Download Review of Vector Analysis

Review of
Vector Analysis
EEL 3472
Review of Vector Analysis
Review of Vector Analysis
Vector analysis is a mathematical tool with which
electromagnetic (EM) 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 A is a scalar written as A or A
A
2
EEL 3472
Review of Vector Analysis
A unit vector eA along
is defined as a vector whose
magnitude is unity (that is,1) and its direction is along
eA 
A
A

A
A
( eA  1)
Thus
A  AeA
which completely specifies A in terms of A and its
direction eA
3
EEL 3472
Review of Vector Analysis
A vector A in Cartesian (or rectangular) coordinates may
be represented as
(A x , A y , A z )
A x ex  A y ey  A z ez
or
where AX, Ay, and AZ are called the components of A in the
x, y, and z directions, respectively; e x , e y , and e z are unit
vectors in the x, y and z directions, respectively.

4


EEL 3472
Review of Vector Analysis
Suppose a certain
vector V is given by
V  2ex  3ey  4ez
The magnitude or
absolute value of
the vector V is
V  22  32  42  5.385
(from the Pythagorean theorem)
5
EEL 3472
Review of Vector Analysis
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,
r  x ex  y ey  z ez
The unit vector in the direction of r is
er 
6
x ex  y ey  z ez
x 2  y 2  z2
r

r
EEL 3472
Review of Vector Analysis
Vector Algebra
Two vectors A and B can be added together to give
another vector C ; that is ,
C AB
Vectors are added by adding their individual components.
Thus, if A x ex  A y ey  A z ez and B  Bx ex  By ey  Bz ez
C  (A x  Bx )ex  (A y  By )ey  (A z  Bz )ez
7
EEL 3472
Review of Vector Analysis
Parallelogram
rule
Head to
tail rule
Vector subtraction is similarly carried out as
D  A  B  A  (B )
D  (A x  Bx )ex  (A y  B y )ey  (A z  Bz )ez
8
EEL 3472
Review of Vector Analysis
The three basic laws of algebra obeyed by any given vector
A, B, and C, are summarized as follows:
Law
Commutative
Addition
AB BA
Associative
A  (B  C)  (A  B)  C
Distributive
k(A  B)  kA  k B
Multiplication
kA  Ak
k(lA)  (kl)A
where k and l are scalars
9
EEL 3472
Review of Vector Analysis
When two vectors A and B 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:
AB
2.Vector (or cross) product:
AB
The dot product of the two vectors A and B is defined
geometrically as the product of the magnitude of B and the
projection of A onto B (or vice versa):
A  B  AB cos  A B
where  A B is the smaller angle between A and B
10
EEL 3472
Review of Vector Analysis
If
and
B  (BX , BY , BZ )
then
A  B  A XBX  A YB Y  AZBZ
which is obtained by multiplying A and B component by
component
AB  BA
A  (B  C )  A  B  A  C
2
A A  A  A2
eX  ey  ey  ez  eZ  ex  0


A  (AX , A Y , AZ , )
eX  ex  ey  ey  eZ  ez 1
11
EEL 3472
Review of Vector Analysis
The cross product of two vectors A and B is defined as
A  B  AB sin  A Ben
where en is a unit vector normal to the plane containing A
and B . The direction of en is determined using the righthand rule or the right-handed screw rule.
Direction of A  B
and en using
(a) right-hand rule,
(b) right-handed
screw rule
12
EEL 3472
Review of Vector Analysis
If
A  (AX , A Y , AZ , ) and B  (BX , BY , BZ )
ex
A  B  Ax
ey
Ay
ez
Az
Bx
By
Bz
then
 (A yBz  A zBy )ex  (A zBx  A xBz )ey  (A xBy  A yBx )ez
13
EEL 3472
Review of Vector Analysis
Note that the cross product has the following basic
properties:
(i) It is not commutative:
AB  B A
It is anticommutative:
A  B  B  A
(ii) It is not associative:
A  (B  C)  (A  B)  C
(iii) It is distributive:
A  (B  C )  A  B  A  C
(iv)
14
AA 0
(sin   0)
EEL 3472
Review of Vector Analysis
Also note that
ex  ey  ez
ey  ez  ex
ez  ex  ey
which are obtained in cyclic permutation and illustrated
below.
Cross product using cyclic permutation: (a) moving clockwise leads to positive results;
(b) moving counterclockwise leads to negative results
15
EEL 3472
Review of Vector Analysis
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
16
EEL 3472
Review of Vector Analysis
17
EEL 3472
Review of Vector Analysis
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 V along the path P is defined as
P2
 V  dl   V
P
18
cos  dl
P1
EEL 3472
Review of Vector Analysis
19
EEL 3472
Review of Vector Analysis
Example.
The vector V is given by V  Vo ex where Vo
is a constant. Find the line integral
I
 V  dl
P
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.
20
EEL 3472
Review of Vector Analysis
V  Voex

