1
Lecture 2: Review of Vector Calculus
Instructor:
Dr. Gleb V. Tcheslavski
Contact: gleb@ee.lamar.edu
Office Hours: Room 2030
Class web site:
www.ee.lamar.edu/gleb/em/In
dex.htm
ELEN 3371 Electromagnetics
Fall 2007
2
Vector norm
Forann-dimensionalvectorx  [ x1
x2 ... xn ]
1/ p
thevectornorm: x p  x
Specialcase : x 
p

p
  xi 
 i

; p  1, 2,...
max xi
(2.2.2)
i
Mostcommonlyused L2  norm : x 2  x  x12  x2 2  ...xn 2
Example: v = (1, 2, 3)
Name
Symbol
value
|v|1
6
L2 – norm
|v|2
141/2  3.74
L3 – norm
|v|3
62/3 3.3
L4 – norm
|v|4
21/471/2 3.15
L – norm
|v|
3
ELEN 3371 Electromagnetics
(2.2.3)
Properties:
– norm
L1
(2.2.1)
1. x  0whenx  0; x  0iff x  0
(2.2.4)
2. kx  k x scalark
(2.2.5)
3. x  y  x  y
(2.2.6)
Norm(x,p)
Fall 2007
3
Vector sum
ELEN 3371 Electromagnetics
Fall 2007
4
Scalar (dot) product
Definitions:
A B  AB cos
(2.4.1)
(2.4.2)
A B  A B cos 
A B  Ax Bx  Ay By  Az Bz
(2.4.3)
Property:
A BB A
(2.4.4)
Scalar projection:
dot(A,B)
ELEN 3371 Electromagnetics
Fall 2007
AB
projB A 
 A cos 
B
(2.4.5)
 A  2ux ;B  u x  2u y

 A B  2
(2.4.6)
5
Vector (cross) product
Definitions:
A  B  AB sin  u AB
A  B  B  A
A  B  0 A || B
Properties:
(2.5.1)
(2.5.2)
(2.5.3)
In the Cartesian coordinate system:
ux
uy
uz
A  B  Ax
Ay
Az 
(2.5.4)
Bx By Bz
( Ay Bz  Az By )ux  ( Az Bx  Ax Bz )u y  ( Ax By  Ay Bx )uz
cross(A,B)
ELEN 3371 Electromagnetics
 A  2ux  u y ;B  ux  2u y
 A  Cand B  C

C  A  B  3uz
Fall 2007
(2.5.5)
6
Triple products
1. Scalar triple product:
A  B  C   B  C  A  C  A  B 
(2.6.1)
2. Vector triple product:
A  B  C   B  A C   C  A B
Note: (2.6.1) represents a circular permutation of vectors.
Q: A result of a dot product is a scalar, a result of a vector product is a vector.
What is about triple products?
ELEN 3371 Electromagnetics
Fall 2007
(2.6.2)
7
Vector fields
A vector field is a map f that assigns
each vector x a vector function f(x).
A vector field is a construction, which
associates a vector to every point in
a (locally) Euclidean space.
A vector field is uniquely specified by
giving its divergence and curl within a
region and its normal component over
the boundary.
From Wolfram MathWorld
ELEN 3371 Electromagnetics
Fall 2007
8
Coordinate systems
• In a 3D space, a coordinate system can be specified by the
intersection of 3 surfaces.
• An orthogonal coordinate system is defined when these three
surfaces are mutually orthogonal at a point.
The cross-product of two unit
vectors defines a unit surface,
whose unit vector is the third unit
vector.
A general orthogonal coordinate
system: the unit vectors are
mutually orthogonal
ELEN 3371 Electromagnetics
Fall 2007
Most commonly used coordinate
systems
(a) – Cartesian; (b) – Cylindrical; (c) – Spherical.
In Cartesian CS, directions of unit vectors are independent of their positions;
In Cylindrical and Spherical systems, directions of unit vectors depend on positions.
ELEN 3371 Electromagnetics
Fall 2007
9
10
Coordinate systems: Cartesian
An intersection of 3 planes:
x = const; y = const; z = const
Properties:
u x u x  u y u y  u z u z  1; (2.10.1)
u x u y  u x u z  u y u z  0. (2.10.2)
ux  u y  uz 

