Review of Vector Analysis

Special functions

• Functions

– Determine a functional relationship between two or

more variables

– We have studied many elementary functions such as

polynomials, powers, logarithms, exponentials,

trigonometric and hyperbolic functions.

– Error function

– Gamma function

– Laplace transform

– Fourier transform

2

The error function

• It occurs in the theory of probability, distribution of

residence times, conduction of heat, and diffusion

matter:

dzexx

z

0

22erf

0 x z

erf x

22 ze

z: dummy variable

1erf

Proof in next slide

3

dyedxeIR

yR

x

00

22

x and y are two independent Cartesian coordinates

dydxeIR

yxR

0

)(

0

2 22

drdreI r

R2

1

00

2 2

in polar coordinatesError between the volume determined by x-y and r-

The volume of has a base area which is

less than 1/2R2 and a maximum height of e-R2

22

2

1 ReR

2

4

1

4

12 ReI

4

1, 2 IR

dzexx

z

0

22erf

1erf

4

More about error function

Differentiation of the error function:22

erf xexdx

d

dzexx

z

0

22erf

Integration of the error function:

Cexx

Cdxexxxxdx

x

x

2

2

1erf

2erferf

The above equation is tabulated under the symbol “ ierf x” with

1C

(Therefore, ierf 0 = 0)

Another related function is the complementary error function “erfc x”

dzexxx

z

22

erf1 erfc

5

The Fourier Transformation

The Laplace Transformation

, sxf x s e f x dx

F

0

, sxf x s e f x dx

L

• The Laplace transform is

s

dtett st

F

ff

0

L

The gamma function

dtetn tn

0

1)(

for positive values of n.

t is a dummy variable since the value of the definite integral is independent of t.

(N.B., if n is zero or a negative integer, the gamma function becomes infinite.)

)1()1(

)1(

)(

0

2

0

1

0

1

nn

dtetnet

dtetn

tntn

tn

repeat

)!1(

)1()1)(2)...(2)(1()(

n

nnn

The gamma function is thus a generalized factorial, for positive integer

values of n, the gamma function can be replaced by a factorial.

8

Vector analysis

It is much quicker to manipulate a single symbol than the

corresponding elementary operations on the separate

variables.

This is the original idea of vector.

Any number of variables can be grouped into a single symbol in

two ways:

(1) Matrices

(2) Tensors: array of components that are functions of the

coordinates of a space.The principal difference between tensors and matrices is the

labelling and ordering of the many distinct parts.

Tensors

21 izziyxz Generalized as zm

A tensor of first rank since one suffix m is needed to specify it.

The notation of a tensor can be further generalized by using more than

one subscript, thus zmn is a tensor of second rank (i.e. m, n) .

The symbolism for the general tensor consists of a main symbol such as z

with any number of associated indices. Each index is allowed to take any

integer value up to the chosen dimensions of the system. The number of

indices associated with the tensor is the “rank” of the tensor.

10

Tensors of zero rank (a tensor has no index)

• It consists of one quantity independent of the number of

dimensions of the system.

• The value of this quantity is independent of the

complexity of the system and it possesses magnitude

and is called a “scalar”.

• Examples:

– energy, time, density, mass, specific heat, thermal

conductivity, etc.

– scalar point: temperature, concentration and pressure

which are all signed by a number which may vary with

position but not depend upon direction.

11

Tensors of first rank (a tensor has a single

index)

• The tensor of first rank is alternatively names a “vector”.

• It consists of as many elements as the number of

dimensions of the system. For practical purposes, this

number is three and the tensor has three elements are

normally called components.

• Vectors have both magnitude and direction.

• Examples:

– force, velocity, momentum, angular velocity, etc.

12

Tensors of second rank (a tensor has two

indices)

• It has a magnitude and two directions associated

with it.

• The one tensor of second rank which occurs

frequently in engineering is the stress tensor.

• In three dimensions, the stress tensor consists

of nine quantities which can be arranged in a

matrix form:

333231

232221

131211

TTT

TTT

TTT

Tmn

13

The physical interpretation of the stress tensor

x

zy

pxx

xyxz

zzzyzx

yzyyyx

xzxyxx

mn

p

p

p

T

The first subscript denotes the plane and the second subscript denotes the

direction of the force.

xy is read as “the shear force on the x facing plane acting in the y

direction”.

14

15

Review of Vector Analysis

Vector analysis is a mathematical tool with which

Transport phenomena concepts are most conveniently

expressed and best comprehended.

