Phys. 201 Methods of Theoretical Physics 1 - yu.edu.jo Department, Yarmouk University, Irbid Jordan Phys. 201 Methods of Theoretical Physics 1 Phys. 201 Methods of Theoretical Physics

Physics Department, Yarmouk University, Irbid Jordan

Phys. 201 Methods of Theoretical Physics 1


Supplements

Dr. Nidal M. Ershaidat

Table of Contents

Vector Space ............................................................................................................ 1

Vector Identities...................................................................................................... 2

The Curl ................................................................................................................... 3

Complex Numbers ................................................................................................. 7

Complex Logarithms ........................................................................................... 10

Determinants......................................................................................................... 11

Ordinary Differential Equations (ODE’s) ......................................................... 16

ODE – An Example .............................................................................................. 19

Exact Differential .................................................................................................. 20

Method of Undetermined Coefficient or Guessing Method.......................... 21

Method of Variation of Parameters ................................................................... 24

Fourier Series......................................................................................................... 27

Curvilinear Coordinates...................................................................................... 35



Dr. Nidal M. Ershaidat Doc. 1

1

Vector Space

A. The Concept

One of the fundamental concepts of linear algebra is the concept of vector

space. For example, many function sets studied in mathematical analysis are

with respect to their algebraic properties vector spaces. In analysis the

notion “linear space” is used instead of the notion “vector space”.

B. Definition

A set X is called a vector space over the number field KKKK if to every pair (x,y)

of elements of X there corresponds a sum x + y ∈ X, and to every pair (α,x)

where α ∈ KKKK and x ∈ X, there corresponds an element αx ∈ X, with the

properties 1-8:

1) x + y = y + x (commutability of addition);

2) x + (y + z) = (x + y) + z (associativity of addition);

3) ∃ 0 ∈ X such that 0 + x = x ∈ X (existence of null element);

4) ∀ x ∈ X ⇒∃ - x ∈ X: x + (-x) = 0 (existence of the inverse element);

5) 1 . x = x (unitarism);

6) α (βx) = (αβ) x (associativity with respect to number multiplication);

7) α (x + y) = α x + α y (distributivity with respect to vector addition);

8) (α + β ) x = α x + β x (distributivity with respect to number addition).

The properties 1-8 are called the vector space axioms. Axioms 1-4 show that

X is a commutative group or an Abelian group with respect to vector

addition. The second correspondence is called multiplication of the vector by

a number, and it satisfies axioms 5-8. Elements of a vector space are

called vectors. If KKKK = RRRR, then one speaks of a real vector space, and if KKKK =

CCCC, then of a complex vector space. Instead of the notion “vector space” we

shall use the abbreviative “space”.

Physics Department, Yarmouk University,

Irbid Jordan

Phys. 201 Mathematical Physics 1

Dr. Nidal M. Ershaidat Doc. A1

2

Vector Identities

In the following φ and ψ are scalar functions, →→→→A and

→→→→B are vectors.

→→→→→→→→→→→→→→→→→→→→→→→→→→→→

⋅⋅⋅⋅∇∇∇∇++++⋅⋅⋅⋅∇∇∇∇====

++++⋅⋅⋅⋅∇∇∇∇ BABA (1)

→→→→→→→→→→→→→→→→→→→→→→→→→→→→

××××∇∇∇∇++++××××∇∇∇∇====

++++××××∇∇∇∇ BABA (2)

××××∇∇∇∇φφφφ++++××××φφφφ∇∇∇∇====

φφφφ××××∇∇∇∇

→→→→→→→→→→→→→→→→→→→→→→→→AAA (3)

××××∇∇∇∇⋅⋅⋅⋅−−−−

××××∇∇∇∇⋅⋅⋅⋅====

××××⋅⋅⋅⋅∇∇∇∇

→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→BAABBA (4)

⋅⋅⋅⋅∇∇∇∇−−−−

⋅⋅⋅⋅∇∇∇∇++++

∇∇∇∇⋅⋅⋅⋅−−−−

∇∇∇∇⋅⋅⋅⋅====

××××××××∇∇∇∇

→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ABBABAABBA (5)

0====

××××∇∇∇∇⋅⋅⋅⋅∇∇∇∇

→→→→→→→→→→→→A (6)

φφφφ∆∆∆∆≡≡≡≡φφφφ∇∇∇∇====

φφφφ∇∇∇∇⋅⋅⋅⋅∇∇∇∇

→→→→→→→→ 2 (Laplacian) (7)

0====

φφφφ∇∇∇∇××××∇∇∇∇

→→→→→→→→ (8)

→→→→→→→→→→→→→→→→→→→→→→→→→→→→

∇∇∇∇−−−−

⋅⋅⋅⋅∇∇∇∇∇∇∇∇====

××××∇∇∇∇××××∇∇∇∇ AAA

2 (9)


Phys. 201 Methods of Theoretical Physics 1 Dr. Nidal M. Ershaidat Doc. 3

3

The Curl

A) Geometrical significance of the Curl

The divergence and curl of a vector field are two vector operators whose

basic properties can be understood geometrically by viewing a vector field as

the flow of a fluid or gas. Here we give an overview of basic properties of

curl than can be intuited from fluid flow. The curl of a vector field captures

the idea of how a fluid may rotate. Imagine that the below vector field F�

represents fluid flow (See Fig. 1). The vector field indicates that the fluid is

circulating around a central axis.

B) Measuring the Curl

In order to measure the density of some matter at a point, we measure the

mass (dm) of a small volume (dV) around the point, and then divide by the

volume. Mathematically, this can be expressed by:

dV

dm=ρ (1)

The smaller the volume, the better the approximation. Actually we define

the density ρ as being the limit:

dV

dm

dV 0lim

→=ρ (2)

A similar procedure is used to measure the strength of the rotation of a fluid.

If the vector field is interpreted as velocity of fluid flow, the fluid appears to

flow in circles.