u y  uz  ux 

uz  ux  u y 
(2.10.3)
An arbitrary vector:
A  Axux  Ayu y  Az uz
ELEN 3371 Electromagnetics
Fall 2007
(2.10.4)
11
Coordinate systems: Cartesian
A differential line element:
dl = ux dx + uy dy + uz dz
Three of six differential surface
elements:
dsx = ux dydz
dsy = uy dxdz
dsz = uz dxdy
The differential volume element
dv = dxdydz
ELEN 3371 Electromagnetics
Fall 2007
(2.11.1)
(2.11.2)
(2.11.3)
Coordinate systems: Cylindrical
(polar)
An intersection of a
cylinder and 2 planes
Diff. length:
dl  d  u   du  dzu z
(2.12.1)
Diff. area:
ds   ddzu  d dzu  d du z
(2.12.2)
Diff. volume: dv   d  d dz
An arbitrary vector: A  A u  A u  Az uz
ELEN 3371 Electromagnetics
12
Fall 2007
(2.12.3)
(2.12.4)
13
Coordinate systems: Spherical
An intersection of a sphere of radius r, a plane that makes an angle  to the x
axis, and a cone that makes an angle  to the z axis.
ELEN 3371 Electromagnetics
Fall 2007
14
Coordinate systems: Spherical
Properties:
ur  u  u 

u  u  ur 

u  ur  u 
(2.14.1)
Diff. length:
dl  drur  rd u  rsin  du
(2.14.2)
Diff. area:
ds  r 2 sin  d dur  rsin  drdu  rdrd u
(2.14.3)
Diff. volume: dv  r 2 sin  drd d
(2.14.4)
An arbitrary vector: A  Ar ur  A u  A u
ELEN 3371 Electromagnetics
Fall 2007
(2.14.4)
15
System conversions
1. Cartesian to Cylindrical:
 y
 
  x 2  y 2 ;  tan 1   ;z  z
x
(2.15.1)
2. Cartesian to Spherical:
 x2  y 2 
 y
 ;  tan 1  
r  x  y  z ;  tan 


z
x


3. Cylindrical to Cartesian:
x  cos ; y  sin ;z  z
2
2
1
2
(2.15.2)
(2.15.3)
4. Spherical to Cartesian:
x  rsin  cos ; y  rsin  sin ;z  rcos 
cart2pol, cart2sph, pol2cart, sph2cart
ELEN 3371 Electromagnetics
Fall 2007
(2.15.4)
16
Integral relations for vectors
b
1. Line integrals:
 F dlor  F dl
a
Example: calculate the work required to move a cart along the path from A to B if
the force field is F = 3xyux + 4xuy
B
B
A
A
W  F dl   (3xyu x  4 xu y ) (dxu x  dyu y ) 
B
 (3xydx  4 xdy)
A
forthecircle :x 2  y 2  42
0
4
4
0
W   3x 16  x 2 dx   4 16  y 2 dy  64  16
ELEN 3371 Electromagnetics
Fall 2007
17
Integral relations for vectors (cont)
2. Surface integrals:
 F dsor  F ds
F – a vector field
s
At the particular location of the loop, the component of A
that is tangent to the loop does not pass through the
loop. The scalar product A • ds eliminates its contribution.
There are six differential surface vectors ds
associated with the cube.
Here, the vectors in the z-plane:
ds = dx dy uz and ds = dx dy (-uz)
are opposite to each other.
ELEN 3371 Electromagnetics
Fall 2007
18
Integral relations for vectors (cont 2)
Example: Assuming that a vector field A = A0/r2 ur
exists in a region surrounding the origin,
find the value of the closed-surface integral.
We need to use the differential surface area
(in spherical coordinates) with the unit vector
ur since a vector field has a component in this
direction only. From (2.14.4):

