = ( ∂u
∂x ∂x2 ∂x3
div = ∇·
curl = ∇×
∆ = div ∇ = ∇ · ∇
∇·A=
∇×
3
X
∂Aj
∂xj
j=1
∂A3
2
− ∂A
∂x2
∂x3
1
3
− ∂A
A =  ∂A
∂x3
∂x1
∂A2
∂A1
−
P∂x3 1 ∂ 2 u∂x2
∆u = j=1 ∂x2
j



The following formulas, which can be obatined from straightforward calculations, will be
used often and have been used already:
19
1.5. VECTOR CALCULUS
For x, y, z ∈ C3 , λ : C3 → C und A, B : C3 → C3 we have
x · (y × z) = y · (z × x) = z · (x × y)
x × (y × z) = (x · z)y − (x · y)z
(1.19)
(1.20)
curl ∇u = 0
(1.21)
div curl A = 0
(1.22)
curl curl A = ∇ div A − ∆A
(1.23)
div (λA) = A · ∇λ + λ div A
(1.24)
curl(λA) = ∇λ × A + λ curl A
(1.25)
∇(A · B) = (A · ∇)B + (B · ∇)A + A × (curl B) + B × (curl A)
(1.26)
div (A × B) = B · curl A − A · curl B
(1.27)
curl(A × B) = A div B − B div A + (B · ∇)A − (A · ∇)B
(1.28)
For completeness we add the expressions of the differential operators ∇, div , curl, and ∆ in
other coordinate systems. Let f : R3 → C be a scalar function and F : R3 → C3 a vector
field.
Cylindrical Coordinates

r cos ϕ
x =  r sin ϕ 
z

Let ẑ = (0, 0, 1)⊤ and r̂ = (cos ϕ, sin ϕ, 0)⊤ and ϕ̂ = (− sin ϕ, cos ϕ, 0)⊤ be the coordinate
unit vectors. Let F = Fr r̂ + Fϕ ϕ̂ + Fz ẑ. Then
∇f (r, ϕ, z) =
∂f
1 ∂f
∂f
r̂ +
ϕ̂ +
ẑ ,
∂r
r ∂ϕ
∂z
1 ∂Fϕ
∂Fz
1 ∂(rFr )
+
+
,
r ∂r
r ∂ϕ
∂z
1 ∂Fz ∂Fϕ
∂Fr ∂Fz
1 ∂(rFϕ ) ∂Fθ
curl F (r, ϕ, z) =
−
−
−
r̂ +
ϕ̂ +
ẑ ,
r ∂ϕ
∂z
∂z
∂r
r
∂θ
∂ϕ
1 ∂
∂2f
∂f
1 ∂2f
∆f (r, ϕ, z) =
+
.
r
+ 2
r ∂r
∂r
r ∂ϕ2
∂z 2
div F (r, ϕ, z) =
Spherical Coordinates


r sin θ cos ϕ
x =  r sin θ sin ϕ 
r cos θ
Let r̂ = (sin θ cos ϕ, sin θ sin ϕ, cos θ)⊤ and θ̂ = (cos θ cos ϕ, cos θ sin ϕ, − sin θ)⊤ and
ϕ̂ = (− sin θ sin ϕ, sin θ cos ϕ, 0)⊤ be the coordinate unit vectors. Let F = Fr r̂ + Fθ θ̂ + Fϕ ϕ̂.
20
CHAPTER 1. INTRODUCTION
Then
∇f (r, θ, ϕ) =
∂f
1 ∂f
1 ∂f
θ̂ +
r̂ +
ϕ̂ ,
∂r
r ∂θ
r sin θ ∂ϕ
1 ∂(sin θ Fθ )
1 ∂Fϕ
1 ∂(r2 Fr )
+
+
,
2
r
∂r
r sin θ
∂θ
r sin θ ∂ϕ
1
∂(sin θ Fϕ ) ∂Fθ
1 ∂Fr ∂(rFϕ )
1
curl F (r, θ, ϕ) =
−
−
θ̂ +
r̂ +
r sin θ
∂θ
∂ϕ
r sin θ ∂ϕ
∂r
1 ∂(rFθ ) ∂Fr
−
ϕ̂ ,
+
r
∂r
∂θ
∂2f
∂
1 ∂
1
∂f
1
2 ∂f
.