This macroscopic circulation of fluid around circles actually is not what curl

measures. But, it turns out that this vector field also has curl, which we

might think of as “microscopic circulation”.

To test for curl, imagine that you immerse a small sphere into the fluid flow,

and you fix the center of the sphere at some point so that the sphere cannot

follow the fluid around (See Fig. 2).

4

(a) Time t (b) Time t’

Figure 1: The “rotating” vector field F�

at two different times.

Although you fix the center of the sphere, you allow the sphere to rotate in

any direction around its center point. The rotation of the sphere measures

the curl of the vector field F�

at the point in the center of the sphere. (The

sphere should actually be really really small, because, remember, the curl is

microscopic circulation.)

(a) Time t (b) Time t’

Figure 2: A small sphere immersed into the fluid flow.

The vector field F�

determines both in what direction the sphere rotates, and

the speed at which it rotates. We define the curl of F�

, denoted curl F�

, by a

vector that points along the axis of the rotation and whose length

corresponds to the speed of the rotation.

We can draw the vector corresponding to curl F�

as follows. We make the

length of the vector curl F�

proportional to the speed of the sphere's

rotation. The direction of curl F�

points along the axis of rotation, but we

need to specify in which direction along this axis the vector should point. We

will (arbitrarily?) set the direction of the curl vector by using the following

5

“right hand rule.” To see where curl F�

should point, curl the fingers of your

right hand in the direction the sphere is rotating; your thumb will point in

the direction of curl F�

. For our example, curl F�

is shown by the green

arrow.

Figure 3: The direction of curl F

�

is conventionally chosen using a right-hand rule.

The curl is a three-dimensional vector, and each of its three components

turns out to be a combination of derivatives of the vector field F�

. Once you

have the formula, calculating the curl of a vector field is a simple matter.

The curl is sometimes called the rotation, or "rot".

C) The Curl in different system of coordinates

The curl of a vector function is the vector product of the del operator with

this vector function:

The Curl in Cartesian coordinates

ky

F

x

Fj

x

F

z

Fi

z

F

y

FF xyzxyz ˆˆˆ

∂∂

−∂

∂+

∂∂

−∂

∂+

∂

∂−

∂∂

=×∇→→

(3)

where kji ˆ,ˆ,ˆ are unit vectors in the x, y, z directions. It can also be

expressed in determinant form:

zyx FFF

zyx

kji

∂∂

∂∂

∂∂

ˆˆˆ

(4)

The Curl in cylindrical polar coordinates

6

The curl in cylindrical polar coordinates, expressed in determinant form is:

zθr

θr

FFrF

zθr

r

k

r

∂∂

∂∂

∂∂

ˆ1

1

(5)

The Curl in spherical polar coordinates

The curl in spherical polar coordinates, expressed in determinant form is:

φθr FθrFrF

φθr

rθrθr

F

sin

1

sin

1

sin

12

∂∂

∂∂

∂∂

=×∇→→

(6)

Reference:

- http://hyperphysics.phy-astr.gsu.edu/hbase/curl.html

- (See the applets in) http://mathinsight.org/curl_idea


Phys. 201 Methods for Theoretical Physics 1


7

Complex Numbers

A. Definition of a complex number

A complex number z is defined by z = x + i y

where x and y are real numbers and 1−−−−====i .

x is called the real part of z and denoted x = Re(z)

y is called the imaginary part of z and denoted y = Im(z)

The form z = x + i y is called the rectangular form of the complex number z.

Note: In this document the letter z refers to a complex number and any

other letter, in particular, x and y, refer to real numbers or variables.

B. The Complex (Numbers) Space

Complex numbers define a space called the space of complex numbers or

simply the complex space. The eight axioms necessary to define a space in

linear analysis are verified, as follows:

1. Commutability of addition

For any two complex numbers z1 and z2 we define

z1 + z2 = z2 + z1 (∀∀∀∀ z1, z2)

2. Associativity of addition

For any set of complex numbers z1, z2 and z3 we define

z1 + (z2 + z3) = (z1 + z2) + z3

3. Existence of null element:

∃∃∃∃ 0 ∈∈∈∈ C such that 0 + z = z ∈ C

4. Existence of the inverse element:

∀ z ∈ C ⇒ ∃ - z ∈ C such that z + (-z) = 0

5. Unitarism:

∀ z ∈ C ⇒ 1 . z = z

And for (α, β and z ∈ C) we have:

6. Associativity with respect to number multiplication

α (β z) = (α β) z

7. Distributivity with respect to complex number addition

α (z1 + z2) = α z1 + α z2

8. Distributivity with respect to number addition

(α + β ) z = α z + β z

8

C. Complex conjugate of a complex number

The complex conjugate of a complex number z (z = x + i y) is defined by

z* = x - i y

D. Basic Operations in the Complex Space

Consider the complex numbers z1 = x1 + i y1 and z2 = x2 + i y2

- Addition

z = z1 + z2 = (x1 + x2) + i (y1 + y2) = z2 + z1

- Subtraction

z = z1 - z2 = (x1 - x2) + i (y1 - y2)

z' = z2 – z1 = (x2 – x1) + i (y2 – y1) # z1 - z2

-Multiplication

z = z1 . z2 = (x1 + i y1) . (x2 + i y2) = x1x2 + i x1 y1+ i x2 y2 - y1y2

-Division

22

11

yix

yix

2

1

z

zz

++++++++

========

(((( ))))22

22

21211221

22

22

22

11 .yx

yyyxiyxixx

yix

yix

yix

yix

++++

++++−−−−++++====

−−−−

−−−−

++++

++++====

22

22

2112

22

22

2121

yx

yxiyxi

yx

yyxx

++++

−−−−++++

++++

++++++++====

E. Graphical Representation of complex numbers

A complex number z = x + i y is represented in a 2D plane, called the Argand

Plane, by a point P whose coordinates are (x and y) (Fig. 1).

