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7/31/2019 SM-55Partial Differential Equations Opt 1
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Figure 6.1: Characteristic coord
simple string.
Note that is constant alon
constant along the cha
the boundary segment ,
intersecting . Similarly, if
characteristics intersecting
on determine the values of
Being the sum of the
nate lines and as determined by the wav
the characteristics (i.e., where const
racteristics. It follows that if is
then is known along all the
is known along , then is known
. And this is precisely the case because t
both and on that segment.
two functions, the solution to the
equation for a
ant), while is
known on
-characteristics
long all the -
he Cauchy data
waveequation is
(616)
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Thus one sees that any disturb
right the other to the left.
characteristicsrelative to the -
string is that these two parts do n
A general linear hyperbolic syst
a simple string is that its solutio
two sets of characteristics whic
unique solution at every p
dependenceof . To justify t
general second order linear hype
System of Partial Diffe
Equations Using Linea
The theme of the ensuing develo
system of partial differential equ
solve them via the method of eig
the Maxwell system of p.d. equa
scalar equation64
where is a time and space de
master equation, or its manifesta
and magnetic fields are entirely
The starting point of the develop
the density of charge
and the charge flux
nce on a string consists of two parts: one pr
The propagation speeds are , the
coordinate system. The idiosyncratic aspe
ot change their shape as they propagate along t
m does not share this feature. However, what i
is uniquely determined in the common region
intersect . In fact, the Cauchy data on
oint in the region . This is why it is call
hese claims it is neccessary to construct this un
rbolic differential equation.
ential Equations: How to Solve
Algebra:
pment is linear algebra, but the subject is an ov
ations, namely, the Maxwell field equations. Th
envectors and eigenvalues. The benefit is that t
ions is reduced to solving a single inhomogene
endent source. The impatient reader will find t
ion in another coordinate system, has been solv
etermined as in Tables -6.9.
ment is Maxwell's equations. There is the set of
opagating to the
slopes of the
t of the simple
e string.
t does share with
traversed by the
determine a
d the domain of
que solution to a
axwell's
rdetermined
e objective is to
e task of solving
us
at once this
ed, the electric
four functions,
(637)
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which are usually given. These
electric and magnetic fields,
Maxwell's gift to twentieth centu
and
Maxwell's field equations65
.
Exercise 62.1 (Charge Flux-Den
Microscopic observations show tthe other hand, macroscopic obs
fluid which is continuous. Dirac
single perspective. This fact is hi
Consider the current-charge den
pace and time dependent charge distributions g
and . The relationship is captur
ry science and technology,
sity of an Isolated Charge)
hat charged matter is composed of discrete poirvations show that charged matter is the carrier
delta functions provide the means to grasp both
ghlighted by the following problem.
ity due to an isolated moving charge,
(638)
ive rise to
d by means of
(639)
(640)
(641)
(642)
t charges. Onof an electric
attributes from a
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a) Show that this current-charge
Remark. The four-vector
parameter is the `wristwatch'
charge.
b) By taking advantage of the fa
expressions for the components
Answer:
where
density satisfies
is the charge's four-velocity in spac
' time (as measured by a comoving clock) attac
t , evaluate the -integrals, and
and .
time. The
ed to this
btain explicit
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Maxwell Wave Equation:
The first pair of Maxwell's equat
vector potential and scalar p
theelectric and magnetic fields,
Conversely, the existence of thes
satisfied automatically. By apply
pair of Maxwell's equations, (6.4
where
It follows that Maxwell's field e
ions, (6.39) and (6.40), imply that there exists a
tential from which one derives
e potentials guarantees that the first pair of thes
ing these potentials to the differential expressio
1-6.42) one obtains the mapping
uations reduce to Maxwell's four-component w
(643)
(644)
e equations is
ns of the second
(645)
(646)
ave equation,
7/31/2019 SM-55Partial Differential Equations Opt 1
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Maxwell's wave operator is the l
the following properties:
1. It is a linearmap from th
at each point event .
2. The map is singular. This
and
3.
4. In particular, one has
1. the fact that
nch pin of his theory of electromagnetism. This
space of four-vector fields into itself, i.e.
means that there exist nonzero vectors and
(647)
is because it has
such that
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for all three-times differentiable
The null space of is therefor
2. the fact that
for all 4-vectors . Thus
or
scalar fields . Thus
nontrivial and 1-dimensional at each .
(649)
(650)
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The left null space of is ther
In light of the singular nature of
has no solution unless the source
This is the linear algebra way of
the differential law of charge c
law of charge conservatio
Maxwell's equations would be
mathematical way of expressin
established knowledge, and that
fore also 1-dimensional at each .