2
A ds 

 A0  2
 0 0  r 2 ur   r sin  d dur   4 A0
ELEN 3371 Electromagnetics
Fall 2007
19
Integral relations for vectors (cont 3)
3. Volume integrals:
  dv
v
v – a scalar quantity
v
Example: Find a volume of a cylinder of radius a and height L
v   dv 
v
ELEN 3371 Electromagnetics
L
2
a
  
d ddz   a 2 L
z 0 0 0
Fall 2007
20
Differential relations for vectors
1. Gradient of a scalar function:
 1 a
1 a 
grad  a   a  
u1 ,...,
un 
hn xn 
 h1 x1
(2.20.1)
Gradient of a scalar field is a vector field which points in the direction of the
greatest rate of increase of the scalar field, and whose magnitude is the
greatest rate of change.
Two equipotential surfaces with potentials V
and V+V. Select 3 points such that distances
between them P1P2  P1P3, i.e. n  l.

V V

n l
V 
Assume that separation
between surfaces is small:
V
dV
un  un
n
dn
Projection of the gradient in the ul direction:
V
dV
dV dn dV
dV
ul  ul 
 cos   un ul  V ul
l
dl
dn dl dn
dn
ELEN 3371 Electromagnetics
Fall 2007
21
Differential relations for vectors (cont)
Gradient in different coordinate systems:
 a
a
a 
1.Cartesian :a   u x , u y , u z 
y
z 
 x
(2.21.1)
 a
1 a
a 
2.Cylindrical :a   u ,
u , u z 
 
z 
 
(2.21.2)
 a
1 a
1 a 
3.Spherical :a   ur ,
u ,
u 
r 
rsin   
 r
Example:
a( x, y, z )  2 x  3 y 2  sin( z );
 a
a
a 
a   ux , u y , uz    2u x , yu y , cos( z )u z 
y
z 
 x
gradient
ELEN 3371 Electromagnetics
Fall 2007
(2.21.3)
22
Differential relations for vectors (cont 2)
2. Divergence of a vector field:
A ds

div  A     lim
v 0
v
(2.22.1)
Divergence is an operator that measures the magnitude of a vector field's
source or sink at a given point.
In different coordinate systems:
Ax Ay Az


x
y
z
1  (  A ) 1 A Az
2.Cylindrical : A 


 
  z
1.Cartesian : A 
1  (r 2 Ar )
1  ( A sin  )
1 A
3.Spherical : A  2


r
r
rsin 

rsin  
divergence
ELEN 3371 Electromagnetics
Fall 2007
(2.22.2)
(2.22.3)
(2.22.3)
23
Differential relations for vectors (cont 3)
Example:  A( x, y, z )  2 xu x  3 y 2u y  sin( z )u z ;
Ax Ay Az
 A 


 2  6 y  cos( z )
x
y
z
Some “divergence rules”:
Divergence (Gauss’s)
theorem:
ELEN 3371 Electromagnetics
 a  0;
(2.23.1)
 ( A1  A2 )   A1   A2 ;
(2.23.2)
 cA  c A
(2.23.3)
 A ds     A dv
v
Fall 2007
(2.23.4)
24
Differential relations for vectors (cont 4)
Example: evaluate both sides of Gauss’s theorem
for a vector field: A = x ux within the unit cube
Ay
Ax
A

 1;
 0; z  0.
x
y
z
Thevolumeintegralis:
   Adv 
v
1/2
1/2
1/2
  
dxdydz  1
x 1/2 y 1/2 z 1/2
Theclosed-surfaceintegralis:
  A ds 
1/2
1/2
  x |
y 1/2 z 1/2
ELEN 3371 Electromagnetics
1/ 2
x 1/2
 ux dzdyux )  
1/2
 x |
y 1/2 z 1/2
Fall 2007
x 1/2
 ux  dzdyux )  1
25
Differential relations for vectors (cont 5)
V
V
Assume we insert small paddle wheels in a flowing river.
The flow is higher close to the center and slower at the edges.
Therefore, a wheel close to the center (of a river) will not rotate since velocity of
water is the same on both sides of the wheel.
Wheels close to the edges will rotate due to difference in velocities.
The curl operation determines the direction and the magnitude of rotation.
ELEN 3371 Electromagnetics
Fall 2007
26
Differential relations for vectors (cont 6)
3. Curl of a vector field:
curl( A) A lim
s 0
un  A dl
(2.26.1)
s
Curl is a vector field with magnitude equal to the maximum "circulation" at each
point and oriented perpendicularly to this plane of circulation for each point. More
precisely, the magnitude of curl is the limiting value of circulation per unit area.
1.Cartesian :  A 
ux
uy