Figure 1: Graphical Representation of z

9

The vector →

OP represents the complex number z.

Fig. 1 representing z is the so-called Argand diagram of z.

The complex conjugate z* is represented by the vector →

*OP in Fig. 2.

Figure 2: Graphical Representation of z*

From Fig. 1 one can see that:

22 yxr +=

x

y1tan−=θ

r and θθθθ are called the polar coordinates of P and alternatively we have:

x = r cos θθθθ

y = r sin θθθθ

z can be written in the form:

z = r (cos θθθθ + i sin θθθθ)

Using Maclaurin’s series for eiθ, cosθ and sin θ we can write:

eiθ

= cos θ + i sin θ

z can be rewritten in the form:

z = r eiθ

Which we call the polar format of z. This form is very useful when defining

and using functions of complex variables.

F. Modulus (Magnitude) and Argument of z

r represents the magnitude or modulus of z and θ is called the argument of z.

And we write:

( ) 22 yxzModr +==

( )x

yz 1tanarg −==θ

© Nidal M. Ershaidat 2013

10

Complex Logarithms

The exponential function can be extended to a function which gives a

complex number as ex for any arbitrary complex number x; simply use the

infinite series with x complex. This exponential function can be inverted to

form a complex logarithm that exhibits most of the properties of the

ordinary logarithm. There are two difficulties involved: no x has ex = 0; and it

turns out that e2ππππi = 1 = e

0. Since the multiplicative property still works for the

complex exponential function, inzzee

ππππ++++==== 2 , for all complex z and integral n.

So the logarithm cannot be defined for the whole complex plane, and even

then it is multi-valued – any complex logarithm can be changed into an

"equivalent" logarithm by adding 2πi at will. The complex logarithm can only

be single-valued on the cut plane. For example, ln i = ½ πi or 5/2 π i or −3/2 π i,

etc.; and although i4 = 1, 4 log i can be defined as 2π i, or 10π i or −6 π i, and so

on.

z = Re(ln(x+iy)) z = |Im(ln(x+iy))| z = |ln(x+iy)|

Plots of the natural logarithm function on the complex plane (principal

branch)


Phys. 201 Methods for Theoretical Physics 1


11

Determinants

Consider the square matrix of order 2:

=

dcba A

The matrix A is invertible if and only if 0#cbda − . This number is called the

determinant of A. It is clear from this, that we would like to have a similar

result for bigger matrices (meaning higher orders). So is there a similar

notion of determinant for any square matrix, which determines whether a

square matrix is invertible or not?

In order to generalize such notion to higher orders, we will need to study the

determinant and see what kind of properties it satisfies. First let us use the

following notation for the determinant:

Determinant of cbdadcba

dcba

dcba −==

=

det

Properties of the Determinant

1. Any matrix A and its transpose have the same determinant, meaning

det A = det AT

This is interesting since it implies that whenever we use rows, a similar

behavior will result if we use columns. In particular we will see how row

elementary operations are helpful in finding the determinant. Therefore, we

have similar conclusions for elementary column operations.

2. The determinant of a triangular matrix is the product of the entries on the

diagonal, that is

dadc

a dba == 0

0.

3. If we interchange two rows, the determinant of the new matrix is the

opposite of the old one, that is

badc

dcba −= .

4. If we multiply one row with a constant, the determinant of the new matrix

is the determinant of the old one multiplied by the constant, that is

12

dλcλba

dcbaλ

dc

bλaλ == .

In particular, if all the entries in one row are zero, then the determinant is

zero.

5. If we add one row to another one multiplied by a constant, the

determinant of the new matrix is the same as the old one, that is

bλdaλcba

dcba

dc

dλbcλa++==++

.

Note that whenever you want to replace a row by something (through

elementary operations), do not multiply the row itself by a constant.

Otherwise, you will easily make errors (due to Property 4).

6. We have

det (AB) = det(A) det (B)

In particular, if A is invertible (which happens if and only if det (A) #0), then

( ) ( )AA

det1det 1- =

If A and B are similar, then det (A) = det(B).

Let us look at an example, to see how these properties work.

Example 1. Evaluate 3112

− .

Let us transform this matrix into a triangular one through elementary

operations. We will keep the first row and add to the second one the first

multiplied by ½. We get

2

70

12

3112 =− .

Using the Property 2, we get

72

72

2

70

12=⋅= .

Therefore, we have

3112

− = 7

which one may check easily.

13

Determinants of Matrices of Higher Order

As we said before, the idea is to assume that previous properties satisfied by

the determinant of matrices of order 2, are still valid in general.

So let us see how this works in case of a matrix of order 4.

Example 2. Evaluate

2113846287654321

.

We have

2113423187654321

2

2113846287654321

= .

If we subtract every row multiplied by the appropriate number from the first

row, we get

10850

0110

128404321

2113423187654321

−−−−

−−−=

We do not touch the first row and work with the other rows. We interchange

the second with the third to get

10850

12840

01104321

10850

0110

128404321

−−−−−−

−−=

−−−−

−−−.

If we subtract every row multiplied by the appropriate number from the

second row, we get

101300

121200

01104321

10850

12840

01104321

−−−−

−=

−−−−−−

−.

Using previous properties, we have

1013001100

01104321

12

101300

121200

01104321

−−

−−=

−−−−

−.

If we multiply the third row by 13 and add it to the fourth, we get

14

30001100

01104321

1013001100

01104321

−=

−−

−

which is equal to 3. Putting all the numbers together, we get

( ) ( ) .7231212

2113846287654321

=⋅−⋅−⋅=

These calculations seem to be rather lengthy. We will see later on that a

general formula for the determinant does exist.


111021

− .

In this example, we will not give the details of the elementary operations.

We have

.9300130021

321111021

==−


110211

−.

