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Fields and Waves
Lesson 2.1
VECTORS and VECTOR CALCULUS
Darryl Michael/GE CRD
VECTORS
Today’s Class will focus on:
• vectors - description in 3 coordinate systems
• vector operations - DOT & CROSS PRODUCT
• vector calculus - AREA and VOLUME INTEGRALS
VECTOR NOTATION
VECTOR NOTATION:
Rectangular or
Cartesian
Coordinate
System

A  Axaˆx  Ayaˆy  Azaˆz
z
 
A  B  AxBx  AyBy  AzBz
aˆx
 
A  B  Ax
aˆy
Ay
aˆz
Az
Bx
By
Bz
y
x

A  Ax2  Ay2  Az2


1
2
Dot Product
(SCALAR)
Cross Product
(VECTOR)
Magnitude of vector
VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
• RECTANGULAR
• CYLINDRICAL
• SPHERICAL
Choice is based on
symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
VECTOR REPRESENTATION: CYLINDRICAL COORDINATES
Cylindrical representation uses: r ,f , z

A  Araˆr  Afaˆf  Azaˆz
UNIT VECTORS:
aˆ
 
A  B  ArBr  AfBf  AzBz
z
r
P
z
x
f
y
r
aˆf aˆz 
Dot Product
(SCALAR)
VECTOR REPRESENTATION: SPHERICAL COORDINATES
Spherical representation uses: r ,q , f

A  Araˆr  Aaˆq  Afaˆf
UNIT VECTORS:
aˆ
 
A  B  ArBr  AqBq  AfBf
z
q
P
r
x
f
y
r
aˆ aˆf 
Dot Product
(SCALAR)
VECTOR REPRESENTATION: UNIT VECTORS
Rectangular Coordinate System
aˆ z
z
Unit Vector
Representation
for Rectangular
Coordinate
System
aˆ x
aˆ y
y
x
The Unit Vectors imply :
aˆ x
Points in the direction of increasing x
aˆ y
Points in the direction of increasing y
aˆ z
Points in the direction of increasing z
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
z
aˆ z
r
P
aˆf
z
x
f
aˆr
y
The Unit Vectors imply :
aˆr
Points in the direction of increasing r
aˆf
Points in the direction of increasing j
aˆ z
Points in the direction of increasing z
VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
aˆf
z
P
q
aˆr
r
x
f
aˆq
y
The Unit Vectors imply :
aˆr
aˆq
aˆf
Points in the direction of increasing r
Points in the direction of increasing q
Points in the direction of increasing j
VECTOR REPRESENTATION: UNIT VECTORS
Summary
RECTANGULAR
Coordinate
Systems
aˆ
x
aˆ y aˆz 
CYLINDRICAL
Coordinate
Systems
aˆ
r
aˆf aˆz 
SPHERICAL
Coordinate
Systems
aˆ
r
aˆ aˆf 
NOTE THE ORDER!
r,f, z
r,q ,f
Do Problem 1
Note: We do not emphasize transformations between coordinate systems
METRIC COEFFICIENTS
Unit is in “meters”
1. Rectangular Coordinates:
When you move a small amount in x-direction, the distance is dx
In a similar fashion, you generate dy and dz
( dx, dy, dz )
Generate:
2. Cylindrical Coordinates:
Differential Distances:
y
Distance = r df
df
r
x
( dr, rdf, dz )
METRIC COEFFICIENTS
3. Spherical Coordinates:
Differential Distances:
y
Distance = r sinq df
( dr, rdq, r sinq df )
z
df
P
q
r sinq
x
r
x
f
y
METRIC COEFFICIENTS
Representation of differential length dl in coordinate systems:
rectangular
cylindrical
spherical

dl  dx  aˆx  dy  aˆy  dz  aˆz

dl  dr  aˆr  r  df  aˆf  dz  aˆz

dl  dr  aˆr  rdq  aˆq  r sin qdf  aˆf
AREA INTEGRALS
• integration over 2 “delta” distances
dy
dx
Example:
y
6 7
AREA =
6
  dy  dx
= 16
2 3
2
Note that: z = constant
3
7
x
In this course, area & surface integrals will be
on similar types of surfaces e.g. r =constant
or f = constant or q = constant et c….
PROBLEM 2
PROBLEM 2A
What is constant?
How is the integration performed? What are the
differentials?
Representation of differential surface element:
Vector is NORMAL
to surface

ds  dx  dy  aˆ z
DIFFERENTIALS FOR INTEGRALS
Example of Line differentials
dl  dx  aˆ x

or
dl  dr  aˆr

or
dl  rdf  aˆf

Example of Surface differentials

ds  dx  dy  aˆ z
Example of Volume differentials
or

ds  rdf  dz  aˆr
dv  dx  dy  dz