A quantity is called a scalar if it has only magnitude (e.g.,

mass, temperature, electric potential, population).

A quantity is called a vector if it has both magnitude and

direction (e.g., velocity, force, electric field intensity).

The magnitude of a vector is a scalar written as A or

AA A

16

A unit vector along is defined as a vector whose

magnitude is unity (that is,1) and its direction is along

A

A

A

AeA )e( A 1

Thus

Ae

which completely specifies in terms of A and its direction Ae

A

AeAA

17

A vector in Cartesian (or rectangular) coordinates may

be represented as

or

where AX, Ay, and AZ are called the components of in the

x, y, and z directions, respectively; , , and are unit

vectors in the x, y and z directions, respectively.

zzyyxx eAeAeA )A,A,A( zyx

A

A

xe

ze

ye

kAjAiAA zyx

Suppose a certainvector is given by

The magnitude or absolute value of the vector is

(from the Pythagorean theorem)

zyx e4e3e2V

V

385.5432V 222

V

18

19

The Radius Vector

A point P in Cartesian coordinates may be represented by

specifying (x, y, z). The radius vector (or position vector) of

point P is defined as the directed distance from the origin O

to P; that is,

The unit vector in the direction of r is

zyx ezeyexr

r

r

zyx

ezeyexe zyx

r

222

zkyjxir

20

Vector Algebra

Two vectors and can be added together to give

another vector ; that is ,

Vectors are added by adding their individual components.

Thus, if and

A B

C

BAC

zzyyxx eAeAeA zzyyxx eBeBeBB

zzzyyyxxx e)BA(e)BA(e)BA(C

21

Parallelogram Head to rule tail rule

Vector subtraction is similarly carried out as

zzzyyyxxx e)BA(e)BA(e)BA(D

)B(ABAD

22

The three basic laws of algebra obeyed by any given vector

A, B, and C, are summarized as follows:

Law Addition Multiplication

Commutative

Associative

Distributive

where k and l are scalars

ABBA

C)BA()CB(A

kAAk

A)kl()Al(k

BkAk)BA(k

23

When two vectors and are multiplied, the result is

either a scalar or a vector depending on how they are

multiplied. There are two types of vector multiplication:

1. Scalar (or dot) product:

2.Vector (or cross) product:

The dot product of the two vectors and is defined

geometrically as the product of the magnitude of and the

projection of onto (or vice versa):

where is the smaller angle between and

A

ABcosABBA

BA

B

AB

A

BA

A B

B

B

A B

A

B

24

If and then

which is obtained by multiplying and component by

component

),A,A,A(A ZYX )B,B,B(B ZYX

ZZYYXXBABABABA

A B

ABBA

CABACBA )(

A A A2

A2

eX ex ey ey eZ ez 1

eX ey ey ez eZ ex 01

0

kkjjii

ikkjji

25

The cross product of two vectors and is defined as

where is a unit vector normal to the plane containing

and . The direction of is determined using the right-

hand rule or the right-handed screw rule.

A

A

nABesinABBA

B

B

ne

ne

BA Direction of and using (a) right-hand rule,(b) right-handed

screw rule

ne

26

If and then

zyx

zyx

zyx

BBB

AAA

eee

BA

),A,A,A(A ZYX )B,B,B(B ZYX

zxyyxyzxxzxyzzy e)BABA(e)BABA(e)BABA(

27

Note that the cross product has the following basic

properties:

(i) It is not commutative:

It is anticommutative:

(ii) It is not associative:

(iii) It is distributive:

(iv)

ABBA

ABBA

C)BA()CB(A

CABACBA )(

0AA )0(sin

28

Also note that

which are obtained in cyclic permutation and illustrated

below.

yxz

xzy

zyx

eee

eee

eee

Cross product using cyclic permutation: (a) moving clockwise leads to positive results;

(b) moving counterclockwise leads to negative results

29

Scalar and Vector Fields

A field can be defined as a function that specifies a particular

quantity everywhere in a region (e.g., temperature

distribution in a building), or as a spatial distribution of a

quantity, which may or may not be a function of time.

Scalar quantity scalar function of position scalar field

Vector quantity vector function of position vector field

30

Line Integrals

A line integral of a vector field can be calculated whenever a

path has been specified through the field.

The line integral of the field along the path P is defined asV

2

1

P

PP

dl cos Vdl V

31

The cart is constrained to move along the prescribed path

from points a to b.

32

Example. The vector is given by where Vo

is a constant. Find the line integral

where the path P is the closed path below.