We have

.5500

010211

510110211

112110211

−=−

=−−

=−

General Formula for the Determinant

Let A be a square matrix of order n. Write A = (aij), where aij is the entry on

the row number i and the column number j, for i = 1, 2, 3, … n and j = 1, 2,

… n. For any i and j, set Cij (called the cofactors) to be the determinant of

the square matrix of order (n-1) obtained from A by removing the row

number i and the column number j multiplied by (-1)i+j. We have

( ) ∑=

=

=nj

jijijAdet

1

Ca

for any fixed i, and

( ) ∑=

=

=ni

i

ijij Ca1

Adet

15

for any fixed j. In other words, we have two types of formulas: along a row

(number i) or along a column (number j). Any row or any column will do.

The trick is to use a row or a column which has a lot of zeros.

In particular, we have along the rows

hgedc

kg

fdb

kh

fea

khg

fedcba

+−=

or

hgbaf

kgcae

khcbd

khg

fedcba

−+−= ,

or

edbak

fdcah

fecbg

khg

fedcba

+−= .

As an exercise write the formulas along the columns.


312123

− .

We will use the general formula along the third row. We have

12231

32130

31124

104

312123

+−−−=− ( ) ( ) 29431164 −=−+−−=

Which technique to evaluate a determinant is easier? The answer depends

on the person who is evaluating the determinant. Some like the elementary

row operations and some like the general formula. All that matters is to get

the correct answer.

Note that all of the above properties are still valid in the general case. Also

you should remember that the concept of a determinant only exists for

square matrices.

http://www.sosmath.com/matrix/determ0/determ1.html Author: M.A. Khamsi




Author’s email: [email protected] Address: Physics Department, Yarmouk University 21163 Irbid Jordan

16

Ordinary Differential Equations (ODE’s)

A) Definition

An ordinary differential equation is any equation which has exact

differentials or exact derivatives in both sides. The order of the highest

derivative is called the order of the differential equations. The solution of an

ODE is any relationship between the unknown function with its variable

which verifies the ODE.

B) First Order Differential Equations

Separable 1st order DE

A 1st order differential of the form:

( ) ( ) 0=+ dyyGdxxF (1)

Is called “separable 1st order DE”.

The general solution of Eq. 1 is obtained by integrating both sides, i.e.

( ) ( ) CdyyGdxxF =+ ∫∫ (2)

Linear 1st order DE

( ) ( )xQyxPdx

dy=+ (3)

If Q(x) = 0 then Eq. 3 can be written as:

( ) 0=+ dxxPy

dy (4)

which is a separable 1st order ODE (See Eq. 1), where ( ) ( )xPxF = and

( )y

yG1

= .

- Homogeneous Linear 1st order DE

Eq. 4 is also called homogeneous 1st order DE.

( ) 0=+ yxPdx

dy

The general solution of Eq. 4 is obtained by integrating both sides, i.e.

Author’s email: [email protected] Address: Physics Department, Yarmouk University 21163 Irbid Jordan

17

( ) ( ){ }∫−= dxxPAxy exp (5)

- Inhomogeneous Linear 1st order DE

( ) ( )xQyxPdx

dy=+ (6)

The general solution of Eq. 6 is given by:

( ) ( ) ( ) ( ) ( )xIxIxI eCdxxQeexy −− += ∫ (7)

where

( ) ( ) ( )dxxPexI xI∫= (8)

Exact Equations

A first order ordinary differential is called exact equation if its lhs can be

expressed as an exact differential dU of a function U(x,y) such that:

( ) ( ) 0=+= dyy,xNdxy,xMdU (9)

This is verified if and only of:

yxx

N

y

M

∂∂

=

∂

∂ (10)

xy

M

∂

∂ is the partial derivative of the function M(x,y) with respect to y. This

means that we derive the function M(x,y) considering x as a constant (which

we symbolize by the subscript x.

yx

N

∂∂ is the partial derivative of the function N(x,y) with respect to x. This

means that we derive the function N(x,y) considering y as a constant (which

we symbolize by the subscript y.

Example: if ( ) 232, yxyxM = then 226 yxx

M

y

=

∂

∂ and yx

y

M

x

34=

∂

∂.

Homogeneous First Order DE

A first order ordinary differential is called exact equation if the lhs can be

expressed as an exact differential dU of a function U(x,y) such that:

=x

yF

dx

dy (11)

© Nidal M. Ershaidat 2013 18

The general solution is:

( )∫∫ +−

= CvvF

dv

x

dx (12)

Where

( )xvxy = (13)

Second Order Differential Equations

Linear 2nd order DE

A 2nd order DE equation is linear if and only of it can be written in the form:

( ) ( ) ( )xRyxQdx

dyxP

dx

yd=++

2

2

(14)

Where P(x), Q(x) and R(x) are general functions of the independent variable

x,

Various methods are used to solve such equations. These methods depend

on the nature of P(x), Q(x) and R(x).

Reference: Phys. 201 Methods of Theoretical Physics 1 – Textbook Introduction to Mathematical Physics, 2nd Edition 2004, Authors: N. Laham and N. Ayoub




Author’s email: [email protected], Address: Physics Department, Yarmouk University 21163 Irbid Jordan © Nidal M. Ershaidat 2013

19

ODE – An Example

Example 1:

Solve 054 =+−′′ yy

This is an inhomogeneous linear 2nd order differential equation.

( )xRybyay =+′+′′ (1)

with:

4,0 −== ba and ( ) constantxR =−= 5

Its general solution y(x) is the sum yC(x) + yP(x) where yC(x) is the

complementary solution i.e. the solution of the corresponding homogeneous

equation ( 04 =−′′ yy ) and yP(x) is the (unique) particular solution obtained

using the method of undetermined coefficients.