∆f (r, θ, ϕ) = 2
r
+ 2
sin θ
+ 2 2
r ∂r
∂r
r sin θ ∂θ
∂θ
r sin θ ∂ϕ2
div F (r, θ, ϕ) =
Elementary Facts from Differential Geometry
Before we recall the basic integral identity of Gauss and Green we have to define rigourously
the notion of domain with C n −boundaries. We denote by B(x, r) := Bj (x, r) := {y ∈ Rj :
|y − x| < r} and B[x, r] := Bj [x, r] := {y ∈ Rj : |y − x| ≤ r} the open and closed ball,
respectively, of radius r > 0 centered at x in Rj for j = 2 or j = 3.
Definition 1.4 We call a region D ⊂ R3 to be C n -smooth (i.e. D ∈ C n ), if there exists a
finite number of open sets Uj ⊂ R3 , j = 1, . . . , m, and bijective mappings Ψ̃j from the closed
unit ball B3 [0, 1] := {u ∈ R3 : |u| ≤ 1} onto U j such that
(i) ∂D ⊂
m
S
Uj ,
j=1
n
(ii) Ψ̃j ∈ C n (B[0, 1]) and Ψ̃−1
j ∈ C (U j ) for all j = 1, . . . , m,
(iii) det Ψ̃′j (u) 6= 0 for all |u| ≤ 1, j = 1, . . . , m, where Ψ̃′j (u) ∈ R3×3 denotes the Jacobian
of Ψ̃j at u,
(iv) it holds that
3
+
Ψ̃−1
j (Uj ∩ D) = B3 (0, 1) := {u ∈ R : |u| < 1, u3 > 0} ,
3
Ψ̃−1
j (Uj ∩ ∂D) = {u ∈ R : |u| < 1, u3 = 0} .
We call {Uj , Ψ̃j : j = 1, . . . , m} a coordinate system of ∂D.
The restriction Ψj of the mapping Ψ̃j to B2 [0, 1] × {0} ⊂ B3 [0, 1] yields a parametrization
of ∂D ∩ U j in the form x = Ψj (u) = Ψ̃j (u1 , u2 , 0), u ∈ B2 [0, 1].
∂Ψ
∂Ψ
(u) and ∂u
(u), are tangential vectors at x = Ψ(u).
If Ψ is one of the mappings Ψj then ∂u
1
2
21
1.5. VECTOR CALCULUS
They are linearly independent since det Ψ̃′ (u1 , u2 , 0) 6= 0 and, therefore, span the tangent
plane at x = Ψ(u). The unit vectors
∂Ψ
(u) ×
ν(x) = ± ∂u1
∂Ψ
(u) ×
∂u
1
∂Ψ
(u)
∂u2
∂Ψ
(u)
∂u2
for x = Ψ(u) ∈ ∂D ∩ Uj are orthogonal to the the tangent plane, thus normal vectors. The
sign is chosen such that ν(x) is directed into the exterior of D (i.e. x + tν(x) ∈ Uj \ D for
small t > 0). The unit vector ν(x) is called the exterior unit normal Rvector. For such
domains and continuous functions f : ∂D → C the surface integral ∂D f (x) ds exists.
First, we need the following tool:
∞
3
Lemma 1.5 For every covering
Pm {Uj : j = 1, . . . , m} of ∂D there exist φj ∈ C (R ) with
supp(φj ) ⊂ Uj for all j and j=1 φj (x) = 1 for all x ∈ ∂D (Partition of Unity).
R
R
P R
With such a partion of unity we write ∂D f (x) ds in the form ∂D f ds = m
j=1 ∂D∩Uj φj f ds =
Pm R
˜
˜
j=1 ∂D∩Uj fj (x) ds with fj = φj f . Using a coordinate system {Uj , Ψj : j = 1, . . . , m} of
∂D as in Definition 1.4 the integral over the surface patch Uj ∩ ∂D is given by
Z
Z
∂Ψj
∂Ψ
j
f˜j (x) ds =
f˜j Ψj (u) (u) ×
(u) du .
∂u1
∂u2
Uj ∩∂D
B2 (0,1)
We collect important properties of the smooth domain D in the following lemma.
Lemma 1.6 Let D ∈ C 2 . Then there exists c0 > 0 such that
(a) ν(y) · (y − z) ≤ c0 |z − y|2 for all y, z ∈ ∂D,
(b) ν(y) − ν(z) ≤ c0 |y − z| for all y, z ∈ ∂D.
(c) Define
Hρ :=
z + tν(z) : z ∈ ∂D , |t| < ρ .