It is convenient to break the path P up into the four parts P1,

P2, P3 , and P4.

dl VIP

V xoeVV

33

For segment P1, Thus

For segment P2, and

xedxdl

o o

1

xx

0x

x

0

ooooxxoxxo

P

xV)0x(Vdx)ee(V)edx()eV(dl V

yedydl

)e (since )()(dl x 0002

y

yy

y

yxo

P

eedyeVVo

V Voe x

34

For segment P3,

left) the to points dl length aldifferenti (the xedxdl

oo

xx

x

xxo

P

xV- )edx()eV(dl Vo

03

0

4

dl VP

field) ive(conservat 00xV0xV I oooo

P P PP 2 3 41

35

Surface Integrals

Surface integration amounts to adding up normal

components of a vector field over a given surface S.

We break the surface S into small surface elements and

assign to each element a vector

is equal to the area of the surface element

is the unit vector normal (perpendicular) to the surface

element

ne dsds

ne

ds

The flux of a vector field A through surface S

36

(If S is a closed surface, is by convention directed

outward)

Then we take the dot product of the vector field at the

position of the surface element with vector . The result is

a differential scalar. The sum of these scalars over all the

surface elements is the surface integral.

is the component of in the direction of (normal

to the surface). Therefore, the surface integral can be

viewed as the flow (or flux) of the vector field through the

surface S

(the net outward flux in the case of a closed surface).

ds

ds

ds

V

cosV

SS

cos ds VdsV

V

37

Example. Let be the radius vector

The surface S is defined by

The normal to the surface is directed in the +z direction

Find

V

dyd

dxd

cz

S

dsV

zyx ezeyexV

38

V is not perpendicular to S, except at one point on the Z axis

Surface S

39

SS

cosdsVdsV

c4d(-d)]-2dc[d

dx)]d(d[cdydxcyx

ccyxdsV

cyx

ccos dxdyds cyxV

2

dx

dx

dscos

222

dx

dx

dy

dy

V

222

S

222

222

40

Introduction to Differential Operators

An operator acts on a vector field at a point to produce

some function of the vector field. It is like a function of a

function.

If O is an operator acting on a function f(x) of the single

variable X , the result is written O[f(x)]; and means that

first f acts on X and then O acts on f.

Example. f(x) = x2 and the operator O is (d/dx+2)

O[f(x)]=d/dx(x2 ) + 2(x2 ) = 2x +2(x2 ) = 2x(1+x)

41

An operator acting on a vector field can produce

either a scalar or a vector.

Example. (the length operator),

Evaluate at the point x=1, y=2, z=-2

Thus, O is a scalar operator acting on a vector field.

Example. , ,

x=1, y=2, z=-2

Thus, O is a vector operator acting on a vector field.

)]z,y,x(V[O

O(A ) A A yx ezey3V

)V(O

scalar32.640zy9VV)V(O 22

A2AAA)A(O yx ezey3V

vectore65.16e49.95

e4e1240)e2e(6

ez2ey6zy9)ezey3()V(O

yx

yxyx

yx22

yx

42

Vector fields are often specified in terms of their rectangular

components:

where , , and are three scalar features functions of

position. Operators can then be specified in terms of ,

, and .

The divergence operator is defined as

zzyyxx ezyxVezyxVezyxVzyxV ),,(),,(),,(),,(

xV yVzV

zyx Vz

Vy

Vx

V

xV

yV zV

43

Example . Evaluate at the

point x=1, y=-1, z=2.zyx

2 e)x2(eyexV V

0Vz

1Vy

x2Vx

x2VyVxV

zyx

zy2

x

31x2V

Clearly the divergence operator is a scalar operator.

Ain Aout

0 A

The flux leaving the one end must exceed the flux entering at the other end.

The tubular element is “divergent” in the direction of flow.

Therefore, the operator is frequently called the “divergence” :

AA divDivergence of a vector

44

The operator is of vector form, a scalar product can be obtained as :

z

A

y

A

x

A

kAjAiAz

ky

jx

iA

zyx

zyx

)(

application

The equation of continuity :

0)()()(

tw

zv

yu

x

where is the density and u is the velocity vector.