( ) ( ) ( )xyxyxy PC += (2)

with

( ) ( ) ( )dxxFeexy xxP ∫ β−αβ= (3)

and

( ) ( )dxxRexF x∫ α−= (4)

Complementary Solution

( ) xxC eCeCxy βα += 21 (5)

Where α and β are the roots of the auxiliary equation (r2 – 4 = 0), i.e.

2=α and 2−=β (6)

Particular Solution

( ) ( ) xx edxexF α−α−

α=−= ∫

55 (7)

⇒⇒⇒⇒( ) ( )

4

555

5

−=βα

−=α

=

α=

∫

∫β−β

α−β−αβ

dxee

dxeeexy

xx

xxxP

(8)

which gives:

( )4

522

21 ++= − xx eCeCxy (9)




Author’s email: [email protected], Address: Physics Department, Yarmouk University 21163 Irbid Jordan © Nidal M. Ershaidat 2013

20

Exact Differential A differential of the form

( ) ( )dyyxQdxyxPdf ,, += (1)

is exact (also called a total differential) if ∫df is path-independent. This will

be true if

.dyy

fdx

x

fdf

∂∂

+∂∂

= (2)

So P and Q must be of the form

( ) ( )y

fyxQand

x

fyxP

∂∂

=∂∂

= ,, (3)

But

,2

xy

f

y

P

∂∂∂

=∂∂

(4)

.2

yx

f

x

Q

∂∂∂

=∂∂

(5)

which yields

.x

Q

y

P

∂∂

=∂∂

(6)

The notion of exact differentials plays an important role in physics. In

thermodynamics, properties of a thermodynamic system (for example, the

pressure, volume and temperature for a gas) should be exact differentials.




21

Method of Undetermined Coefficient or Guessing Method

Principle

This method is based on a guessing technique. That is, we will guess the form

of yP and then plug it in the equation to find it. However, it works only under

the following two conditions:

Condition 1: the associated homogeneous equations has constant coefficients

Condition 2: the nonhomogeneous term R(x) is a special form

( ) ( ) ( )xexPxRx β= α

cos (1)

Or

( ) ( ) ( )xexLxRx β= α

sin (2)

where P(x) and L(x) are polynomial functions. Note that we may assume that

R(x) is a sum of such functions (see the remark below for more on this).

Assume that the two conditions are satisfied. Consider the equation

( )xRycybya =+′+′′ (3)

where a, b and c are constants and

( ) ( ) ( )xexPxRx

n β= αcos or ( ) ( ) ( )xexPxR

xn β= α

sin (4)

where Pn(x) is a polynomial function with degree n. Then a particular solution yP

is given by

( ) ( ) ( ) ( ) ( )( )xexUxexTxxyx

nx

ns

P β+β= ααsincos (5)

Where

( ) 22210 xAxAxAAxT nn ++++= … (6)

And

( ) 22210 xBxBxBBxU nn ++++= …

Where the constants Ak and Bk are to be determined.

The value of the power s depends on α + iβ in the following manner:

� if α + iβ is not a root of the characteristic equation then s=0.

� if α + iβ is a simple root of the characteristic equation then s=1.

� if α + iβ is a double root of the characteristic equation then s=2.


Remark: If the nonhomogeneous term R(x) satisfies the following

( ) ( )∑=

=

=Ni

i

i xRxR1

(7)

where Ri(x) are of the forms cited above, then we split the original equation

into N equations:

( ) ( )NixRycybya i ,,2,1 …==+′+′′ (8)

Then find a particular solution yi. A particular solution to the original

equation is given by

( ) ∑=

=

=Ni

i

iP yxy1

. (8)

Summary

Let us summarize the steps to follow in applying this method:

(1) First, check that the two conditions mentioned above are satisfied;

(2) If the equation is given as

( )∑=

=

=+′+′′Nk

k

k xRycybya1

, (9)

( ) ( ) ( )xexPxRx

n β= αcos or ( ) ( ) ( )xexPxR

xn β= α

sin (10)

where Pn is a polynomial function with degree n, then split this equation into

N equations

( )xRycybya k=+′+′′ , (11)

where

(3) Write down the characteristic equation a r2 + b r + c = 0 and find its roots;

(4) Write down the number αk+ i βk. Compare this number to the roots of the

characteristic equation found in the previous step.

(4.1) If αk+ iβk is not one of the roots, then set s = 0;

(4.2) If αk+ iβk is one of the two distinct roots, then set s = 1;

(4.3) If αk+iβk is equal to both roots (which means that the

characteristic equation has a double root, then set s = 2;

In other words, s measures how many times αk+ i βk is a root of the

characteristic equation;

(5) Write down the form of the particular solution


( ) ( ) ( ) ( ) ( )( )xexUxexTxxyx

nx

ns

P β+β= ααsincos , (12)

where

( ) 22210 xAxAxAAxT nn ++++= … (13)

and

( ) .22

210 xBxBxBBxU nn ++++= … (14)

(6) Find the constants Ai and Bi by plugging Rk into the equation

( )xRycybya k=+′+′′ ,

(7) Once all the particular solutions yk are found, then the particular solution

of the original equation is the series

( ) ∑=

=

=Nk

k

kP yxy1

(15)

Reference: http://www.sosmath.com/diffeq/second/guessing/guessing.html




24

Method of Variation of Parameters

Principle

This method has no prior conditions to be satisfied. Therefore, it may sound

more general than the method of undetermined coefficient or guessing

method. We will see that this method depends on integration while the other

one cited is purely algebraic which, for some at least, is an advantage.

Consider the equation

( ) ( ) ( )xRyxQyxPy =+′+′′ (1)

In order to use the method of variation of parameters we need to know that

{y1, y2} is a set of fundamental solutions of the associated homogeneous

equation ( ) ( ) 0=+′+′′ yxQyxPy . We know that, in this case, the general

solution of the associated homogeneous equation is 2211 ycycyh += . The

idea behind the method of variation of parameters is to look for a particular

solution such as

( ) ( ) ( ) ( ) ( )xxx 2211 yxuyxuyP += , (2)

where u1 and u2 are functions. From this, the method got its name.