, the four-component Maxwell waveequatio
also satisfies
expressing
onservation. Thus Maxwell's equations apply i
holds. If charge conservation did
silent. They would not have a solution. S
g the fact that at its root theory is based on
arbitrary hypotheses must not contaminate the t
n
(651)
(652)
and only if the
ot hold, then
ch silence is a
observation and
eoretical.
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The Over determined System
The linear algebra aspects of Ma
following problem from linear al
Solve for , under t
The fact that is singular and
but consistent. This means that t
One solves the problem in two st
Step I:
Let be the set of eige
of finding three vectors that satis
and
:
xwell's wave operator are illustrated by the
gebra:
he stipulation that
belongs to the range of makes the system
ere are more equations than there are unknown
eps.
nvectors having non-zero eigenvalues. Whateve
fy
(653)
over-determined
s.
r is, the task
(654)
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Being spanned by the three eige
is well-determined. However, th
Step II:
Continuing the development, rec
and that if
then
vectors with non-zero eigenvalues, the range s
scalars are at this stage as yet undetermine
all that quite generally
(655)
ace of ,
.
(657)
(656)
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It is appropriate to alert the r
eigenvalues become differe
subscript labels will refer to
respectively.
Equating (6.56) and (6.57), one
following equations for ,
Consequently, the solution is
where is an indeterminate
If one represents the stated probl
in Figure 6.3,
ader that in the ensuing section the vec
tial operators which act on scalar fields a
the TE, TM, and TEM eletromagnetic66
finds that the linear independence of
, and :
multiple of the null space vector .
em ( determines ) as an input-o
tors and the
nd that the three
ector potentials
implies the
(658)
(659)
(660)
utput process, as
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Figure 6.3: The matrix defin
then its solution is represented b
Figure: The solution to
In general, the task of finding th
However, given the fact that the
is already known, one finds that
make the task quite easy, if not t
es an input-output process.
y the inverse input-output process as in Figure
defines an inverse input-output process.
e eigenvectors ofa 4 4 matrix can be a nont
solution to
he associated constraints on the eigenvectors,
ivial.
.4.
ivial task.
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Maxwell Wave Equation (cont
The above linear algebra two-ste
system is an invaluabl
Cylindrical Coordinates:
The benefits of the linear algebrbe extended by inspection from
the four-dimensional coordinate
sets of coordinate surfaces. For c
the transverse coordinates
the longitudinal coordinates
The transition from a rectilinear
the following symbols:
and
nued):
p analysis of an over determined (but consistent
e guide in solving Maxwell's wave equation
viewpoint applied to Maxwell'sequations canectilinear cartesian to cylindrical coordinates. T
system lends itself to being decomposed into tw
ylindricals these are spanned by
in the transverse plane, and
in the longitudinal plane.
o a cylindrical coordinate frame is based on the
)
(661)
his is because
o orthogonal
replacement of
(677)
(678)
(679)
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Such a replacement yields the ve
tangent to the coordinate lines. T
vector components.
This replacement is very powerf
calculations that went into exhib
and systems of equatio
in the next section when we appl
Applying it within the context o
potential four-vectors are as foll
1. for a source
2. the solution to the Maxw
for a source
the solution to the Maxwell field
and
for a source
ctor field components relative to an orthonorma
o emphasize this orthonormality, hats ( ) are
l. It circumvents the necessity of having to rep
ting the individual components of Maxwell's
s. We shall again take advantage of the power
y it to Maxwell's system relative to spherical co
cylindrical coordinates, one finds that the sour
ws:
ll field equations has the form
equations has the form
l (o.n.) basis
laced over the
at the previous
, ,
f this algorithm
ordinates.
e and the vector
(680)
(681)
(682)
(683)
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3. the solution to the Maxw
Spherical Coordinates:
One of the chief virtues of the ldirects attention to the system'
way to identify them in a comp
permits a 2+2 decompositi
surfaces. Spherical coordinates p
a sphere spanned by , w
The distinguishing feature
cylindrical coordinates, is that
spheres) are not congruent. Insdistance from the origin. This sc
and hence the Maxwell wave o
TE-TM-TEM decomposition of
ll field equations has the form
near algebra viewpoint applied to Maxwell's efundamental vector spaces and their prope
utational way happens when the underlying c
n into what amounts to longitudinal
rovide a nontrivial example of this. There a tra
ile the longitudinalcoordinates are .
of spherical coordinates, as compared to
oordinate rectangles on successive transverse
tead, they have areas that scale with the squaling alters the representation of the divergence
perator. Nevertheless, the eigenvalue method
he e.m. field readily accomodates these alterati
(684)
(685)
uations is that itties. The easiest
ordinate system
and transverse
sverse surface is
rectilinear or
surfaces (nested
are of the radialof a vector field
ith its resulting
ns.
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