Then there exists ρ0 > 0 such that for all ρ ∈ (0, ρ0 ] and every x ∈ Hρ there exist
unique (!) z ∈ ∂D and |t| ≤ ρ with x = z + tν(z). The set Hρ is an open neighborhood
of ∂D for every ρ ≤ ρ0 . Furthermore, z − tν(z) ∈ D and z + tν(z) ∈
/ D for 0 < t < ρ
and z ∈ ∂D.
One can choose ρ0 such that for all ρ ≤ ρ0 the following holds:
• |z − y| ≤ 2|x − y| for all x ∈ Hρ and y ∈ ∂D, and
• |z1 − z2 | ≤ 2|x1 − x2 | for all x1 , x2 ∈ Hρ .
If Uδ := x ∈ R3 : inf z∈∂D |x − z| < δ denotes the strip around ∂D then there exists
δ > 0 with
Uδ ⊂ Hρ0 ⊂ Uρ0
(1.29)
22
CHAPTER 1. INTRODUCTION
(d) There exists r0 > 0 such that the surface area of ∂B(z, r) ∩ D for z ∈ ∂D can be
estimated by
|∂B(z, r) ∩ D| − 2πr2 ≤ 4πc0 r3 for all r ≤ r0 .
(1.30)
S
S
Proof: We make use of a finite covering Uj of ∂D, i.e. we write ∂D = (Uj ∩ ∂D) and
use local coordinates Ψj : R2 ⊃ B2 [0, 1] → R3 which yields the parametrization of ∂D ∩ Uj .
First, it is easy to see (proof by contradiction) that there exists δ > 0 with the property
3
that for every pair
(z, x) ∈ ∂D × R with |z − x| < δ there exists Uj with z, x ∈ Uj . Let
diam(D) = sup |x1 − x2 | : x1 , x2 ∈ D be the diameter of D.
(a) Let x, y ∈ ∂D and assume first that |y − x| ≥ δ. Then
ν(y) · (y − x) ≤ |y − x| ≤ diam(D) δ 2 ≤ diam(D) |y − x|2 .
δ2
δ2
Let now |y − x| < δ. Then there exists Uj with y, x ∈ Uj . Let x = Ψj (u) and y = Ψj (v).
Then
∂Ψ
∂Ψj
(u) × ∂u2j (u)
ν(x) = ± ∂u1
∂Ψj
∂Ψj
∂u1 (u) × ∂u2 (u)
and, by the definition of the derivative,
y − x = Ψj (v) − Ψj (u) =
2
X
k=1
(vk − uk )
∂Ψj
(u) + a(v, u)
∂uk
with a(v, u) ≤ c|u − v|2 for all u, v ∈ Uj and some c > 0. Therefore,
2
X
∂Ψj
∂Ψj
∂Ψj 1
ν(x) · (y − z) ≤ (u) ×
(u) ·
(u)
(vk − uk ) ∂Ψj
∂Ψ
∂u1
∂u2
∂uk
∂u1 (u) × ∂u2j (u) k=1
|
{z
}
= 0
∂Ψj
∂Ψj
1
(u)
×
(u)
·
a(v,
u)
+ ∂Ψj
∂Ψj
∂u
∂u
1
2
∂u1 (u) × ∂u2 (u)
2
−1
≤ c0 |x − y|2 .
≤ c |u − v|2 = c Ψ−1
j (x) − Ψj (y)
This proves part (a). The proof of (b) follows analogously from the differentiability of u 7→ ν.
(c) Choose ρ0 > 0 such that
(i) ρ0 c0 < 1/16 and
(ii) ν(x1 ) · ν(x2 ) ≥ 0 for x1 , x2 ∈ ∂D with |x1 − x2 | ≤ 2ρ0 and
S
(iii) Hρ0 ⊂ Uj .
