SM-55Partial Differential Equations Opt 1

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


5/15

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,


6/15

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.


9/15

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)


11/15

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.


13/15

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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SM-55Partial Differential Equations Opt 1