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Review of Review of Vector AnalysisVector Analysis
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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 anddirection (e.g., velocity, force, electric field intensity).
The magnitude of a vector is a scalar written as A or
AA A
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A unit vector along is defined as a vector whose magnitude is unity (that is,1) and its direction is along
AA
A
AeA )e( A 1
Thus
Ae
which completely specifies in terms of A and its direction Ae
A
AeAA
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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
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xe
ze
ye
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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
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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
rr
zyx
ezeyexe zyx
r
222
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Vector Algebra
Two vectors and can be added together to giveanother vector ; that is ,
Vectors are added by adding their individual components.Thus, if and
A BC
BAC
zzyyxx eAeAeA zzyyxx eBeBeBB
zzzyyyxxx e)BA(e)BA(e)BA(C
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Parallelogram Head to rule tail rule
Vector subtraction is similarly carried out as
zzzyyyxxx e)BA(e)BA(e)BA(D
)B(ABAD
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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
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When two vectors and are multiplied, the result iseither a scalar or a vector depending on how they aremultiplied. 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 definedgeometrically as the product of the magnitude of and theprojection of onto (or vice versa):
where is the smaller angle between and
A
ABcosABBA
BA
B
AB
A
BA
A BB
B
A B
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If and then
which is obtained by multiplying and component bycomponent
),A,A,A(A ZYX )B,B,B(B ZYX
ZZYYXX BABABABA
A B
ABBA
CABACBA )(
A A A2A2
eX ex eyey eZ ez 1
eX ey eyez eZ ex 0
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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
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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(
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Note that the cross product has the following basicproperties:(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
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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
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Scalar and Vector Fields
A field can be defined as a function that specifies a particularquantity 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 fieldVector quantity vector function of position vector field
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Line Integrals
A line integral of a vector field can be calculated whenever apath 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
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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
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For segment P1, Thus
For segment P2, and
xedxdl
o o
1
xx
0x
x
0ooooxxoxxo
P
xV)0x(Vdx)ee(V)edx()eV(dl V
yedydl
)0e (since 0)()(dl x
02
y
yy
y
yxo
P
eedyeVVo
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V Voe x
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For segment P3,
dl dxe x (the differential length dl points to the left)
oo
xx
xxxo
P
xV- )edx()eV(dl Vo
03
04
dl VP
field) ive(conservat 00xV0xV I ooooP P PP 2 3 41
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Example. Let the vector field be given by . Find the line integral of over the semicircular path shownbelow
xoeVV V
V
Consider the contribution of the path segment located at the angle
dl dl cose x
dl sine y
Since - 90cos cos( - 90 ) sinsin sin( - 90 ) cos
dl dl sine x
dl cose y
addl
{ (sine x cose y )
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o
oo
yxxxo
yxxo
aV
aVdaV
deeeeaV
adeeeVI
2
)0cos180cos(sin
])(cos)([sin
)cos(sin)(
11
180
0
0
180
0 1
180
0
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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 elementis the unit vector normal (perpendicular) to the surface
element
ne dsds
neds
The flux of a vector field A through surface S
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(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 isa differential scalar. The sum of these scalars over all thesurface elements is the surface integral.
is the component of in the direction of (normal to the surface). Therefore, the surface integral can beviewed as the flow (or flux) of the vector field through thesurface S(the net outward flux in the case of a closed surface).
ds
ds
ds
V
cosV
SS
cos ds VdsV
V
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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
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V is not perpendicular to S, except at one point on the Z axis
Surface S
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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
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Introduction to Differential Operators
An operator acts on a vector field at a point to producesome function of the vector field. It is like a function of afunction.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)
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An operator acting on a vector field can produceeither 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
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Vector fields are often specified in terms of their rectangular components:
where , , and are three scalar features functions ofposition. 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 Vz
Vy
Vx
V
xV
yV zV
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Example . Evaluate at thepoint x=1, y=-1, z=2.
zyx2 e)x2(eyexV V
0Vz
1Vy
x2Vx
x2VyVxV
zyx
zy2
x
31x2V
Clearly the divergence operator is a scalar operator.
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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 subsequentsections.
V
V
V
V2
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Coordinate Systems
In order to define the position of a point in space, an appropriate coordinate system is needed. A considerableamount 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 easyin another system.
We will consider the Cartesian, the circular cylindrical, andthe spherical coordinate systems. All three are orthogonal(the coordinates are mutually perpendicular).
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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
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zyx edzedyedxdl
Differential elements in the right handed Cartesian coordinate system
dxdydzd
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z
y
x
adxdy
adxdz
adydzdS
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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 ofrectangular coordinates and polar coordinates.
)z,,(
z
20
0
2/12z
22
zzz
)AAA(A
eAeAeAor )AA,A(
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Positions in the x-y plane are determined by the values of
Relationship between (x,y,z) and )z,,(
and
zz xy
tan yx 122
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eee
eee
eee
z
z
z
0eeeeee
1eeeeee
z
zz
Point P and unit vectors in the cylindrical coordinate system
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z and ,
semi-infinite plane with its edge along the z - axis
Constant surfaces
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Differential elements in cylindrical coordinates
Metric coefficient
zp adzadaddl
dzdddv
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Planar surface ( = const)
Cylindrical surface
( =const)
dS ddza ddza dda z
Planar surface ( z =const)
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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 AnalysisReview of Vector Analysis
0 r 0
Colatitude( polar angle)
0 2
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2/1222r
rrr
)AAA(A
eAeAeAor )AA,A(
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
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cosrz
sinsinry
cossinrx
22
11-22
1222
yx
xcos
xy
tan z
yxtan zyxr
rz
cosz
tan 11
Relationships between space variables )z,,( and ),,,r(),z,y,x(
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and ,,rConstant surfaces
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Differential elements in the spherical coordinate system
adsinrardadrdl r
ddrdsinrdv 2
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a d rdr
a d rd sinr
a d d sinrdS r2
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1.
2.
3.
POINTS TO REMEMBER
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4.
5.
6.
7.
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