Assume that x has two representation as x = z1 + t1 ν1 = z2 + t2 ν2 where we write νj for
ν(zj ). Then
1
|z1 − z2 | ,
|z1 − z2 | = (t2 − t1 ) ν2 + t1 (ν2 − ν1 ) ≤ |t1 − t2 | + ρ c0 |z1 − z2 | ≤ |t1 − t2 | +
16
23
1.5. VECTOR CALCULUS
thus |z1 − z2 | ≤
16
|t
15 1
− t2 | ≤ 2|t1 − t2 |. Furthermore, since ν1 · ν2 ≥ 0,
(ν1 + ν2 ) · (z1 − z2 ) = (ν1 + ν2 ) · (t2 ν2 − t1 ν1 ) = (t2 − t1 ) (ν1 · ν2 + 1) ,
|
{z
}
≥1
thus
|t2 − t1 | ≤ (ν1 + ν2 ) · (z1 − z2 ) ≤ 2c0 |z1 − z2 |2 ≤ 8c0 |t1 − t2 |2 ,
i.e. |t2 − t1 | 1 − 8c0 |t2 − t1 | ≤ 0. This yields t1 = t2 since 1 − 8c0 |t2 − t1 | ≥ 1 − 16c0 ρ > 0
and thus also z1 = z2 .
Let U be one of the sets Uj and Ψ : R2 ⊃ B2 (0, 1) → U ∩ ∂D the corresponding bijective
mapping. We define the new mapping F : R2 ⊃ B2 (0, 1) × (−ρ, ρ) → Hρ by
F (u, t) = Ψ(u) + t ν(u) ,
(u, t) ∈ B2 (0, 1) × (−ρ, ρ) .
For sufficiently small ρ the mapping F is one-to-one and satisfies det F ′ (u, t) ≥ c̃ > 0 on
B2 (0, 1) × (−ρ, ρ) for some c̃ > 0. Indeed, this follows from
⊤
∂ν
∂Ψ
∂ν
∂Ψ
′
(u) + t
(u) ,
(u) + t
(u) , ν(u)
F (u, t) =
∂u1
∂u1
∂u2
∂u2
and the fact that for t = 0 the matrix F ′ (u, 0) has full rank 3. Therefore,
F is a bijective
S
mapping from B2 (0, 1) × (−ρ, ρ) onto U ∩ Hρ . Therefore, Hρ = (Hρ ∩ Uj ) is an open
neighborhood of ∂D. This proves also that x = z − tν(z) ∈ D and x = z + tν(z) ∈
/ D for
0 < t < ρ.
For x = z + tν(z) and y ∈ ∂D we have
2
|x − y|2 = (z − y) + tν(z) ≥ |z − y|2 + 2t(z − y) · ν(z)
≥ |z − y|2 − 2ρc0 |z − y|2
≥
1
|z − y|2
4
since 2ρc0 ≤
3
.
4
Therefore, |z − y| ≤ 2|x − y|. Finally,
2
|x1 − x2 |2 = (z1 − z2 ) + (t1 ν1 − t2 ν2 ) ≥ |z1 − z2 |2 − 2 (z1 − z2 ) · (t1 ν1 − t2 ν2 )
≥ |z1 − z2 |2 − 2 ρ(z1 − z2 ) · ν1 − 2 ρ(z1 − z2 ) · ν2 ≥ |z1 − z2 |2 − 4 ρ c0 |z1 − z2 |2 = (1 − 4ρc0 ) |z1 − z2 |2 ≥
1
|z1 − z2 |2
4
since 1 − 4ρc0 ≥ 1/4.
The proof of (1.29) is simple and left as an exercise.
(d) Let c0 and ρ0 as in parts (a) and (c). Choose r0 such that B[z, r] ⊂ Hρ0 for all r ≤ r0
(which is possible by (1.29)) and ν(z1 ) · ν(z1 ) > 0 for |z1 − z2 | ≤ 2r0 . For fixed r ≤ r0 and
arbitrary z ∈ ∂D and σ > 0 we define
Z(σ) = x ∈ ∂B(z, r) : (x − z) · ν(z) ≤ σ
24
CHAPTER 1. INTRODUCTION
We show that
Z(−2c0 r2 ) ⊂ ∂B(z, r) ∩ D ⊂ Z(+2c0 r2 )
Let x ∈ Z(−2c0 r2 ) have the form x = x0 + tν(x0 ). Then
(x − z) · ν(z) = (x0 − z) · ν(z) + t ν(x0 ) · ν(z) ≤ −2c0 r2 ,
i.e.
t ν(x0 ) · ν(z) ≤ −2c0 r2 + (x0 − z) · ν(z) ≤ −2c0 r2 + c0 |x0 − z|2
≤ −2c0 r2 + 2c0 |x − z|2 = 0 ,
i.e. t ≤ 0 since |x0 − z| ≤ 2r and thus ν(x0 ) · ν(z) > 0. This shows x = x0 + tν(x0 ) ∈ D.
