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• Physics Department, Yarmouk University, Irbid Jordan

Phys. 201 Methods of Theoretical Physics 1

Phys. 201 Methods of Theoretical Physics 1

Supplements

Dr. Nidal M. Ershaidat

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

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

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

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

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

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

Ordinary Differential Equations (ODEs) ......................................................... 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

• Physics Department, Yarmouk University, Irbid Jordan

Phys. 201 Methods of Theoretical Physics 1

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 (,x)

where KKKK and x X, there corresponds an element x 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) (x) = () x (associativity with respect to number multiplication);

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

8) ( + ) x = x + 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 and are scalar functions, A and

B are vectors.

++++====

++++ BABA (1)

++++====

++++ BABA (2)

++++====

AAA (3)

====

BAABBA (4)

++++

====

ABBABAABBA (5)

0====

A (6)

====

2 (Laplacian) (7)

0====

(8)

====

AAA 2 (9)

• Physics Department, Yarmouk University, Irbid Jordan

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= (1)

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

the density as being the limit:

dV

dm

dV 0lim

= (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 x

yzxyz

+

+

=

(3)

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

expressed in determinant form:

zyx FFF

zyx

kji

(4)

The Curl in cylindrical polar coordinates

• 6

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

zr

r

FFrF

zr

r

k

r

1

1

(5)

The Curl in spherical polar coordinates

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

r FrFrF

r

rrr

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

• Physics Department, Yarmouk University, Irbid Jordan

Phys. 201 Methods for Theoretical Physics 1

Dr. Nidal M. Ershaidat Doc. 4

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, ref

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