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1-1
Engineering Electromagnetics
Chapter 1:
Vector Analysis
1-2
Vector Addition
Associative Law:
Distributive Law:
1-3
Rectangular Coordinate System
1-4
Point Locations in Rectangular Coordinates
1-5
Differential Volume Element
1-6
Summary
1-7
Orthogonal Vector Components
1-8
Orthogonal Unit Vectors
1-9
Vector Representation in Terms of Orthogonal Rectangular Components
1-10
Summary
1-11
Vector Expressions in Rectangular Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the Direction of B:
1-12
Example
1-13
Vector Field
We are accustomed to thinking of a specific vector:
A vector field is a function defined in space that has magnitude and direction at all points:
where r = (x,y,z)
1-14
The Dot Product
Commutative Law:
1-15
Vector Projections Using the Dot Product
B • a gives the component of Bin the horizontal direction
(B • a)a gives the vector component of B in the horizontal direction
1-16
Operational Use of the Dot Product
Given
Find
where we have used:
Note also:
1-17
Cross Product
1-18
Operational Definition of the Cross Product in Rectangular Coordinates
Therefore:
Or…
Begin with:
where
1-19
Circular Cylindrical Coordinates
Point P has coordinatesSpecified by P(z)
z
x
y
1-20
Orthogonal Unit Vectors in Cylindrical Coordinates
1-21
Differential Volume in Cylindrical Coordinates
dV = dddz
1-22
Summary
1-23
Point Transformations in Cylindrical Coordinates
1-24
Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems
1-25
Transform the vector,
into cylindrical coordinates:
Example
1-26
Transform the vector,
into cylindrical coordinates:
Start with:
1-27
Transform the vector,
into cylindrical coordinates:
Then:
1-28
Transform the vector,
into cylindrical coordinates:
Finally:
1-29
Spherical Coordinates
Point P has coordinatesSpecified by P(r)
1-30
Constant Coordinate Surfaces in Spherical Coordinates
1-31
Unit Vector Components in Spherical Coordinates
1-32
Differential Volume in Spherical Coordinates
dV = r2sindrdd
1-33
Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems
1-34
Example: Vector Component Transformation
Transform the field, , into spherical coordinates and components
1-35
Summary Illustrations