2. EEL 34722
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 is a scalar written as A or
A
A A
Review of Vector Analysis
3. EEL 34723
A unit vector along is defined as a vector whose
magnitude is unity (that is,1) and its direction is along
A
A
A
A
eA )e( A 1
Thus
Ae
which completely specifies in terms of A and its
direction Ae
A
AeAA
Review of Vector Analysis
4. EEL 34724
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
Review of Vector Analysis
xe
ze
ye
5. EEL 34725
Suppose a certain
vector is given by
The magnitude or
absolute value of
the vector is
(from the Pythagorean theorem)
zyx e4e3e2V
V
385.5432V 222
V
Review of Vector Analysis
6. EEL 34726
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
ezeyex
e zyx
r
222
Review of Vector Analysis
7. EEL 34727
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
Review of Vector Analysis
8. EEL 34728
Parallelogram Head to
rule tail rule
Vector subtraction is similarly carried out as
zzzyyyxxx e)BA(e)BA(e)BA(D
)B(ABAD
Review of Vector Analysis
9. EEL 34729
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
Review of Vector Analysis
10. EEL 347210
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
Review of Vector Analysis
11. EEL 347211
If and then
which is obtained by multiplying and component by
component
),A,A,A(A ZYX )B,B,B(B ZYX
ZZYYXX
BABABABA
A B
ABBA
CABACBA )(
A A A
2
A2
eX ex ey ey eZ ez 1
eX ey ey ez eZ ex 0
Review of Vector Analysis
12. EEL 347212
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
nAB esinABBA
B
B
ne
ne
BA Direction of
and using
(a) right-hand rule,
(b) right-handed
screw rule
ne
Review of Vector Analysis
13. EEL 347213
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(
Review of Vector Analysis
14. EEL 347214
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
Review of Vector Analysis
15. EEL 347215
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
Review of Vector Analysis
16. EEL 347216
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
Review of Vector Analysis
18. EEL 347218
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
dlcosVdlV
Review of Vector Analysis
20. EEL 347220
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.
dlVI
P
V xoeVV
Review of Vector Analysis
21. EEL 347221
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(dlV
yedydl
)0e(since0)()(dl x
02
y
yy
y
yxo
P
eedyeVV
o
Review of Vector Analysis
V Voex
22. EEL 347222
For segment P3,
dl dxex (the differential lengthdl points to the left)
oo
xx
x
xxo
P
xV-)edx()eV(dlV
o
03
0
4
dlV
P
field)ive(conservat00xV0xVI oooo
P P PP 2 3 41
Review of Vector Analysis
23. EEL 347223
Example. Let the vector field be given by .
Find the line integral of over the semicircular path shown
below
xoeVV
V
V
Consider the contribution of
the path segment located at
the angle
dl dl cosex
dl siney
Since - 90
cos cos( - 90) sin
sin sin( - 90) cos
dl dl sinex
dl cosey
ad
dl
{ (sinex cosey )
Review of Vector Analysis
25. EEL 347225
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
nedsds
ne
ds
The flux of
a vector
field A
through
surface S
Review of Vector Analysis
26. EEL 347226
(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
cosdsVdsV
V
Review of Vector Analysis
27. EEL 347227
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
Review of Vector Analysis
28. EEL 347228
V is not perpendicular to S, except at one point on the Z axis
Surface S
Review of Vector Analysis
30. EEL 347230
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)
Review of Vector Analysis
31. EEL 347231
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
yx
22
yx
Review of Vector Analysis
32. EEL 347232
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 e)z,y,x(Ve)x,y,x(Ve)z,y,x(V)z,y,x(V
xV yV zV
zyx V
z
V
y
V
x
V
xV
yV zV
Review of Vector Analysis
33. EEL 347233
Example . Evaluate at the
point x=1, y=-1, z=2.
zyx
2
e)x2(eyexV V
0V
z
1V
y
x2V
x
x2VyVxV
zyx
zy
2
x
31x2V
Clearly the divergence operator is a scalar operator.
Review of Vector Analysis
34. EEL 347234
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
Each of these will be defined in detail in the subsequent
sections.
V
V
V
V2
Review of Vector Analysis
35. EEL 347235
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).
Review of Vector Analysis
36. EEL 347236
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
Review of Vector Analysis
37. EEL 347237
zyx edzedyedxdl
Differential elements in the right handed Cartesian coordinate system
dxdydzd
Review of Vector Analysis
39. EEL 347239
Cylindrical Coordinates .
- 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
2/12
z
22
zzz
)AAA(A
eAeAeAor)AA,A(
Review of Vector Analysis
40. EEL 347240
Positions in the x-y plane are determined by the values of
Relationship between (x,y,z) and )z,,(
and
zz
x
y
tanyx 122
Review of Vector Analysis
43. EEL 347243
Differential elements in cylindrical coordinates
Metric coefficient
zp adzadaddl
dzdddv
Review of Vector Analysis
44. EEL 347244
Planar surface
( = const)
Cylindrical
surface
( =const)
dS ddza
ddza
ddaz
Planar surface
( z =const)
Review of Vector Analysis
45. EEL 347245
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(
Review of Vector Analysis
0 r
0
Colatitude
(polar angle)
1 24 34
0 2