The functions u1 and u2 are solutions to the system

( )

=′′+′′=′+′

xRyuyu

yuyu

2211

2211 0 (3)

which implies

( ) ( ) ( )( )( )

( ) ( ) ( )( )( )

=

−=

∫

∫

.,

,

21

12

21

21

dxxyyW

xRxyxu

dxxyyW

xRxyxu ,

(4)

Where (((( ))))(((( ))))xyyW 21 , is the wronskian of y1 and y2. Therefore, we have

( ) ( ) ( ) ( )( )( )

( ) ( ) ( )( )( )∫∫ +−= dx

xyyW

xRxyxydx

xyyW

xRxyxyxyP

21

12

21

21

,, (5)

25

Summary

Let us summarize the steps to follow in applying this method:

(1) Find {y1, y2} a set of fundamental solutions of the associated

homogeneous equation (((( )))) (((( )))) 0====++++′′′′++++′′′′′′′′ yxQyxPy ;

(2) Write down the form of the particular solution

(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))xyxuxyxuxyP 2211 ++++==== (6)

(3) Write down the system:

(((( ))))

====′′′′++++′′′′

====′′′′++++′′′′

xRyuyu

yuyu

2211

2211 0 (7)

(4) Solve it. That is, find u1 and u2;

(5) Plug u1 and u2 into the equation giving the particular solution.

Example

Find the particular solution to ( )xyy tan1+=+′′ ; 22

π<<

π− x

Solution

Let us follow the steps in the summary:

(1) A set of fundamental solutions of the associated homogeneous equation

0=+′′ yy is {y1 = cos(x), y2 = sin(x)};

(2) The particular solution is given as

( ) ( ) ( ) ( ) ( ) .sincos 21 xxuxxuxyP +=

(3) We, thus, have the system:

( ) ( )( ) ( ) ( )

+=′+′−=′+′

xxuxu

xuxu

tan1cossin

0sincos

211

21

(4) We solve for 1u′ and 2u′ and get:

( ) ( )( )( ) ( ) ( )( )

+=′+−=′

xxxu

xxu

tan1cossin

tan1sin

2

1

Using Integration techniques, we get:

( ) ( ) ( ) ( ) ( )( ) ,tanseclnsincos1 xxxxxu +−+=

( ) ( ) ( ) .cossin2 xxxu −=

(5) The particular solution is

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )xxxxxxxxyP cossinsintanseclnsincoscos −++−+=

or

26

( ) ( ) ( )( ) .tanseclncos1 xxxyP +−=

Remark: Note that since the equation is linear, we may still split if

necessary. For example, we may split the equation

( )xyy tan1+=+′′

into the two equations 1=+′′ yy (R1) and ( )xyy tan=+′′ (R2)

then, find the particular solutions y1 for (R1) and y2 for (R2), to generate a

particular solution for the original equation by 21 yyyP +=

There are no restrictions on the method to be used to find y1 or y2. For

example, we can use the method of undetermined coefficients to find y1,

while for y2, we are only left with the variation of parameters.

Reference: http://www.sosmath.com/diffeq/second/variation/variation.html




27

Fourier Series

A) Definition

A Fourier series is an expansion of a periodic function f(x) in terms of an

infinite sum of sines and cosines. Fourier series make use of the

orthogonality relationships of the sine and cosine functions. The computation

and study of Fourier series is known as harmonic analysis and is extremely

useful as a way to break up an arbitrary periodic function into a set of simple

terms that can be plugged in, solved individually, and then recombined to

obtain the solution to the original problem or an approximation to it to

whatever accuracy is desired or practical.

B) Illustration

Examples of successive approximations to common functions using Fourier

series are illustrated below.

(a) (b)

(c) (d)

Figure 1: Using Fourier series in order to approximate some functions.

a) The square wave, b) The sawtooth wave, c) the triangle wave and d) the

semicircle.

28

C) Application - solution of ordinary differential equations

In particular, since the superposition principle holds for solutions of a linear

homogeneous ordinary differential equation, if such an equation can be

solved in the case of a single sinusoid, the solution for an arbitrary function

is immediately available by expressing the original function as a Fourier

series and then plugging in the solution for each sinusoidal component. In

some special cases where the Fourier series can be summed in closed form,

this technique can even yield analytic solutions.

D) Generalized Fourier Series

Any set of functions that form a complete orthogonal system have a

corresponding generalized Fourier series analogous to the Fourier series. For

example, using orthogonality of the roots of a Bessel function of the first

kind gives a so-called Fourier-Bessel series.

E) Computation of Fourier series

The computation of the (usual) Fourier series is based on the following

integral identities which represent the orthogonality relationships of the sine

and the cosine functions:

( ) ( )∫π+

π−

δπ= nmdxxnxm sinsin (1)

( ) ( )∫π+

π−

δπ= nmdxxnxm coscos (2)

( ) ( )∫π+

π−

= 0cossin dxxnxm (3)

( )∫π+

π−

= 0sin dxxm (4)

( )∫π+

π−

= 0cos dxxm (5)

For 0, ≠nm , where δmn is the Kronecker delta.

Using the method for a generalized Fourier series, the usual Fourier series

involving sines and cosines is obtained by taking f1(x) = cos x and f2(x) = sin x.

Since these functions form a complete orthogonal system over [- π, π], the

29

Fourier series of a function f(x) is given by

( ) ( ) ( )∑∑∞

=

∞

=

++=11

0 sincos2

1

n

n

n

n xnbxnaaxf (6)

a0, a1 …an are called the Fourier coefficients.