Analogously, for x = x0 − tν(x0 ) ∈ ∂B(z, r) ∩ D we have t > 0 and thus
(x − z) · ν(z) = (x0 − z) · ν(z) − t ν(x0 ) · ν(z) ≤ c0 |x0 − z|2 ≤ 2c0 |x − z|2 = 2c0 r2 .
Therefore, the surface area of ∂B(z, r) ∩ D is bounded from below and above by the surface
areas of Z(−2c0 r2 ) and Z(+2c0 r2 ), respectively. Since the surface area of Z(σ) is 2πr(r + σ)
we have
−4πc0 r3 ≤ |∂B(z, r) ∩ D| − 2πr2 ≤ 4πc0 r3 .
2
Integral identities
Now we can formulate the mentioned integral identities. We do it only in R3 . By C n (D)3 we
denote the space of vector fields F : D → C3 which are n−times continuously differentiable.
By C n (D)3 we denote the subspace of C n (D)3 that consists of those functions F which,
together with all derivatives up to order n, have continuous extentions to the closure D of
D.
Theorem 1.7 (Theorem of Gauss, Divergence Theorem)
Let D ⊂ R3 be a bounded domain which is C 2 −smooth. For F ∈ C 1 (D)3 ∩C(D)3 the identity
ZZ
Z
div F (x) dx =
Fν (x) ds
D
∂D
holds. In particular, the integral on the left hand side exists.
As a conclusion one derives the theorems of Green.
Theorem 1.8 (Green’s first and second theorem)
Let D ⊂ R3 be a bounded domain which is C 2 −smooth. Furthermore, let u, v ∈ C 2 (D) ∩
C 1 (D). Then
ZZ
Z
∂v
(u ∆v + ∇u · ∇v) dx =
u
ds ,
∂ν
D
∂D
ZZ
Z ∂u
∂v
−v
ds .
(u ∆v − ∆u v) dx =
u
∂ν
∂ν
D
∂D
25
1.5. VECTOR CALCULUS
Here, ∂u(x)/∂ν = ν(x) · ∇u(x) for x ∈ ∂D.
Proof: The first identity is derived from the divergence theorem be setting F = u∇v. Then
F satisfies the assumption of Theorem 1.7 and div F = u ∆v + ∇u · ∇v.
The second identity is derived by interchanging the roles of u and v in the first identity and
taking the difference of the two formulas.
2
We will also need their vector valued analoga.
Theorem 1.9 (Integral identities for vector fields)
Let D ⊂ R3 be a bounded domain which is C 2 −smooth. Furthermore, let A, B ∈ C 1 (D)3 ∩
C(D)3 and let u ∈ C 2 (D) ∩ C 1 (D). Then
ZZ
Z
curl A dx =
ν × A ds ,
(1.31a)
D
ZZ
D
∂D
(B · curl A − A · curl B) dx =
Z
(ν × A) · B ds ,
(1.31b)
ZZ
Z
u (ν · A) ds .
(1.31c)
D
(u div A + A · ∇u) dx =
∂D
∂D
Proof: For the first identity we consider the components separately. For the first one we
have


ZZ
ZZ ZZ
0
∂A3 ∂A2
(curl A)1 dx =
−
dx =
div  A3  dx
∂x2
∂x3
D
D
D
−A2


Z
Z
0


A3
=
ν·
ds =
(ν × A)1 ds .
∂D
∂D
−A2
For the other components it is proven in the same way.
For the second equation we set F = A × B. Then div F = B · curl A − A · curl B and
ν · F = ν · (A × B) = (ν × A) · B.
For the third identity we set F = uA and have div F = u div A + A · ∇u and ν · F = u(ν · A).
2
Surface Gradient and Surface Divergence
We have to introduce two more notions from differential geometry, the surface gradient
and surface divergence.
To continue we do need a decomposition of the vector Laplace operator. To this end we
define some tangential differential operators on the boundary ∂D of a C 2 smooth domain
D ⊆ R3 (). As an abbreviation we introduce the notation
E = Eν ν + Eτ
26
CHAPTER 1. INTRODUCTION
for the normal component Eν = E · ν and the projection Eτ = ν × (E × ν) on the tangential
plane of a vectorfield E on a surface ∂D with unit normal vector field ν.
Definition 1.10 Let ϕ ∈ C 1 (U ) be defined in a neighborhood U of the C 2 -boundary boundary
∂D of the domain D and h ∈ C 1 (U ) be a tangential vector field, i.e. hν = 0 for the unit
normal field ν on ∂D ∩ U .