For segment P1, dl  dx ex
x  xo
 V  dl  
P1
x 0
Thus
xo
(Vo ex )  (dx ex )  Vo  (ex  ex )dx  Vo (xo  0)  Voxo
0
For segment P2,
dl  dy e y
y  yo
and
 V  dl   (V e )  (dy e ) 0
o x
P2
21
y
(since ex  e y  0)
y 0
EEL 3472
Review of Vector Analysis
For segment P3,
dl  dxex (the differential length
dl points to the left)
x  xo
 V  dl   (V e )  (dx e )  - V x
o

P3
x
x
o
o
x 0
 V  dl  0
P4
I
       V x
o
P1
22
P2
P3
o
 0  Voxo  0  0 (conservative field)
P4
EEL 3472
Review of Vector Analysis
Example.
Let the vector field V be given by V  Vo ex.
Find the line integral of V over the semicircular path shown
below
Consider the contribution of
the path segment located at
the angle 
dl  dl cos ex
 dl sin ey
Since    - 90
cos  cos( - 90)  sin 
sin   sin(  - 90)  cos
dl  dl sin ex
 dl cos ey
 ad
{  (sin ex  cos ey )
dl
23

EEL 3472
Review of Vector Analysis
 180
I
(V e )  (sin e


o x
x
 cos e y )ad
0
180
 aVo  [sin  (ex  ex )  cos  (ex  e y )]d


0
1
0
180
 aVo  sin d   aVo (cos
180
0)



  cos
0
1
1
 2aVo
24
EEL 3472
Review of Vector Analysis
Surface Integrals
Surface integration amounts to adding up normal
components of a vector field over a given surface S.
The flux of
a vector
field A
through
surface S
We break the surface S into small surface elements and
assign to each element a vector ds  ds en
ds is equal to the area of the surface element
en is the unit vector normal (perpendicular) to the surface
element
25
EEL 3472
Review of Vector Analysis
(If S is a closed surface, ds is by convention directed
outward)
Then we take the dot product of the vector field V at the
position of the surface element with vector ds. The result is
a differential scalar. The sum of these scalars over all the
surface elements is the surface integral.
 V  ds   V
S
ds cos 
S
V cos  is the component of V in the direction of ds (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).
26
EEL 3472
Review of Vector Analysis
Example.
Let V be the radius vector
V  xex  yey  zez
The surface S is defined by
zc
d x  d
d y  d
The normal to the surface is directed in the +z direction
Find
 V  ds
S
27
EEL 3472
Review of Vector Analysis
Surface S
V is not perpendicular to S, except at one point on the Z axis
28
EEL 3472
Review of Vector Analysis
 V  ds   V ds cos 
S
V 
x 2  y 2  c2
S
ds  dxdy
V
cos 
c
x 2  y 2  c2
c os
ds
  
x  d y  d 
x d

c
2
2
2
V

ds

x

y

c

dydx  c  [d  (d)]dx
S


2
2
2
x y c
x  d y  d
x  d
 2dc[d - (-d)]  4d2c
29
EEL 3472
Review of Vector Analysis
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)
30
EEL 3472
Review of Vector Analysis
An operator acting on a vector field O[V(x, y, z)] can produce
either a scalar or a vector.
Example. O(A)  A  A (the length operator), V  3yex  zey
Evaluate O(V) at the point x=1, y=2, z=-2
O(V)  V  V  9y2  z2  40  6.32  scalar

Thus, O is a scalar operator acting on a vector field.
Example. O(A)  A A  A  2A ,
x=1, y=2, z=-2
V  3yex  zey ,
O(V)  (3y ex  z ey ) 9y2  z2  6y ex  2z ey
 (6 ex  2ey ) 40  12ex  4ey
 49.95 ex  16.65ey  vector
Thus, O is a vector operator acting on a vector field.
31
EEL 3472
Review of Vector Analysis
Vector fields are often specified in terms of their rectangular
components:
V(x, y, z)  Vx(x, y, z)ex  Vy (x, y, x)ey  Vz(x, y, z)ez
where Vx , Vy , and Vz are three scalar features functions of
position. Operators can then be specified in terms of Vx ,
Vy , and Vz .
The divergence operator is defined as
V 
32