These coefficients are obtained using the following relations:

( )∫π+

π−π

= dxxfa1

0 (7)

( ) ( )∫π+

π−π

= dxxnxfan cos1

(8)

( ) ( )∫π+

π−π

= dxxnxfbn sin1

(9)

n = 1, 2, 3, …. Note that the coefficient of the constant term a0 has been

written in a special form compared to the general form for a generalized

Fourier series in order to preserve symmetry with the definitions of an and bn.

A Fourier series converges to the function f (equal to the original function at

points of continuity or to the average of the two limits at points of

discontinuity)

( ) ( )

( ) ( )

ππ−=

+

π<<π−

+

=

−+

+−

π→π→

→→

,limlim2

1

limlim2

1

0

000

xforxfxf

xforxfxf

f

xx

xxxx (10)

if the function satisfies so-called Dirichlet conditions.

As a result, near points of discontinuity, a "ringing" known as the Gibbs

phenomenon, illustrated above, can occur.

30

For a function f(x) periodic on an interval [- L, L] instead of [- π, π], a simple

change of variables can be used to transform the interval of integration from

[- π, π] to [- L, L].

Let

xL

x ′π= (11)

xdL

dx ′π= (12)

Solving for x′ gives xL

xπ

=′ , and plugging this in Eq. 6 gives

( ) ∑∑∞

=

∞

=

′π+

′π+=′

11

0 sincos2

1

n

n

n

n xL

nbx

L

naaxf (13)

Therefore,

( )∫+

−

′′=L

L

xdxfL

a1

0 (14)

( )∫+

−

′

′π′=L

L

n xdxL

nxf

La cos

1 (15)

( )∫+

−

′

′π′=L

L

n xdxL

nxf

Lb sin

1 (16)

Similarly, the function is instead defined on the interval [0,2L], the above

equations simply become

( )∫ ′′=L

xdxfL

a

2

0

0

1 (17)

( )∫ ′

′π′=L

n xdxL

nxf

La

2

0

cos1

(18)

( )∫ ′

′π′=L

n xdxL

nxf

Lb

2

0

sin1

(19)

In fact, for f(x) periodic with period 2 L, any interval [x0, x0+2L ] can be used,

with the choice being one of convenience or personal preference (Arfken

1985, p. 769).

The coefficients for Fourier series expansions of a few common functions are

31

given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51). One of the

most common functions usually analyzed by this technique is the square

wave. The Fourier series for a few common functions are summarized in the

table below.

F) Examples

Function f(x) Fourier series

sawtooth wave L

x

2 ∑

∞

=

π−

1

sin1

2

1

n

xL

n

n

square wave 112 −

−−

L

xH

L

xH ∑

∞

=

ππ

…,5,3,1

sin14

n

xL

n

n

triangle wave T(x) ( )∑

∞

=

−

π−

π…,5,3,1

2

21

2sin

18

n

n

xL

n

n

If a function is even so that f(x) = f(- x), then f(x) sin(nx) is odd. (This follows

since sin(nx) is odd and an even function times an odd function is an odd

function.) Therefore, bn=0 for all n. Similarly, if a function is odd so that f(x) = -

f(-x), then f(x) cos(nx) is odd. (This follows since cos(nx) is even and an even

function times an odd function is an odd function.) Therefore, an=0 for all n.

G) Complex Coefficients

The notion of a Fourier series can also be extended to complex coefficients.

Consider a real-valued function f(x). Write

( ) ∑∞

=

=0n

xnin eAxf (20)

Now examine

( ) ∫ ∑∫π+

π−

−∞

∞−=

π+

π−

−

= dxeeAdxexf xmi

n

xnin

xmi (21)

( )∫∑

π+

π−

−∞

∞−=

= dxeA xmni

n

n (22)

( ) ( )[ ]∫∑π+

π−

∞

∞−=

−+−= dxmnimnAn

n sincos (23)

32

nm

n

nA δπ= ∑∞

∞−=

2 (24)

mAπ= 2 , (25)

So

( )∫π+

π−

−

π= dxexfA xni

n2

1 (26)

The coefficients can be expressed in terms of those in the Fourier series

( ) ( ) ( )[ ]∫π+

π−

+π

= dxxnixnxfAn sincos2

1 (27)

( ) ( ) ( )[ ]

( )

( ) ( ) ( )[ ]

>−π

=π

<+π

=

∫

∫

∫

π+

π−

π+

π−

π+

π−

0sincos2

1

02

1

0sincos2

1

ndxxnixnxf

ndxxf

ndxxnixnxf

(28)

( )

( )

<−π

<π

<+π

=

02

1

02

1

02

1

0

nforbia

nfora

nforbia

nn

nn

(29)

For a function periodic in [- L/2, L/2], these become

( ) ( )∑∞

∞−=

π=n

Lxnin eAxf

2 (30)

( ) ( )∫−

π−=2

2

21L

L

Lxnin dxexf

LA (31)

These equations are the basis for the extremely important Fourier transform,

which is obtained by transforming An from a discrete variable to a continuous

one as the length L → ∞.

33

H) References

Arfken, G. "Fourier Series." Ch. 14 in Mathematical Methods for Physicists,

3rd ed. Orlando, FL: Academic Press, pp. 760-793, 1985.

Askey, R. and Haimo, D. T. "Similarities between Fourier and Power Series."

Amer. Math. Monthly 103, 297-304, 1996.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca

Raton, FL: CRC Press, 1987.

Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value

Problems, 5th ed. New York: McGraw-Hill, 1993.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical,

Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in

Mathematical Physics. New York: Dover, 1959.

Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals,

3rd ed., rev. and enl. New York: Dover, 1950.

Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover,

1963.

Dym, H. and McKean, H. P. Fourier Series and Integrals. New York:

Academic Press, 1972.

Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA:

Brooks/Cole, 1992.

Groemer, H. Geometric Applications of Fourier Series and Spherical

Harmonics. New York: Cambridge University Press, 1996.

Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University

Press, 1988.

Körner, T. W. Exercises for Fourier Analysis. New York: Cambridge

University Press, 1993.