x
Ax

y
Ay
uz
In different coordinate systems:
 Ay Ax 
  Az Ay 
 Ax Az 


u


u


 x 
 uz
 y 
z  y z 
 z x 
 x y 
Az
 1    A  1 A 
 1 Az A 
 A Az 
 uz
2.Cylindrical :  A  



 u  
 u  





z

z














(2.26.2)
(2.26.3)
 1   (sin  A ) A  
 1  1 Ar  (r A )  
 1   (r A ) Ar  
3.Spherical :  A  

u




 r  
  u   
  u

  
r  
  
 r  r
 rsin  
 r  sin  
curl
(2.26.3)
ELEN 3371 Electromagnetics
Fall 2007
27
Differential relations for vectors (cont 7)
 A dl    A ds
Stokes’ theorem:
(2.27.1)
s
The surface integral of the curl of a vector field over a surface S equals to the
line integral of the vector field over its boundary.
Example: For a v. field A = -xy ux – 2x uy, verify Stokes’ thm. over
from(2.26.2) A  (2  x)uz
9 y 2
3
   A ds   
3
 
  A dxdyu z 
y 0 x 0
s
9 y 2
(2  x)u z dxdyu z
y 0 x 0

9  y2 
 
2
  2 9 y 
dy  9 1  
2 
 2
y 0 
3
3
0
 A dl   
0
A dxu x 
x 0 y 0
0
 A dl   
x  0 y 3
arc
  xyu
x
 2 xu y   dxu x  dyu y  dzu z 
arc
 
2
(
xy

dx

2
x

dy
)

x
9

x
dx

2
9

y
dy


9
1  
arc
3
0
 2
0

A dyu y 
ELEN 3371 Electromagnetics
3
2
Fall 2007
28
The Laplacian operator:
Repeated vector operations
  A  0
(2.28.1)
  a  0
(2.28.2)
 a   2 a
(2.28.3)
    A  ( A)   2 A
(2.28.4)
2a 2a 2a
 a 2  2  2
x
y
z
2
 a 


1    1  2 a  2 a

 2 2 2


  z
Cartesian
(2.28.5)
Cylindrical
(2.28.6)
a 
 a 

  r2 
  sin 

1  r 
1
1
2a
 

 2
 2
 2 2
r
r
r sin 

r sin   2
ELEN 3371 Electromagnetics
Fall 2007
Spherical
(2.28.7)
29
Phasors
A phasor is a constant complex number representing the complex
j
V
(

)

V
e
0
amplitude (magnitude and phase) of a sinusoidal function of time.
(2.29.1)
forv(t )  V0 cos(t   )  Re V (  )e jt 

(2.29.2)
 1

dv
 Re  jV (  )e jt  ; vdt '  Re  V (  )e jt 
dt
 j

(2.29.3)
Note: Phasor notation implies that signals have the same frequency.
Therefore, phasors are used for linear systems…
Example: Express the loop eqn for a circuit in phasors if v(t) = V0 cos(t)
vL
di
1
 Ri   idt '
dt
C
i(t )  I 0 cos(t   )
1


V0 cos(t )  I 0  L sin(t   )  R cos(t   ) 
sin(t   ) 
C


v(t )  V0 cos(t )  Re V0e j e jt   Re V (  )e jt 

V ( )   R 

ELEN 3371 Electromagnetics
Fall 2007
1 

j L 
 I (  )
C  

30
Conclusions
Questions?
Ready for your first homework??
ELEN 3371 Electromagnetics
Fall 2007