0)(

tu

Output - input : the net rate of mass flow from unit volume

A is the net flux of A per unit volume at the point considered, counting

vectors into the volume as negative, and vectors out of the volume as positive.

zzyyxx BABABABA

46

1. - gradient, acts on a scalar to produce a vector

2. - divergence, acts on a vector to produce a scalar

3. - curl, acts on a vector to produce a vector

4. -Laplacian, acts on a scalar to produce a scalar

V

V

V

V2

Hamilton’s operator

Tz

Tk

y

Tj

x

Ti

which defines the operator for determining the complete

vector gradient of a scalar point function.

The operator is pronounced “del” or “nabla”.

The vector T is often written “grad T” for obvious reasons.

can operate upon any scalar quantity and yield a vector

gradient.

47

zk

yj

xi

More about the Hamilton’s operator ...

z

Tk

y

Tj

x

TidrTdr (vector) · (vector)

dzz

Tdy

y

Tdx

x

T

z

Tk

y

Tj

x

TikdzjdyidxTdr

TT ddrdr

dTT

But T is the vector equilvalent

of the generalized gradient48

Laplacian operator

In Cartesian coordinates

In cylindrical coordinates, the Laplacian operator is

In spherical coordinates, the Laplacian operator is

49

Curl

The curl is a vector operation that can be used to determine whether there is

a rotation associated with a vector field.

The curl operation determines both the sense and the

magnitude of the rotation.50

Another form of the vector product :

zyx AAA

zyx

kji

A

zyx

zyx

BBB

AAA

kji

BA

is the “curl” of a vector ; AcurlA

What is its physical meaning?

Assume a two-dimensional fluid element

uv

x

y xx

vv

yy

uu

O A

B

Regarded as the angular velocity of OA, direction : k

Thus, the angular velocity of OA is ; similarly, the angular velocity of OB is x

vk

y

uk

y

u

x

vk

vu

yx

kji

0

0u

51

52

Coordinate Systems

In order to define the position of a point in space, an

appropriate coordinate system is needed. A considerable

amount of work and time may be saved by choosing a

coordinate system that best fits a given problem. A hard

problem in one coordinate system may turn out to be easy

in another system.

We will consider the Cartesian, the circular cylindrical, and

the spherical coordinate systems. All three are orthogonal

(the coordinates are mutually perpendicular).

53

Cartesian coordinates (x,y,z)

The ranges of the coordinate variables are

A vector in Cartesian coordinates can be written as

The intersection of three orthogonal infinite places

(x=const, y= const, and z = const)

defines point P.

z

y

x

zzyyxxzyx eAeAeAor )A,A,A(

A

Constant x, y and z surfaces

54

zyx edzedyedxdl

Differential elements in the right handed Cartesian coordinate system

dxdydzd

55

z

y

x

adxdy

adxdz

adydzdS

56

Cylindrical Coordinates . (r, , z)

- the radial distance from the z – axis

- the azimuthal angle, measured from the x-axis in the xy – plane

- the same as in the Cartesian system.

A vector in cylindrical coordinates can be written as

Cylindrical coordinates amount to a combination of

rectangular coordinates and polar coordinates.

)z,,(

z

20

0

21222 /)A(A

Aor ),,(

z

zzz

AA

eAeAeAAA

57

Positions in the x-y plane are determined by the values of

Relationship between (x,y,z) and )z,,(

and

zz x

ytan yx 122

58

eee

eee

eee

z

z

z

0eeeeee

1eeeeee

z

zz

Point P and unit vectors in the cylindrical coordinate system

59

z and ,

semi-infinite plane with its edge along the z - axis

Constant surfaces

60

Differential elements in cylindrical coordinates

Metric coefficient

zp adzadaddl

dzdddv

61

Spherical coordinates .

- the distance from the origin to the point P

- the angle between the z-axis and the radius

vector of P

- the same as the azimuthal angle in cylindrical coordinates

),,r(

0 r

0

Colatitude( polar angle)

0 2

62

21222 /)A(A

Aor ),,(

AA

eAeAeAAA

r

rrr

eee

eee

eee

r

r

r

0eeeeee

1eeeeee

rr

rr

A vector A in spherical coordinates may be written as

Point P and unit vectors in spherical coordinates

63

cosrz

sinsinry

cossinrx

22

11-22

1222

yx

xcos

x

ytan

z

yxtan zyxr

r

zcos

ztan 11

Relationships between space variables )z,,( and ),,,r(),z,y,x(

64

and ,,rConstant surfaces

65

Differential elements in the spherical coordinate system

adsinrardadrdl r

ddrdsinrdv 2

66

Summary of the Transformation

between Coordinate Systems

67

1.

2.

3.

POINTS TO REMEMBER

68

4.

5.

6.

7.