Krantz, S. G. "Fourier Series." §15.1 in Handbook of Complex Variables.

Boston, MA: Birkhäuser, pp. 195-202, 1999.

Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions.

Cambridge, England: Cambridge University Press, 1958.

Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994.

Sansone, G. "Expansions in Fourier Series." Ch. 2 in Orthogonal Functions,

rev. English ed. New York: Dover, pp. 39-168, 1991.

34

Weisstein, E. W. "Books about Fourier Transforms."

http://www.ericweisstein.com/encyclopedias/books/FourierTransforms.html.

Whittaker, E. T. and Robinson, G. "Practical Fourier Analysis." Ch. 10 in The

Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New

York: Dover, pp. 260-284, 1967.

Source internet

Weisstein, Eric W. "Fourier Series." From MathWorld - A Wolfram Web

Resource. http://mathworld.wolfram.com/FourierSeries.html

© 1999 CRC Press LLC, © 1999-2007 Wolfram Research, Inc. | Terms of

Use




35

Curvilinear Coordinates

What are curvilinear coordinates?

A curvilinear coordinate system is composed of intersecting surfaces. If the intersections are all at right angles, then the curvilinear coordinates are said to form an orthogonal coordinate system. If not, they form a skew coordinate system.

Curvilinear q1 q2 q3 h1 h2 h3 →1a

→2a

→3a

Cartesian x y z 1 1 1 i j k

Spherical r θ φ 1 r r sin θ 0r θ φ

Cylindrical r (or ρ) φ z 1 r 1 0r or ρ φ k

→→→++= 333222111 adqhadqhadqhld

�

∑∑=

→

=

→→

∂φ∂

=∂

φ∂=φ∇

3

1

3

1 i iii

i ii

qha

la

( ) ( ) ( )

∂∂

+∂

∂+

∂∂

=⋅∇→→

3

213

2

312

1

321

321

1

q

hhA

q

hhA

q

hhA

hhhA

332211

321

332211

321

1

AhAhAh

qqq

hahaha

hhhA

∂∂

∂∂

∂∂

=×∇

→→→

→→

∂φ∂

∂∂

+

∂φ∂

∂∂

+

∂φ∂

∂∂

=φ∇=φ∇⋅∇→→

33

21

322

31

211

32

1321

2 1

qh

hh

qqh

hh

qqh

hh

qhhh




36

Cartesian to Cylindrical Cylindrical to Cartesian Cartesian to Spherical Spherical to Cartesian

φφ−φ= sinˆcosˆˆ0ri φ+φ= sinˆcosˆ

0 jir φφ−φθθ+φθ= sinˆcoscosˆcossinˆˆ0ri θ+φθ+φθ= cosˆsinsinˆcossinˆ

0 kjir

φφ+φ= cosˆsinˆˆ0rj φ+φ−=φ cosˆsinˆˆ ji φφ+φθθ+φθ= cosˆsincosˆsinsinˆˆ

0rj θ−φθ+φθ=θ sinˆsincosˆcoscosˆˆ kji

kk ˆˆ = kk ˆˆ = θθ−θ= sinˆcosˆˆ0rk φ+φ−=φ cosˆsinˆˆ ji

Cartesian Spherical Cylindrical

ld�

kdzjdyidxld ˆˆˆ ++=�

φφθ+θθ+= ˆsinˆ0 drdrrdrld

�

kdzdrrdrld ˆˆ0 +φφ+=

�

ψ∇→

zk

yj

xi

∂ψ∂

+∂ψ∂

+∂ψ∂ ˆˆˆ

φ∂ψ∂

φθ

+θ∂ψ∂

θ+∂ψ∂

=ψ∇→

ˆsin

11ˆ0

rrrr

zk

rrr

∂ψ∂

+φ∂ψ∂

φ+∂ψ∂

=ψ∇→

ˆ1ˆ0

→→⋅∇ A z

A

y

A

x

A zyx

∂∂

+∂

∂+

∂∂

( ) ( ) ( )

φ∂

∂+θ

θ∂∂

+θ∂∂

θ=⋅∇ φθ

→→ArArAr

rrA r sinsin

sin

1 2

2

( ) ( )

∂

∂+

φ∂

∂+

∂

∂=⋅∇ φ→→

z

ArA

r

Ar

rA zr1

→→×∇ A

zyx AAA

zyx

kji

∂∂

∂∂

∂∂

ˆˆˆ

φθ

→→

θφ∂∂

θ∂∂

∂∂

φθθ

θ=×∇

ArArAr

rrr

rA

r sin

ˆsinˆˆ

sin

10

2

zr AArAzr

krr

rA

φ

→→

∂∂

φ∂∂

∂∂

φ

=×∇

ˆˆˆ

10

ψ∇2 2

2

2

2

2

22

zyx ∂

ψ∂+

∂

ψ∂+

∂

ψ∂=ψ∇

φ∂

ψ∂

θ+

θ∂ψ∂

θθ∂

∂

θ+

∂ψ∂

∂∂

=ψ∇2

2

222

2

2

2

sin

1sin

sin

11

rrrr

rr

2

2

2

2

22

22 11

zrrrr ∂

ψ∂+

φ∂

ψ∂+

∂ψ∂

+∂

ψ∂=ψ∇

Or

φ∂

ψ∂θ

+

θ∂ψ∂

θθ∂

∂+

∂ψ∂

θ∂∂

θ=ψ∇

2

22

2

2

sin

1sinsin

sin

1

rr

rr

∂ψ∂

∂∂

+

φ∂ψ∂

φ∂∂

+

∂ψ∂

∂∂

=ψ∇z

rzrr

rrr

112

Documents

Phys. 201 Methods of Theoretical Physics 1 - yu.edu.jo Department, Yarmouk University, Irbid Jordan Phys. 201 Methods of Theoretical Physics 1 Phys. 201 Methods of Theoretical Physics