(a) The surface gradient of ϕ is defined by
∇τ ϕ = (∇ϕ)τ = ∇ϕ −
∂ϕ
ν.
∂ν
(1.32)
(b) The surface divergence of h is given by
Div(h) = div(h) − ν · Jh ν.
(1.33)
where Jh denotes the Jacobian matrix of h.
Example 1.11 We parametrize the boundary of the unit sphere S 2 = {x ∈ R3 : kxk = 1}
by spherical coordinates
Ψ(θ, φ) =
sin θ cos φ, sin θ sin φ, cos θ)⊤ .
Then the surface gradient and surface divergence on S 2 are given by
Grad f (θ, φ) =
Div F (θ, φ) =
∂f
1 ∂f
(θ, φ) θ̂ +
(θ, φ) φ̂ ,
∂θ
sin θ ∂φ
1 ∂
1 ∂Fφ
sin θ Fθ (θ, φ) +
(θ, φ) ,
sin θ ∂θ
sin θ ∂φ
where θ̂ = (cos θ cos φ, cos θ sin φ, − sin θ)⊤ and φ̂ = (− sin θ sin φ, sin θ cos φ, 0)⊤ are the tangential vectors which span the tangent plane and Fθ , Fφ are the components of F w.r.t. these
vectors, i.e. F = Fθ θ̂ + Fφ φ̂. From this we note that
1 ∂2f
∂f
1 ∂
(θ, φ) .
sin θ (θ, φ) +
Div Grad f (θ, φ) =
sin θ ∂θ
∂θ
sin2 θ ∂φ2
This differential operator is called the Laplace-Beltrami operator and will be denoted by
∆S = Div Grad .
Since ∂D is the closed boundary of a bounded domain applying the divergence theorem leads
to teh surface divergence theorem.
Lemma 1.12 The surface divergence defined in (1.33) satisfies
Z
Div(h) ds = 0.
∂D
27
1.5. VECTOR CALCULUS
Proof Let ν ∈ C 2 (D) ∩ C 1 (D) be an extension of the normal vector ν to the bounded
domain D ⊆ R3 . The identity (1.28) leads to
ν · curl(ν × h) = div(h) − hν div(ν) + ν · Jν h − ν · Jh ν.
By differentiation of ν · ν = 1 on ∂D we obtain ν · Jν h = 0. And h · ν = 0 on ∂D yields
Div(h) = ν · curl(ν × h).
(1.34)
Finally the divergence theorem implies
Z
Z
Div(h) ds =
div(curl(ν × h)) dx = 0 .
∂D
D
2
From the product rule it follows that
Div(ϕh) = (∇τ ϕ) · h + ϕ Div(h).
The previous Lemma implies the partial integration formula
Z
Z
Div(h)ϕ ds = −
(∇τ ϕ) · h ds.
∂D
(1.35)
∂D
Lemma 1.13 The tangential gradient and the tangential divergence depend only on the
values ϕ|∂D and the tangential field h|∂D on ∂D, respectively.
Proof Let ϕ1 , ϕ2 ∈ C ∞ (U ) be such that ϕ = ϕ1 − ϕ2 vanishes on ∂D. The identity (1.35)
shows
Z
(∇τ ϕ) · h ds = 0
∂D
for all h ∈ C 1 (U ). Thus, choosing h ∈ C 1 (U ) with h|∂D = ∇τ ϕ yields ∇τ ϕ = 0 on ∂D. By a
density argument follows the equation for continuously differentiable functions ϕ. Similarly
the assertion for the tangential divergence is obtained.
2
...
Corollary 1.14 Let w ∈ C 1 (D)3 such that w, curl w ∈ C(D)3 . Then the surface divergence
of ν × w exists and is given by
Div(ν × w) = −ν · curl w
on ∂D .
(1.36)
Proof: Let ϕ ∈ C 1 (D) be arbitrary. By Gauss’ theorem:
Z
ZZ
ZZ
ϕ ν · curl w ds =
div (ϕ curl w) dx =
∇ϕ · curl w dx
∂D
D
=
ZZ
D
=
Z
∂D
D
div (w × ∇ϕ) dx =
Z
∂D
(ν × w) · Grad ϕ ds = −
The assertion follows since ϕ is arbitrary.
ν · (w × ∇ϕ) ds
Z
∂D
Div(ν × w) ϕ ds .
2