Vx 
Vy 
Vz
x
y
z
EEL 3472
Review of Vector Analysis
Example V  x2ex  yey  (2  x)ez
point x=1, y=-1, z=2.
Vx  x2

Vx  2x
x
. Evaluate   V at the
Vy  y
Vz  2  x


Vy  1
Vz  0
y
z
  V  2x  1  3
Clearly the divergence operator is a scalar operator.
33
EEL 3472
Review of Vector Analysis
1.
V
2.
  V - divergence, acts on a vector to produce a scalar
- gradient, acts on a scalar to produce a vector
3.   V - curl, acts on a vector to produce a vector
4.
2V
-Laplacian, acts on a scalar to produce a scalar
Each of these will be defined in detail in the subsequent
sections.
34
EEL 3472
Review of Vector Analysis
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).
35
EEL 3472
Review of Vector Analysis
Cartesian coordinates (x,y,z)
The ranges of the coordinate variables are
 x 
 y 
z 
A vector A in Cartesian coordinates can be written as
(A x , A y , A z )
or
A x ex  A y ey  A z ez
The intersection of three
orthogonal infinite places
(x=const, y= const, and z =
const)
defines point P.
Constant x, y and z surfaces
36
EEL 3472
Review of Vector Analysis
dl  dxex  dyey  dz ez
d  dxdydz
Differential elements in the right handed Cartesian coordinate system
37
EEL 3472
Review of Vector Analysis
dS  dydz ax
dxdz ay
dxdy az
38
EEL 3472
Review of Vector Analysis
Cylindrical Coordinates (, , z) .
0
0    2
z 
A vector
- the radial distance from the z – axis
- the azimuthal angle, measured from the xaxis in the xy – plane
- the same as in the Cartesian system.
in cylindrical coordinates can be written as
(A , A Az )
or
A e  A e  Az ez
2
2
2
A  (A  A   Az )1 / 2
Cylindrical coordinates amount to a combination of
rectangular coordinates and polar coordinates.
39
EEL 3472
Review of Vector Analysis
Relationship between (x,y,z) and (, , z)
Positions in the x-y plane are determined by the values of
 and 
y
  x2  y2
  tan1
zz
x
40
EEL 3472
Review of Vector Analysis
Point P and unit vectors
in the cylindrical
coordinate system
e  e  ez
e  ez  e
ez  e  e
e  e  e  e  ez  ez  1
e  e  e  ez  e  e  0
41
EEL 3472
Review of Vector Analysis
semi-infinite
plane with its
edge along
the z - axis
Constant ,  and z surfaces
42
EEL 3472
Review of Vector Analysis
Metric coefficient
dl  d ap  da  dz az
dv  dddz
Differential elements in cylindrical coordinates
43
EEL 3472
Review of Vector Analysis
dS  ddza
ddza
d  d  a z

Cylindrical
surface
(  =const)
Planar surface
(  = const)
Planar surface
( z =const)

44
EEL 3472
Review of Vector Analysis
Spherical coordinates (r, , ) .
0r
0
Colatitude
( polar angle)
0    2
- 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

45
EEL 3472
Review of Vector Analysis
er  e  e
e  e  er
e  er  e
er  er  e  e  e  e  1
Point P and unit vectors in spherical
coordinates
er  e  e  e  e  er  0
A vector A in spherical coordinates may be
written as
(Ar , A  A  )
or
Ar er  A  e  A  e
2
2
2
A  (Ar  A   A  )1 / 2
46
EEL 3472
Review of Vector Analysis
r
2
2
2
x y z
1
  tan
x2  y2
z
  tan1
  tan-1
y
 cos1
x
x
x2  y2

z
 cos1
z
r
x  r sin  cos 
y  r sin  sin 
z  r cos 
Relationships between space variables (x, y, z), (r, , ), and (, , z)
47
EEL 3472
Review of Vector Analysis
Constant
48
r, , and  surfaces
EEL 3472
Review of Vector Analysis
dl  dr ar  rda  r sin d a
dv  r2 sindrdd
Differential elements in the spherical coordinate system
49
EEL 3472
Review of Vector Analysis
dS  r 2 sin  d d ar
r sin  dr d a
rdr d a
50
EEL 3472
Review of Vector Analysis
51
EEL 3472
Review of Vector Analysis
52
EEL 3472
Review of Vector Analysis
53
EEL 3472
Review of Vector Analysis
54
EEL 3472
Review of Vector Analysis
POINTS TO REMEMBER
1.
2.
3.
55
EEL 3472
Review of Vector Analysis
4.
5.
6.
7.
56
EEL 3472